Technology Computer Aided Design (TCAD) Laboratory. Lecture 2, A simulation primer

Technology Computer Aided Design (TCAD) Laboratory Lecture 2, A simulation primer [Source: Synopsys] Giovanni Betti Beneventi E-mail: gbbeneventi@arces.unibo.it ; giobettibeneventi@gmail.com Office: Engineering faculty, ARCES lab. (Ex. 3.2 room), viale del Risorgimento 2, Bologna Phone: +39-051-209-3773 Advanced Research Center on Electronic Systems (ARCES) University of Bologna, Italy 1

Outline Introduction Definition of equilibrium and out-of-equilibrium Static, Transient and AC simulations Simplify the simulation domain Numerical methods from a TCAD user perspective Meshing Numerical methods Synopsys Sentaurus TCAD Solvers 2

Introduction In this lecture we will talk about different topics, all of them to be discussed before starting the laboratory activity. Some content of this class is related to the review of basic concepts, like the choice of a simulation domain, the equilibrium and out-of-equilibrium definitions, the definitions of static, transient, and AC simulations. In addition, a TCAD user perspective background on the numerical methods and on the solvers used in Synopsys Sentaurus TCAD is provided. 4

Outline Introduction Definition of equilibrium and out-ofequilibrium Static, Transient and AC simulations Simplify the simulation domain Numerical methods from a TCAD user perspective Meshing Numerical methods Synopsys Sentaurus TCAD Solvers 5

Equilibrium and out-of-equilibrium In a system under equilibrium condition, each process and its inverse must selfbalance independently from any other process that may occur in the system. In other words, equilibrium is the special case where each fundamental process and its inverse self-balance, it is also known as detailed balance principle. If a net current flows, or if light is shined upon the semiconductor creating free carriers, we are out-of-equilibrium. Applied Voltage e I semiconductor Shined light E= hn e electrode V electrode A system under steady-state, transient, or ac signal is always out-of-equilibrium. The only exception is the presence of a capacitor, than can be biased but in which the net current flow is zero due to the presence of the oxide 6

Static, transient, and AC simulations (1) Static condition or steady-state (DC bias point). In steady-state conditions, for each property of the systems, its partial derivative with respect to time is zero, i.e., nothing is changing with time. To perform steady-state simulations, the Sentaurus TCAD keyword is Quasistationary. The Quasistationary command is used to ramp a device from a solution to another through the modification of the boundary conditions (that can be Voltage, Current, or Temperature). For example, ramping a drain voltage of a device from 0 to 5V is performed by Quasistationary( Goal {Voltage=5 Name=Drain } ){ Coupled { Poisson Electron Hole } } Step Boundary Re-solve the Device 8

Static, transient, and AC simulations (2) Transient response. A transient response or natural response is the timevarying response of a system to a change from equilibrium. In Sentaurus, the keyword that must be used to perform transient simulation is Transient. The command must start with a device that has already been solved under stationary conditions. The simulation then proceeds by iterating between incrementing time and re-solving the device. An example of performing a transient simulation is: Transient( InitialTime = 0.0 FinalTime=1.0e-5 ){ Coupled { Poisson Electron Hole } } Increment time Re-solve the Device 9

Static, transient, and AC simulations (3) Small signal AC analysis. Performing a small signal or AC analysis means simulate the behavior of system when a relatively small harmonic signal is superimposed to a steady-state condition or DC bias point. Small-signal modeling is a common analysis technique in electrical engineering which is used to approximate the behavior of nonlinear devices with linear equations. This linearization is formed about the DC bias point of the device (that is, the voltage/current levels present when no AC signal is applied), and can be accurate for small excursions about this point. The keyword for AC analysis in Sentaurus Sdevice is ACCoupled. Graphical, qualitative, illustration of steady-state, transient, and AC signals. 10

Example M. A. Alam (2008), "ECE 606: Principles of Semiconductor Devices, http://nanohub.org/resources/5749. Italy, France, Germany and UK form our system. Italian population is the quantity of interest. Immigration/emigration in/from the other countries are our processes. France 4 4 Italy France 5 3 Italy France 2 6 Italy Germany 4 4 2 2 UK EQUILIBRIUM 10 people going out and 10 people coming in Italy (population of Italy is stationary ). In addition, immigration and emigration counterbalance country-wise (each process counterbalance its opposite in each country): we are in equilibrium condition. Germany 3 4 2 3 UK STEADY STATE 10 people going out and 10 people coming in Italy (population of Italy is stationary) But immigration and emigration process do not counterbalance countrywise: we are out-ofequilibrium. Germany 1 4 5 2 UK TRANSIENT 12 people going out and 8 people coming in Italy. Italian population is not conserved, i.e. Italian population is not stationary. In addition, immigration and emigration do not counterbalance country-wise: we are out-ofequilibrium. 11

Simulation on 1D, 2D and 3D domains (1) Reality is always 3D. However, some problems can be thought (approximated) as they would occur in less than 3D. This can be done when there is invariance of the problem on some directions. In other words, a dimension can be suppressed from the simulation if the physical internal properties of the simulated device would not change if the dimension itself would be made infinite. To make such approximations (not always easy to see!), an a-priori knowledge of the problem is needed. Example Standard planar MOSFET FinFET z y x L 2D L Possible to think about a 2D equivalent (on the yz plane) of what happens in the transistor channel. That is the device features can be thought to be invariant in the x direction (within the channel). The larger the L the better the 2D approximation. L very thin and raised source and drain. Intrinsically a 3D device, very difficult to simplify into a 2D equivalent! 13

Simulation on 1D, 2D and 3D domains (2) The simplification of a 3D problem into 2D or even 1D, if possible, is strongly encouraged. Indeed, the simplification of the simulation domain, means a lower number of nodes in which the numerical simulation must be computed. Reducing the number of the mesh nodes decreases the computational burden and therefore the time needed for the simulation to be performed. e.g.: same device, for 2D simulation ~ 10 3 nodes, for a 3D simulations ~10 5 nodes ~ 2 orders of magnitude of difference in the node numbers! One of the most frequently employed simplification is simulating a 3D device having cylindrical symmetry using a 2D domain and solving the model equations in cylindrical coordinates. Example Nanowire (NW) MOSFET source gate drain equivalent 2D domain to be simulated in cylindrical coordinates To simulate a 2D domain in cylindrical coordinates, the partial-differential equations must be expressed in the cylindrical coordinate system. In Sentaurus Synopsys this is accomplished by using the Cylindrical keyword in the Math section of the Sdevice input file (see later) Because of the simple geometry, all the devices simulated in the course need only a 2D planar (*) simulation domain. (*) planar = Cartesian coordinates vs. cylindrical coordinates 14

Scope In this tutorial we discuss the basic features of the numerical method used in TCAD Sentaurus Devices from a TCAD user perspective. The purpose of this tutorial is not describing into details the numerical methods implemented in the software, but discussing their general features, in order for the user to understand which numerical methods is most useful to be employed for a given problem and to cope with possible convergence issues. In general, in an iterative solution method, we start with a guess for the solution (often the zero vector) and then we successively renew this guess, getting closer to the solution at each stage. The iterations continue until the solution converges to a desired accuracy x i x i-1 < e, where x is the solution vector, that is the vector of the values of the problem unknowns, i is the index counting the iteration number, and e is the vector which defines the needed accuracy. 16

Considerations on numerical mesh (1) Define and effectively optimize numerical meshes in which the problem can be solved assuring convergence and, at the same time, reasonable simulation times, is a challenge. However, some general rules can be applied: The grid spacing must be sufficiently dense so that all the relevant features of the geometry (e.g. doping) are accurately represented. Points must be allocated to accurately approximate the physical quantities of interest (e.g. potentials, fields, carrier concentrations, currents). This means that high grid densities must be allocated in regions where the geometrical and physical quantities of interest undergo rapid changes (e.g. junctions). Conversely, the spacing between points could be relaxed in the areas where values are expected to stay relatively constant without adding any significant contribution to the overall error (e.g. quasi-neutral regions, deep regions inside the device). In fact, because the overall computation time depends on the total number of grid points, grid point number must be minimized for computational efficiency. In advanced finite-element software, as well as in Synopsys Sentaurus there are tools for automated grid generation and automatic adaptation of grids. 18

Considerations on numerical mesh (2) Example. Numerical grid in a pn diode junction p region n region Once that convergence is achieved within reasonable simulation time, an operation that must always been performed prior to elaborate the results of a simulation is to check the invariance of the solution with respect to variations of the numerical mesh. That is, once a mesh has been created to assure convergence in reasonable time, try to further reduce the mesh spacing and see if the solutions does not change. If the solution does not change mesh is good! 19

Generalities The power of an iterative method lies in its ability to achieve convergence efficiently, i.e. as fast as possible. In general, convergence and speed are two conflicting issues, since more robust convergence means a higher number of iterations and heavier mesh (i.e. an higher number of points in which the solution must be computed). For semiconductor devices, two different approaches are usually employed to solve the coupled set of equations that comprises the Drift-Diffusion model, depending upon the problem (it is also possible to combine the two methods for the same problem when needed): 1. The Newton iteration (or fully-coupled method) and 2. The Gummel iteration (or de-coupled method) For both of them, once the non-linear partial-differential-equations of the Drift-Diffusion model are discretized in space, the Newton s method for the solution of non-linear systems is applied. For both of them, in general, the solution converges non-linearly, meaning that the error, defined as the difference in the unknown between two subsequent iteration is a non-linear function of the iteration counter. 21

Newton method In both Newton and Gummel iteration schemes, the Newton iterative method is applied to solve the non-linear system. Let s briefly review the concept of the Newton method by considering a single-variable equation. The first step of the method consists in manipulating the equation in the residual form f x = 0, being f a function of some real unknown x and let its derivative being f. Thus, to solve the equation one needs to find out the zeroes of the function. To get successively better approximation of the zeroes we start with a first guess x 0. Then, provided that the function is reasonably well-behaved, a better approximation of x 0, x 1, can be easily calculated. In fact: Being x 0 a first guess as a zero of f x. Suppose x 1 is a zero of f x then f x 0 = 0 f(x 0) x 1 x 0 x 1 = x 0 f(x 0) f (x 0 ) Geometrically (x 1,0) is the intersection with the x-axis of a line tangent to f at x 0, f x 0. The process is repeated as x n+1 = x n f(x n) f (x n ) until a sufficiently accurate value is reached. f(x) x 1 x 0 x two iterations enough to get good accuracy 22

Newton iteration In the fully-coupled Newton iteration (also called Bank-Rose scheme in semiconductor device simulators) the total system of unknowns is solved together, meaning that there is only one system to be solved. The systems derives from the discretization of (i.e. includes) all equations to be solved: Poisson equations, Continuity equations and Transport equations. Initial guess of the solution Solve a whole system including Poisson, Continuity and Transport equations Newton iteration n converged?? y The keyword for the Newton iteration in Sentaurus Device is Coupled. 23

Gummel iteration Each iteration of the Gummel method treats one equation at a time, solving for the given equation (i.e. Poisson equation or Continuity and Transport equations) with respect to its primary unknown (i.e., for Poisson equation the electric field, for the Continuity and Transport equations the carrier densities) updating at each step only the values of the primary unknown. An iteration is completed when the procedure has been performed on each independent variable. Initial guess of the solution Solve Poisson equation n converged?? y Solve Continuity and Transport equations Gummel iteration n converged?? y The keyword for the Gummel iteration in Sentaurus Device is Plugin. 24

Combined Newton-Gummel iteration A combined scheme is often used where heating effects (accounted for with the Fourier heating postulate) come into play. First of all, the Drift-Diffusion model is computed until convergence is achieved with a Newton iteration, then electric field and current density solutions are plugged into the Fourier equation, which is computed until convergence is achieved in a Gummel scheme. Initial guess of the solution Solve a whole system including Poisson, Continuity and Transport equations Newton iteration n converged?? y Gummel iteration Solve Heat equation n converged?? y 25

Comparison between Newton and Gummel iterations Newton s iteration converge with a lower number of iteration compared to the Gummel iteration (quadratic rate of convergence vs. linear rate of convergence). However the single Newton iteration takes more time than a Gummel iteration. Gummel iteration converges relatively slowly compared to Newton iteration but the method will often tolerate poor initial guess. In certain problems where it is difficult to choose good initial guess, starting with Gummel to refine an initial guess, then switching to Newton after some iterations to achieve quicker convergence can be useful. In general use Gummel only when the transport problem can be decoupled from the electrostatic problem (i.e. at low fields where diffusion dominates, no band-to-band-tunneling, no avalanche, no field-dependent mobility). In those cases, Gummel is quicker than Newton, since Newton keeps updating quantities that are essentially constant or weakly changing. As initial guess, one often relies on the equilibrium solution. 26

Error handling/accuracy of simulations During a Solve statement, Sentaurus Device tries to determine the value of an equation variable x, such that the computed updated x (after the n-th iteration) is small enough. That is, it iterates until ε R x x x < 1 where ε R = 10-Digits and ε A = Error x + ε A and x is a scaling constant equal to 0.1 What does it mean? Let s simplify the inequality in two limiting cases: For x x x < ε R Relative Error criterion For x 0 x < ε A x Absolute Error criterion 27

Choosing the appropriate solver At each step of the simulation a linear system is solved Three linear solvers are available in Sentaurus Sdevice, which basic features are listed in the following table solver type memory requirements SUPER direct (systematic triangularization of the matrixes) high good for PARDISO direct parallel high 2D modification of default parameters not recommended ILS (based on the GMRES) iterative parallel (first guess than refinement) low 1D 3D requires modification of default parameters 29