Книги2 / 1988 Kit Man Cham, Soo-Young Oh, John L. Moll, Keunmyung Lee, Paul Vande Voorde, Daeje Chin (auth.) Computer-Aided Design and VLSI Device Development 1988

device technology to sub-micron dimensions has extended the required knowledge of devices in a new technology beyond the capability of GEMINI. There is a great need to migrate to a more general analysis program such as PISCES-II. A difficulty, however, is that users consider it as a time-consuming job. Moreover, PISCES-II needs several key enhancements which will make it more useful. A mobility model that is better than the default model should be implemented to provide the accuracy acceptable for generation of I-V characteristics and SPICE parameter extractions. For the study of hot electron related reliability problems, an impact-ionization model is needed.

To practically eliminate the convergence problem and make it easy for users to begin with or migrate from GEMINI to PISCES-II, a terminated rectangular grid by eliminations has been developed and tested. A paralleland vertical-field dependent mobility model is implemented. It enables the accurate simulation of the linear and saturation operation of MOSFETs. With these provisions, PISCES-II will be as simple as GEMINI but with flexible geometry and more capabilities. PISCES will calculate the drain current and VT correctly and eliminate the complication of the quasi-fermi level specification in GEMINI because the quasi-fermi level is calculated from the current continuity equations.

A simple tutorial will be presented on the application of PISCES-II to a n-channel device. All the doping profiles used are generated by SUPREM-III.

Basic equations and numerical algorithm

The operation of a semiconductor device can be completely specified by both Poisson equation and the electron and hole continuity equations with proper boundary conditions. The basic equations are

where

104 Computer-Aided Design

The Poisson equation governs the electrostatic potential and the electron and hole continuity equations govern the carrier concentrations. N is the net concentration of ionized impurity. U (n,p) represents net electron and hole recombination. PISCES-II supports both Shockley-Read-Hall and Auger recombination. JJ.n and J.Ip are electron and hole mobilities. The same electron mobility model used in CADDET has been implemented in PISCES-II. A similar model for hole mobility has been developed and implemented. These mobility models are

where Ep is the field parallel to the current flow and En is the field normal to the current flow.

For electrons, the En dependence and coefficients are

To solve the governing equations on a computer, they must be discretized on a grid. In PISCES-II, each equation is integrated over a small polygon enclosing each node, which is formed by the bisectors of the lines connecting to the neighboring nodes. Using current and electric flux conservation, the integration equates the net flux into the polygon with the sources and sinks inside it. If the number of nodes is N, this method generates 3 N non-linear algebraic equations for unknown potentials and carrier concentrations. To avoid negative concentrations, the Gummel-Scharfetter formula [3.11] is used to discretize the current flux in the continuity equations. For further details of discretization, see Price's thesis [3.17].

By discretization, the semiconductor device equations are transformed into 3 N coupled non-linear algebraic equations. These equations should be solved by a non-linear iteration method. There are two approaches; one is the decoupled method (Gummel method) and the other is the coupled method (Newton method). In the Gummel method, these equations are solved sequentially. First, the Poisson equation is solved for potential with fixed quasi-fermi levels. Then, the new potential is substituted into the continuity equations. The new carrier concentrations are fed-back into the Poisson equation. The same procedure is repeated until it converges. In this way, only one equation is solved at a time. Thus, the matrix size is only N. This decoupled method converges quickly when the interaction of the Poisson and continuity equations is small such as in the subthreshold region or when VDS is small. However, the convergence slows down when the interaction is significant such as in the linear and saturation regions of a MOSFET. ICCG (Incomplete Cholesky Conjugate Gradients) iterative matrix solution method is recommended to solve the equations in the Gummel method. It is the fastest iterative method known, which can be used on an irregular grid.

In the Newton method, all the equations are solved simultaneously. The single matrix equation includes all the coupling between the equations. Due to this, the Newton algorithm converges at a rate that is nearly independent of bias conditions. The size of the matrix, however, is 3 times as large (or twice as large when solving for only one carrier) as the matrix size of the Gummel method. Thus, the cpu time per iteration is long but the number of iterations is smaller. The single biggest acceleration of a Newton iteration is the

Newton-Richardson procedure which only refactors the Jacobian matrix when necessary. With this procedure, the Newton method with one-carrier is faster than the Gummel method in the linear and saturation regions of a MOSFET. The direct matrix solution method is recommended for the Newton method.

At the beginning of the iterative process, PISCES must have an initial guess for the potential and carrier concentrations at each node point. The initial guess directly affects the rate of convergence. There are four types of initial guesses used in PISCES-II. The first is init initial guess which uses the charge neutral assumption and is only valid when VDS is zero. Any later solution begins with an initial guess obtained by modifying one or two previous solutions. For a previous initial guess, the previous solution is used as an initial value only modified by changing the applied bias at the contacts. Sometimes, a better guess can be obtained by local guess. In this case, the previous solution is used and the majority fermi levels is changed to be equal to the bias applied at the contact connected to this region. Frequently, the best initial guess is obtained by projection. In this method, two previous solutions with appropriate bias conditions are used and extrapolated to the new bias condition. In a MOSFET, the projection method yields rapid convergence when the gate bias is stepped except near the subthreshold and linear region transition. The previous works better than local or projection when the drain bias is stepped.

Example

An n-channel transistor structure is used for this example. The grid generation has been implemented through macros so that they are transparent to the users. The information that must be supplied by the user is similar to that required by GEMINI. There are three phases of PISCES-II simulations. First is the specification of the device structure. Doping profiles should be specified and optimum grid should be generated. All information is saved in a mesh file. Then, device characteristics at specified bias conditions are simulated and the solutions are saved. Finally, graphical displays of I-V characteristics or internal distributions are performed to extract the desired information from the solutions. The detailed input of grid generation (expanded by a macro processor) will be explained.

Grid generation

The correct generation of grid is a critical issue in device simulations. To minimize the discretization errors, the grid should be fine enough to resolve rapid changes of internal distributions. The grid should also be a reasonable fit to the device shape to accurately represent the device geometries. However, the number of nodes in a grid should be minimized because simulation time is proportional to N a where a usually varies between 1.5 and 2. Above all, inadequate grid is the main cause of the non-convergence in PISCES-II. To satisfy the above conditions, PISCES-II adopts a flexible triangular grid structure. The most important disadvantage of this grid is the difficulty of user specification. To overcome this, PISCES-II supports two grid generation methods and automatic regrid capability. The first is the rectangular grid generated using input statements. The second is a stand-alone interactive grid generator. Based on our experience with the automatic regrid, it is difficult to control the number and location of nodes. It also generates obtuse triangles in critical areas, such as the channel in a MOSFET structure. These obtuse triangles can disturb the solution accuracy and in the worst case cause convergence problems. Thus, the terminated rectangular grid with elimination is recommended for planar MOS structures. The PISCES-II input of this grid will be illustrated for a n-channel device. The full input from the macro processor is shown in Fig. 3.25. The following will explain the input file in detail.

title cmos n-channel device : n4a.i

$

$••• derme NMOSGRD for device structure and regrid option hp2623 x.s =7

$

$ mosgrid x.dim=3 y.dim=3 tox=0.025 x.l =0.9 x.4=2.1 meshf=n4msh3

$

$ ••• DEFINE THE RECTANGULAR MESH •••

MESH RECT NX=37 NY=34 OUTF=n4msh3

The first statement of all inputs should be a title statement. In title statement, heading should follow the title. The comment statement in PISCES-II starts with $. Comment statements can be placed anywhere. The program ignores whatever is written in title or comment statements. The option statement can also be placed anywhere. It specifies the various feature in PISCES simulation and postprocessing. In the input file, the option x.s specifies the x length of subsequent plots as 7 inches. This is necessary for hp2623 terminals. Convenient feature of PISCES-II input parameter name is that only the minimum number of characters to ensure unique identification are sufficient. For more details of input format and sequence, refer to overview (A.l) in PISCES-II manual [3.16].

A rectangular grid is specified by mesh, x.mesh, y.mesh, and eliminate statements. In the mesh statements, the number of grids in the x direction (ox) is 37 and in the y direction (ny) is 34. Outfile gives the name of a output mesh file as n4msh3.

$

X.M N= 1 L=O.OO R=1.00 X.M N=6 L=0.81 R=0.70 X.M N= 7 L=O.8S R=l.00 X.M N=12 L=0.95 R=l.00 X.M N=13 L=0.98 R=1.00 X.M N=19 L=1.50 R=1.10 X.M N=25 L=2.02 R=0.91 X.M N=26 L=2.05 R=l.00 X.M N=31 L=2.15 R=1.00 X.M N=32 L=2.19 R=1.00 X.M N=37 L=3.00 R=1.60

The actual position of x and y grids are specified by x.mesh and y.mesh statements. In both statements, node is the identification number of the grid point and location is the coordinate of that grid point in micron. Ratio is the

ratio of the adjacent spacing in the previous grids. The spacing decreases with proportion to ratio when ratio is less than 1 and increases when ratio is greater than 1. The first x.mesh mesh statement defines the starting point. The ratio parameter in the first statement is always equal to one. The second statement allocated 5 uniform grids in the uniform source region. The third statement provides a transition from large spacing to small spacing near the source junction at 0.9 JJ m. The fourth statement puts 5 grid lines near the source junction. The fifth statement provides an intermediate spacing for the transition. The sixth and seventh statements allocate coarse grid in the channel region. The eighth statement is a transition line. The ninth statement allocates 5 fine grid lines near the drain junction. The tenth and eleventh statements are for the drain region.

$

The first and second y.mesh statements allocate 4 grids in the gate oxide region. To resolve the change of the vertical field and carrier concentration in

the inversion layer, the first spacing in the inversion layer is 0.5 nm. The ratio of the adjacent spacing should be not greater than 2 to prevent numerical errors or convergence problems. Thus, the statements from the fourth to ninth increase the spacing only by 2 times. The tenth statement provides relatively fine grids for the depletion region below the channel. The next three statements gradually increase the spacing and allocate coarse grids in the substrate.

To reduce the number of nodes in the area where they are not needed, a eliminate statement will remove every other line in the specified area. The direction of grids to be eliminated is specified by y.direction or x.direction. The area to be eliminated is specified by four parameters (ix.low, ix.high, iyJow, iy.high). The first and second eliminate statements selectively remove the x grids (Y.DIR) below the source and drain junctions. The next four statements eliminate the x grids (Y.DIR) in the substrate. The last two statements remove the fine y grids (X.DIR) in the source and drain regions. These eliminations reduce the total number of nodes from 1258 to 939.

The material type of the sub-area is specified by the region statement. Number is the identification (ID) number of the region. The boundary of the region is specified by ix.low, ix.high, iy.low, and iy.high. The first region statement specifies the gate oxide and the second statement specifies the silicon substrate. Next, electrode statements specify the contacts. In PISCES-II, a contact is a rectangular region specified by ix.low, ix.high, iy.low, and iy.high. The contact can collapse to a line if ix.low=ix.high or iy.low=iy.high. It can be located inside as well as on the boundary. Here, drain, gate, source, and substrate are specified as the first, second, third, and fourth electrodes. In the subsequent input, these contacts are referred by their ID numbers.

To specify the doping profiles in the device, the next statements should be a set of doping statements.