By: Engr. Irfan Ahmed Halepoto

Modeling & Simulation of

Semiconductor Devices

LECTURE#03

Device Simulation Techniques • Majority of semiconductor devices are subject to high electric

fields, carrier gradients (direction) and current densities which give rise to hot electron transport.

• The basic transport equations considers only the field dependent mobility of carriers which does not account for carrier heating, so they are not suitable for modeling VLSI and high-frequency devices.

• Current interest in low dimensional and high mobility semiconductor devices (MOSFET, MESFET (metal semiconductor field-effect transistor),HBT, HEMT (high electron mobility transistor)) is providing a strong incentive to develop models capable of accounting for high field effects, non-equilibrium transport and relaxation mechanisms.

• This problem is being addressed using modified classical models (accounting for energy & momentum relaxation mechanisms) like, Monte Carlo techniques, quantum mechanical models and Boltzmann Transport Equation .

The Monte Carlo Technique (Classical) • Monte Carlo techniques are one of the only practical way to

evaluate difficult integrals or to sample random variables governed by complicated probability density functions.

– i.e. an electron travelling through a semiconductor sample, under the influence of an electric field.

• The electron is accelerated by the field, but its motion is punctuated by scattering 'events', which result in changes in the electron's speed and direction.

• Three elements of randomness may be identified in such a motion:

i) The time for which the electron travels until a scattering event occurs.

ii) The nature of the scattering event (impurity or electron scattering)

iii) The scattering angle; i.e., the direction of motion after scattering.

Monte Carlo Technique…

Monte Carlo Calculation of Pi

Monte Carlo Computation of Population Distribution

Quantum Transport Theory (Modern) • It is the theory in physics based on the principle that matter & energy

have the properties of both particles and waves.

– Created to explain the radiation of energy from a blackbody, the photoelectric effect, and the Bohr theory.

– Used to account for a wide range of physical phenomena like, existence of discrete packets of energy & matter, uncertainty principle, and the Pauli exclusion principle.

• The trend towards very small devices has lead an important application of quantum transport theory (to analyze device structures) in semiconductor devices.

• Quantum transport theory has been used in semiconductor modeling to authenticate and determine the range of validity of Boltzmann transport models (study the non-equilibrium behavior of a collection of particles).

• Quantum transport theory can clarify device operation in the optical frequency system, where Boltzmann transport theory is not valid, even for linear field distributions.

Black body is an idealized object that absorbs all electromagnetic radiation falling on it.

Quantum: Range, How much ?

Quantum Transport Theory

Quantum Transport: Atom to transistor

Quantum Transport: Energy distribution

Boltzmann Transport Equation

• Boltzmann Transport equation is a powerful tool for analyzing transport phenomena within systems that involve density and temperature gradients.

• An equation used to study the non-equilibrium behavior of a collection of particles.

• The equation is applied to analysis the currents within a system, transport coefficients and the relationships b/w them.

• The Boltzmann equation applies to a quantity known as the distribution function, which describes this non-equilibrium state mathematically and specifies how quickly and in what manner the state changes when the disturbing forces are varied.

Boltzmann Transport Equation

Boltzmann Transport Equation in one dimension

Boltzmann Transport Equation : non-equilibrium state

Boltzmann Transport Equation –Statement • It states that the rate of change of a function which specifies the

probability of finding a particle in a unit volume of phase space is equal to the sum of terms arising from external forces, diffusion of particles, and collisions of the particles. – Also known as Maxwell-Boltzmann equation.

• Here f is the unknown distribution function which depends on a position

vector r , a velocity vector v , and the time t. • ∂f/∂t is the rate of change of f at fixed values of r and v . • The equation expresses this rate of change as the sum of three

contributions: – first, (∂f/∂t)force arises when the velocities of the particles change

with time as a result of external driving forces; – second, (∂f/∂t)diff is the effect of the diffusion of the particles from

one region in space to the other; – third, (∂f/∂t)coll is the effect of the collisions of the particles with each

other or with other kinds of particles.

Semiconductor Simulation Approaches

• In the semiconductor industry, simulation work generally falls into three categories.

– Process

– Device

– Circuit simulation.

SEMICONDUCTOR PROCESS SIMULATION • Semiconductor process simulation is the modeling of the

fabrication of semiconductor devices such as diode, transistors etc.

• It is a branch of electronic design automation, and part of a sub-field known as TCAD (Technology Computer Aided Design).

• This figure shows a result from

semiconductor process.

• The input is a description of the

semiconductor fabrication process.

• The result as shown here is the

final geometry & concentrations of

all the dopants.

• This will then be used by other

programs to predict the electrical

properties of the devices formed

Semiconductor process simulation • The ultimate goal of process simulation

is an accurate prediction of the active dopant distribution, stress distribution and the device geometry.

• Process simulation is typically used as an input for device simulation, the modeling of device electrical characteristics.

• Collectively process and device simulation form the core tools for the design phase known as TCAD (Technology Computer Aided Design). FET: Simulated model

Process simulation –Process flow • The fabrication of integrated circuit devices requires a series of

processing steps called a process flow. • Process simulation involves modeling all essential steps in the

process flow in order to obtain dopant , stress profiles and device geometry. • The input for process simulation is the layout and process flow.

• Layout is selected as a linear cut in a full layout for a 2D simulation or a rectangular cut from the layout for a 3D simulation.

• The process steps most often associated with process simulation are – Ion implantation – Annealing (diffusion and dopant activation) – Etching – Deposition – Oxidation – Epitaxy .

IC Fabrication process flow

CVD: chemical vapor deposition

PVD: physical vapor deposition

Process Simulation Elements • All process simulators use a combination of the finite element

analysis (FE) and/or finite volume methods (FV) methods, which are important for the requirements for process simulation to achieve accurate results.

– A possible difficulty can be the change in the geometry during the simulated fabrication of the device.

• Process simulation uses an FE/FV mesh to compute and store the dopant and stress profiles.

– Each geometrical change in the simulation domain requires a new mesh which fits to the new boundaries.

– Since geometry can be modified, so as process simulation depends on the cumulative results of previous steps, which make process simulation an especially challenging application of the FE/FV technique.

Process Simulation Elements

Meshing domains

Finite Element Analysis

• Finite element analysis (FEA) is a computer simulation technique used in engineering analysis.

• It uses a numerical technique called the finite element method (FEM).

• There are three phases in any computer-aided engineering task:

– Pre-processing: defining the finite element model and environmental factors to be applied to it

– Analysis solver : solution of finite element model

– Post-processing: result accuracy using visualization tools.

Finite Volume Method • Finite volume method is a method for representing and evaluating

partial differential equations as an algebraic equations.

• Similar to the finite difference method, values are calculated at discrete places on a meshed geometry.

• "Finite volume" refers to the small volume surrounding each node point on a mesh.

• In the finite volume method, volume integrals in a partial differential equation that contain a divergence term which are converted to surface integrals, using the divergence theorem.

• These terms are then evaluated as fluxes at the surfaces of each finite volume.

– Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative.

• Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes.

Process Simulation - dopant profile • An important results of process simulation is the dopant profile after

processesing.

• The accuracy of the profile strongly depends on maintaining a proper density of mesh points at any time during the simulation.

• A typical full flow CMOS process simulation can have more than 50 mesh changes and the number of mesh changes can increase dramatically if adaptive meshing is performed.

• For each mesh change, interpolation is used to obtain data values on the new mesh.

• It is important to manage the mesh changes in such a way to avoid accuracy degradation due to interpolation error.

• Maintaining a balance between interpolation error, computational expense, and minimization of required user input is important for obtaining accurate results with a minimum of computational expense.

– This is especially true when simulating devices in 3D.

Process Simulation - dopant profile

Doping profile doping profile (red n+, blue p+) and geometry of the MOS transistor.

SEMICONDUCTOR DEVICE SIMULATION

• Device simulation: modeling of a individual active semiconductor device, such as a diode or transistor.

– The simplest approach to device modeling is the construction of an equivalent circuit.

• Equivalent circuit models have been widely used; there advantages are their simplicity and the consequent ease with which they can be incorporated in a complete circuit description.

• However, such models generally represent a very crude approximation to the actual device performance under various conditions, and give no physical insight into the operation of the device.

Device simulation Approaches

• Drift-diffusion approach

• Energy transport approach

• Monte Carlo Approach

Device Simulation Approaches- drift-diffusion approach

• The most widely used physical device model is called drift-diffusion simulation .

• Behavior of a population of electrons inside a semiconductor device when subject to excitation

– such as that caused by an electric field

– can be represented by the Boltzmann transport equation.

• Solution of the Boltzmann equation gives the electron energy distribution function.

• In the drift-diffusion approach, a first order approximation to the distribution function is obtained.

• Main disadvantage is that it is based on an equilibrium approximation so electron heating effects are not included.

Device Simulation Approaches- drift-diffusion approach

Flow diagram of a Drift-Diffusion simulation

Device Simulation Approaches- Energy transport approach

• Drift-diffusion restriction can be circumvented by using an energy transport simulation.

• energy transport simulation method provides a second order solution of the Boltzmann equation

– the second moment equation gives an expression for energy transport within the device.

• Whilst the energy transport simulation extends the possible range of device feature sizes which can be considered, it still represents an approximation to the true electron energy distribution.

Device Simulation Approaches- Energy transport approach

Device Simulation Approaches- Monte Carlo Approach

• The Monte Carlo approach to semiconductor device simulation yields essentially an exact solution to the Boltzmann equation.

• The method is therefore extremely well suited to modeling ultra-small devices: in which transport occurs under conditions far from equilibrium.

• In 1966, Kurosawa first introduced Monte Carlo techniques into semiconductor modeling for analyzing both material properties and device operation.

• Monte Carlo techniques has a great popularity due to detailed understanding of semiconductor properties .

• At present Monte Carlo techniques are applied to the study of a wide range of semiconductors (including Si, GaAs and InP) and to the direct analysis of device operation.

Device Simulation Approaches- Monte Carlo Approach

Statistical Prediction of Integrated Circuit Performance

Monte Carlo Method for Semiconductor Device Simulation

• As the device structures becomes more complex with variable design possibilities, empirical routes to device designs become impractically expensive.

• In this scenario, device simulation represents an extremely cost effective tool in the design process; a few hours CPU time can save many weeks of clean-room time.

• Monte Carlo (MC) method is employed to investigate the transport phenomena in semiconductors and semiconductor devices.

• The development of sophisticated techniques for semiconductor device fabrication, such as molecular beam epitaxy (MBE) & electron beam lithography, has enabled the realization of complex multi-layer device structures, with feature sizes previously considered impossible.

– Examples include the High Electron Mobility Transistor (HEMT) and the Heterojunction Bipolar Transistor (HBT).

SEMICONDUCTOR CIRCUIT SIMULATION

• Semiconductor circuit simulation uses mathematical models to replicate the behavior of an actual electronic device or circuit.

• Simulating a circuit’s behavior before actually building it, can greatly improves efficiency and provides insights into the behavior of electronics circuit designs.

• In particular, for integrated circuits the tooling (photo masks) is expensive, breadboards are impractical, and probing the behavior of internal signals is extremely difficult.

• Therefore almost all IC design relies heavily on simulation.

• known analog simulator is SPICE (Simulation Program with Integrated Circuit Emphasis) .

SPICE – Circuit Simulator • SPICE is an acronym for Simulation Program with Integrated Circuit

Emphasis • SPICE is a computer simulation and modeling program used by

engineers to mathematically predict the behavior of electronics circuits.

• SPICE is a powerful program that is used in IC and board-level design to check the integrity of circuit designs and to predict circuit behavior.

• The increased utilization of PCs has led to the production of PSPICE, a widely available PC version distributed by the MicroSim Corporation.

• SPICE is generally used to predict the behavior of low to mid frequency (DC to around 100MHz) circuits.

SPICE-Background • Integrated circuits are impossible to bread board before

manufacture. • High costs of photolithographic masks and other manufacturing

prerequisites make it essential to design the circuit to be as close to perfect as possible before the integrated circuit is first built.

• Simulating the circuit with SPICE is the industry-standard way to verify circuit operation (transistor level) before committing to manufacturing an integrated circuit.

• Even with a breadboard, some circuit properties may not be accurate compared to the final printed wiring board, such as parasitic resistances and capacitances.

• These parasitic components can often be estimated more accurately using SPICE simulation.

• Device characteristics can be affected by component manufacturing tolerances, in these cases it is common to use SPICE to perform Monte Carlo simulations of the effect of component variations on performance, a task which is impractical using calculations by hand for a circuit of any appreciable complexity.

SPICE- Device models • SPICE included many semiconductor device compact models:

– MOSFET model

– combined Ebers–Moll and Gummel-Poon bipolar model

– JFET model

– Model for a junction diode.

• In addition, it had many other elements:

– resistors

– capacitors

– inductors (including coupling),

– independent voltage and current sources

– ideal transmission lines

– voltage and current controlled sources.

SPICE-Types of Analysis supported • D.C sweep of current/voltage source (.DC)

• Operating point determination (.OP)

• Thevenin's equivalents (.TF)

• Time domain (transient) response (.TRAN)

• Fourier analysis (.FOUR)

• Small signal frequency response (.AC)

• Noise analysis (.NOISE)

• Sensitivity analysis (.SENS)

SPICE Programming

• Circuit simulation programs, of which SPICE and derivatives are the most prominent, take a text netlist describing the circuit elements (transistors, resistors, capacitors, etc) and their connections, and translate this description into equations to be solved.

• The general equations produced are nonlinear differential algebraic equations which are solved using implicit integration methods, Newton's method and sparse matrix techniques.

• SPICE combined operating point solutions, transient analysis, and various small-signal analyses with the circuit elements and device models needed to successfully simulate many circuits.

SPICE Programming Steps

• Basically, SPICE operates like this:

• 1. Describe a circuit in a text file (“.cir” extension) called a netlist OR draw the circuit using graphical symbols on a schematic page.

• 2. Run a simulation SPICE reads the netlist and then performs the requested analysis: AC, DC, or TRANSIENT RESPONSE. The results are stored in a text output file (“.out” extension) or a binary data file.

• 3. View the results of the simulation in a text output file (“.out” ) using a text editor. Most SPICE programs provide a graphical viewer to plot the waveforms stored in the binary data file.

SIMULATING YOUR OWN CIRCUITS

• Simulating and testing your own circuits can be easy.

– 1. Draw the circuit and number each node.

– 2. Label each component and give it a value.

– 3. Create a text file (netlist) listing all of the components and node connections.

– 4. Decide on the type of analysis you want performed (AC, Transient, DC, Noise, etc.) and include the appropriate statements. Run the simulation and view the results.

SPICE EXAMPLE –Circuit & Input file

SPICE EXAMPLE- Programming

SPICE v/s PSPICE Simulation Models • One of the most common errors made by even seasoned

engineers is confusing a SPICE model with a PSPICE model. • PSPICE is a commercially available program that uses proprietary

languages to define components and models. • While a circuit must be presented to SPICE in the form of a netlist. • The netlist is a text description of all circuit elements such as

transistors and capacitors, and their corresponding connections. • Modern schematic capture and simulation tools such as Multisim

allow users to draw circuit schematics in a user-friendly environment, and automatically translate the circuit diagrams into netlists.

• Consider as an example the simple voltage divider circuit in next slide, which include both netlist and corresponding circuit schematic.

Voltage Divider Schematic with Voltage Divider Netlist

* Any text after the asterisk '*' is ignored by

SPICE

* Voltage Divider

vV1 1 0 12

rR1 1 2 1000

rR2 2 0 2000

.OP * perform a DC operating point analysis

.END