# Presentation: 826993

## Advanced Device Simulation at Agere Systems

*Presentation: C.Rafferty*
**M.S. Hybertsen, J. Bude, R.K. Smith,**
**B. Witzigmann, M.A. Alam **
- Agere Systems, Murray Hill, NJ 07974

- Agere Systems

- 10/15/01

## Overview

- Monte Carlo Device Simulation
- Introduction
- Problem formulation
- Numerical methods
- Laser simulation
- Optics
- Electronics
- Simulation

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## Device simulation - introduction

- Traditional drift-diffusion models describe the behavior of the mean-energy carriers
- Many important effects depend on high energy carriers
- Ballistic transport
- Damage from hot-carrier injection into dielectrics
- Memory programming by injection of hot carriers into floating gates

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## BTE for device simulation

- Full problem is described by distribution of electrons in space and momentum
- 6-dimensional problem, 3 space dimensions, 3 momentum dimensions
- Boltzmann Transport Equation
- Direct solution extremely challenging
- Monte Carlo approach generally adopted

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## Monte Carlo device simulation

- Inject electrons and holes, one at a time, close to contacts, with momentum randomly distributed so that energy mean is at thermal equilibrium
- Integrate equations of motion, subject to random scattering events with lattice vibrations (phonons) and dopant atoms
- Follow until particle exits or comes to equilibrium
- Use statistical enhancement techniques to avoid following ‘boring’ particles
- Accumulate statistics over many particles

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## Monte Carlo simulation - discretization

- Represent phase space by a Cartesian product of a space (x) grid and a momentum (k) grid
- X-grid is typical transistor grid with simplices of dimension 1-100nm
- K-grid is a tetrahedral decomposition of unit momentum cell
- Main focus of this presentation

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## Equations of motion

- Applied during free-flight time
- Time-step inversely proportional to scattering rate
- Time-step also limited by time to reach cell edge
- Inside a Tx-Tk cell, energy and potential are linear, gradients are constant, equations can be integrated exactly
- Much better than numerical integration, no gain/loss of energy

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**Notes:**
Velocity is derivative of E vs k
Rate of change of momentum is the space derivative of potential (how steep a hill you are climbing)
Exact is not true in all codes, but a result of chosing linear basis functions on simplices.

## Scattering rates 

- Scatter from a state with momentum     to a state with a different momentum       with or without change in energy
- Requires selection of final state *with specific energy*
- Requires knowledge of density of states            ~ area of energy isosurface with energy

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**Notes:**
Free-flights are interrupted by scattering

## Band structure (1)

- Electrons in free space: momentum and energy are simply related by
-  Electrons in a periodic crystal: momentum is characterized by a band index and a momentum which varies over a minimum periodic cell in reciprocal space

- Brillouin Zone
- (k-space)

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## Band structure (2)

- Surfaces of constant energy no longer spherical
- For low energy electrons or holes, parabolic ok
- For higher energies, analytic bands *maybe* ok
- For hot carriers, must take full band shape into account
- Difficult to identify states with particularly energy
- Difficult to calculate area of iso-energy surface
- But must be accurate since

**Cu**

**Si**

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## Hole bands

**Cross section of k-space at **
** ****Energy contours**

**B2**

**B3**

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## Electron bands

**B4**

**B6**

**B5**

**B7**

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## Full band representation

- Decompose Brillouin zone into 48 wedges, all of which are related to the first by symmetry (8 axis inversion, 3 cyclic permutation, 2 axis switching)
- Discretize irreducible wedge into tetrahedral mesh in k-space
- Linear shape function in tet –> constant E surface easy
- Further important simplification if tet is energy aligned so that nodes are on constant energy surfaces
- Simplifies choice of       with specific energy
- Greatly simplifies iso-energy surface area calculation

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## Interpolation

- define

- Type A

- Type B

- Type C

- The three quantities
-  are functions of the tet geometry only and can be precomputed

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**Notes:**
This is not true in all codes, but a result of chosing linear basis functions on simplices.

## Total DOS of a constant energy surface

- The integrated DOS are pre-computed for each contour level of the mesh

- A complete DOS calculation requires under a dozen arithmetic operations

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**Notes:**
This is not true in all codes, but a result of chosing linear basis functions on simplices.

## Final state selection

- Given a desired final state energy         , choosing a new momentum state            after scattering is also easy
- Identify tets with
- Select tetrahedron type randomly among A,B,C
-                 defines isosurface inside tet
- Choose two random variables distributed randomly over a right triangle (type A or C) or a square (type B)
- Treat as barycentric coordinates to choose       on isosurface

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**Notes:**
This is not true in all codes, but a result of chosing linear basis functions on simplices.

## Mesh generation

- Special purpose mesh-generation carried out once for each material (silicon, GaAs, InP, ...)
- Calculate fine 3d mesh of E(k) using band-structure code
- Build skeleton from critical points                   and ridge lines
- Use ridge lines to divide into regions
- Within each region, generate vertices along contours
- Adhere to ridge lines
- Adhere to wedge boundaries
- Piecewise linear approximation to region boundaries
- Contour-aligned grid produced by refining grid edges
- Edges crossing more than one countour subdivided
- Curvature used to monitor discretization error

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**Notes:**
This is not true in all codes, but a result of chosing linear basis functions on simplices.

## Mesh generation

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**Notes:**
This is not true in all codes, but a result of chosing linear basis functions on simplices.

## Mesh generation statistics

| Band | Vertices | Tetrahedra | Contours | Quality |
| --- | --- | --- | --- | --- |
| 1(h) | 349 | 1198 | 24 | 0.567 |
| 2(h) | 530 | 2268 | 19 | 0.611 |
| 3(h) | 414 | 1617 | 18 | 0.590 |
| 4(e) | 1343 | 6143 | 16 | 0.529 |
| 5(e) | 1658 | 8077 | 17 | 0.564 |
| 6(e) | 613 | 2431 | 13 | 0.456 |
| 7(e) | 419 | 1542 | 11 | 0.421 |

- Error tolerance of 0.02eV, and 0.002eV near minima

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**Notes:**
This is not true in all codes, but a result of chosing linear basis functions on simplices.

## Results

- Simulation times in hours, not weeks
- Huge dynamic range in carrier distribution functions
- Smooth variation of output with input (low statistical noise)
- Availability of new simulation methods has led to the development of new types of flash memory using physical mechanisms not previously understood

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**Notes:**
This is not true in all codes, but a result of chosing linear basis functions on simplices.

## Good spatial resolution available - 

- essential for accurate damage modeling

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## Simulation Versus Measurement – Resolution of High Energy Tail Essential to Match Experiment

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## Microscopic Laser Simulation

- Fundamental Processes
- Quantum optics
- Carrier transport
- Carrier recombination
- Quantum well capture
- Quantum well gain
- Length scales (10Å - 10m)

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## Optical processes

- Photons generated when electrons meet holes (spontaneous emission)
- Photons absorbed by free carriers
- Photons lost at mirror ends of device (absorbed or escaped)
- Photons stimulate emission of photons (basis of laser action - stimulated emission)
- In steady state

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## Shape of mode is determined by scalar wave equation

Lateral length scale much smaller than longitudinal, write

Solve for lateral mode (eigenvalue problem)

Generally only one lateral mode active, determines center energy (~1eV), with a number of longitudinal modes separated by ~1meV energy
Laser is designed so center energy (determined by device geometry) corresponds to energy emitted when e-h pair recombines
Product         determines local optical power at each point in space

- Lateral length scale much smaller than longitudinal, write
- Solve for lateral mode (eigenvalue problem)
- Generally only one lateral mode active, determines center energy (~1eV), with a number of longitudinal modes separated by ~1meV energy
- Laser is designed so center energy (determined by device geometry) corresponds to energy emitted when e-h pair recombines
- Product         determines local optical power at each point in space

- Optical modes

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## Free carriers

- Charge balance
- Electron transport
- Hole transport
-                     discretized as scalar variables, one per mesh point
- Recombination         algebraic function
- Capture of electrons into wells        a (messy, but local) function of electron density, electrostatic potential and captured electron density
- Exponential upwinding used (Scharfetter-Gummel)

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## Bound carriers - modes

- Solutions
- Various scattering processes broaden sharp levels, accounted for empirically
- Density of states ~

- E.g. parabolic bands:
- (if life were easy, in fact
-    is much more complex)

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## Bound carriers - density

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## Bound carriers - interaction with free carriers

- Either       or        serves to characterize bound carrier population
- Convenient to keep both variables as solution variables and add the relationship between them as an extra equation

- Cn(n3D,Fn2D,V) appears in n3D equations near quantum well
- But F2D (V) non-locally through the quantum well region
- Solution: introduce F2D as an auxiliary solution variable

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## Capture and gain expressions

- Capture rate
-                                        analagous to
- Gain

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## Gain: Qualitative Physics

- Carrier scattering: broadening
- Probability of emission
- Integrate:
- Ah (Eh)fh (Eh)Ae (Ee)fe(Ee)
- for h = Ee - Eh
- Key point
-  Usually simplify:  << kBT
- common Lorentzian approximation:
- fh (Ehk) fe(Eek)L(h - (Eek - Ehk))
-  But for lasers:  ~ kBT
-  full convolution required

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## Assemble all the pieces:Fully Coupled Device Equations

- Free Carriers (3D)
- Bound Carriers (2D)
- Photon Modes

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**Notes:**
Bound electrons can have arbitrarily large energy.  Their ky is fixed but their kx can be whatever, so kx2+ky2 can be as high as you like.  Once they are over the height of the barrier, they find it very easy to scatter and interact with the free electrons.

## Key Numerical Issue - Laser Threshold

- Threshold for photon intensity
- Pole as gain approaches loss
- I ~ exp(V/kT)
- 2X in current happens over 0.04V at V~0.8V
- Solution: introduce slack variable to avoid negative photon number

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## Solution Flowchart

- Solve optical mode

- Solve bound carrier eigenvalue system

- Assemble Newton RHS and Jacobian for 3d carriers
- Assemble Newton RHS and Jacobian for 2d carriers
- Assemble Newton RHS and Jacobian for photon number and slack variable

- Direct solve

- Converge?

- New bias point

**Electrical and optical PDEs discretized on same grid**
**Fully self-consistent solution of bound and free electrons**
**Slack variable for photon number**

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**Notes:**
Bound electrons can have arbitrarily large energy.  Their ky is fixed but their kx can be whatever, so kx2+ky2 can be as high as you like.  Once they are over the height of the barrier, they find it very easy to scatter and interact with the free electrons.

## System Jacobian

- All derivatives evaluated
- Derivatives of eigenvalues wrt parameters of eigenvalue problem expanded using first-order perturbation theory

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## Steady State Results:Simulation vs Experiment

- Gain spectra near threshold
- Spectral shape
- Temperature dependence
- Direct verification of key QW model

- Threshold current density
- Change with doping
- Temperature dependence
*J**th** ****  exp(T/T**0**)*
*Break in T**0** near 50 C*

- low doped

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## Effective Differential Gain:Simulation vs Experiment

- Modulation efficiency increases with active layer doping
- But ... Temperature sensitivity also increases !
- Simulation accounts for both the doping and the temperature dependence
***No ***parameter is adjusted for the modulation efficiency

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## Conclusion

- Demonstrate efficient, integrated* LASER *simulator
- Fully self-consistent calculation of electron and photon processes
- Solved using full Newton method
- Slack variables solve threshold problem
- Excellent agreement with steady state and dynamic performance data for 1.3m MQW lasers
- Explain temperature dependent threshold (T0)
- Explain temperature & density dependent modulation efficiency
- Explains diminishing returns of extra quantum wells

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