# Presentation: 148364

## Practical Physics-Based Device Modeling: The Numerical Boltzmann/Spherical Harmonic Method

- Neil Goldsman
- Electrical and Computer Engineering
- University of Maryland
- College Park, MD USA

## Outline

- Overview of Device Simulation
- Benefits
- Approaches
- Numerical Boltzmann/Spherical Harmonic Method: A Balanced Approach
- Formulation and Application of Semiclassical to MOSFETs
- Quantum Correction (Wigner)
- Transient Application to BJT

## Benefits of Device Simulation

- _Allows you to probe inside device where measurements can not be made._
- Explore new experimental devices and device phenomena without fabricating device.
- Educate researchers, engineers and students to expand the frontiers of mathematical and computational skills germane to microelectronics.
- Help develop CAD tools so industry can continue its march forward in the miniaturization of electronic devices and circuits.

## Semiconductor Simulation Models:Hierarchy

- Hydrodynamic Model

- Drift Diffusion Model

- Quantum Boltzmann/Wigner Equation

- Quantum Hydrodynamic Model

- Quantum Drift Diffusion Model

- Schrödinger Poisson System

- Semiconductor Simulation Models:Hierarchy

**SEMICLASSICAL**

**QUANTUM**

- Circuit (Spice) Model

- Circuit (Spice) Model?

## Semiclassical Models

**All contain Poisson Equation + Transport Equation**

- Boltzmann Transport Equation Model:

- Hydrodynamic Model:

- Drift-Diffusion Model:

- Transport equations are shown here only for electrons. Transport equations for holes are analogous and understood.

## The Model Trade-Offs

- CAD tools usually represent a trade-off between physical rigor and computational efficiency.
- Generally speaking, the more physically rigorous, the less computationally efficient.
- Drift-Diffusion: Most CPU Efficient (Commercial Codes)
- Monte Carlo: Least CPU Efficient
- One of my personal goals in device simulation is to optimize this trade-off.

## Numerical Boltzmann/Spherical Harmonics Method

## Boltzmann-Device Model

- _Advantages:_
- Gives distribution function for entire device
- From distribution function virtually all device characteristics can be obtained.
- Explicitly includes scattering
- Expandable to quantum domain (Schrodinger or Wigner).
- _Difficulties:_
- Boltzmann equation is 7 dimensional (3 real-space, 3 momentum, 1 time)
- Direct discretization virtually impossible due to high dimensionality.
- Collision integral can cause global coupling of discretized system.

## Boltzmann-Device Model

- Poisson Equation:

- Boltzmann Transport Equation:

- Hole-Current Continuity Equation:

- Solve the system for :

- Potential :
- Electron Distribution Function :
- Hole Concentration :

## Collision Term

- The summation over *i* represents the different scattering processes which occur in MOSFETs.

- = the transition probability per unit time for the *i*th type of scattering.

**Optical Phonon Scattering**
**Acoustic Phonon Scattering**
**Surface Roughness Scattering**

**Ionized Impurity Scattering**
**SHR Recombination**
**Impact Ionization**

- The different scattering mechanisms incorporated in the collision term can be :

## BTE: Spherical Harmonics Method

- Solution to ‘Curse of Dimensionality Problem’
- Represent Distribution Function ***f(******r******,k,t)*** using Spherical Harmonic basis functions.
- Reduces dimension from 7 to 5 for 3D transient

## System of Equations for flm

- Boltzmann Transport Equation:

- Spherical Harmonic Form for BTE:

## Differential Difference Equation Form

- The equations for each unknown coefficient *f**l**m**(**r**,****)* described in the SH form of BTE are truncated.
- Substitute equations for *f**1**-1*, *f**1**0**, f**1**1* into equation for *f**0**0*
- We get a 3-D self-adjoint, differential difference equation from the BTE:
- Solve SH formulated BTE numerically: finite diffs, CG

- Where, the collision term is a summation over all the scattering mechanisms:

## Flowchart

- Poisson Equation           k+1(x,y)

- Electron BTE           f k+1(x,y

- Hole Current Continuity           pk+1(x,y)

- Converged?

- k(x,y), f k(x,y), pk(x,y)

- 0(x,y), n0(x,y), p0(x,y)

- k+1(x,y), f k(x,y

- k+1(x,y), f k+1(x,y), pk(x,y)

- k+1(x,y), f k+1(x,y), pk+1(x,y)

- No

- Yes

## Numerical Boltzmann/Spherical Harmonic Device CAD Results: Device Structure and Distribution Function

- Y=0.0001m

- Y=0.4m

- Distribution Function

- Electron Concentration

- MOS Cross Section

## Numerical Boltzmann/Spherical Harmonic Device CAD Results: Benchmark I-V with Experiment

- Doping Profile

*Leff = *0.88m

*Leff = *0.35m

*Leff = *0.15m

## Numerical Boltzmann/Spherical Harmonic Device CAD Results: Impact Ionization and Substrate Current

- Generation Rate

- Agreement with experiment: No fitting parameters!

*Leff = *0.88m

*Leff = *0.35m

*Leff = *0.15m

## Gate Current: Mathematical Model

- The final gate leakage current will be the summation of the tunneling and
- thermionic current
- Where
- Tunneling current
- Thermionic current

*J**gate **= Gate Current Density** *
*f      = Distribution Function*
*g     = Density of States*
*T**tun  **= Tunneling Probability*
*T**ther **= Thermionic Probability** *

## Numerical Boltzmann/Spherical Harmonic Device CAD Results: Gate Tunneling and Thermal Emission Current 

## Extending the Numerical Boltzmann Spherical Harmonic to the Quantum DomainSpherical Harmonics Boltzmann/Wigner Equation Method

## Quantum Correction

- Keeping only the first order expansion of potential, the Boltzmann/Wigner transport equation reduces to the Semiclassical Boltzmann Transport Equation.
- The higher order terms are hence the Quantum Correction to the BTE. The potential expansion can be truncated at the first two terms as the higher order terms quickly become negligible for many electronic systems.
**The STEADY STATE BOLTZMANN/WIGNER EQUATION CAN NOW BE WRITTEN AS:**

**Quantum Correction Term**

## Quantum Effects: Boltzmann/Wigner Results

- Doping profile

- Quantum Dist. Fn.

- Carrier Con. Ratio: Class/QM

- I~V Comparison

## Extending the Numerical Boltzmann Spherical Harmonic to the Transient RegimeTransient Spherical Harmonics Boltzmann Equation Method

## E

- B

- C

**Vbe**

**Vcb**

- Input Signal: 20 psec step

- BJT Structure: 50nm Base

**Transient Numerical Boltzmann/Spherical Harmonic****BJT Simulation**

- Electric Field vs Position (μm)

- 0

- 0.3

- B

- C

## E

- B

- C

**Vbe**

**Vcb**

- Input Signal: 20 psec step

- BJT Structure: 50nm Base

**Transient Numerical Boltzmann/Spherical Harmonic****BJT Simulation**

- 0.0

- 0.3

- microns

- Electron Concentration (logscale)
-  vs Position

- n

- B

- C

## E

- B

- C

**Vbe**

**Vcb**

- Input Signal: 20 psec step

- BJT Structure: 50nm Base

**Transient Numerical Boltzmann/Spherical Harmonic BJT Simulation**

- B

- C

- Average Electron Energy vs Position

- eV

- microns

- 0.0

- 0.3

## E

- B

- C

**Vbe**

**Vcb**

- Input Signal: 20 psec step

- BJT Structure: 50nm Base

**Transient Numerical Boltzmann/Spherical Harmonic****BJT Simulation**

- Distribution Function vs Position & Energy

- B

- C

- eV

- microns

- f(x,E)

## E

- B

- C

**Vbe**

**Vcb**

- Input Signal: 20 psec step

- BJT Structure: 50nm Base

**Transient Numerical Boltzmann/Spherical Harmonic****BJT Simulation**

## Summary

- In hierarchy of models, Numerical Boltzmann gives good tradeoff between accuracy and efficiency (1000 times faster than Monte Carlo)
- Numerical Boltzmann excellent for places where distribution function is required (impact ionization, gate leakage current, reliability).
- As devices shrink to nanoscale, quantum effects becoming very important, use either Wigner, density gradient or coupled Schrodinger
- Transient very important as well with clock speeds approaching 10GHz.