nanoMOS 4.0: A Tool to Explore Ultimate Si Transistors and Beyond

Network for Computational Nanotechnology (NCN)Purdue, Norfolk State, Northwestern, UC Berkeley, Univ. of Illinois, UTEP

Xufeng WangSchool of Electrical and Computer Engineering

Purdue UniversityWest Lafayette, IN 47906

nanoMOS 4.0: A Tool to Explore Ultimate Si Transistors and

Beyond

Xufeng Wang

Outline

• Why nanoMOS simulator?• Device geometries in nanoMOS.• nanoMOS development history and my involvement• Overview of nanoMOS code structure• Overview of nanoMOS software development• Conclusion• Acknowledgement

Xufeng Wang

Introduction

Xufeng Wang

Featured devices

Flexible and efficient modeling is needed to explore these device proposals.

Si/III-V double gate MOSFET

SOI MOSFET

HEMTspinFET

Xufeng Wang

Why nanoMOS?

• It studies a very general structure: double gate, thin body, n-MOSFET with fully depleted channel.

• It features several transport models: drift-diffusion, semiclassical ballistic, quantum ballistic, and quantum dissipative.

• It is computationally efficient, easily modified, written in MATLAB, and freely available on nanohub.org with Rappture interface

• Well documented on various thesis and papers.

Xufeng Wang

X. Wang, D. Nikonov

Xufeng Wang

Xufeng WangXufeng Wang

nanoMOS 1.0

Development history

Creation

nanoMOS 2.0

Rappture interface

nanoMOS 3.0

Quantum transport for III-V material

Asymmetrical gate configuration

nanoMOS 3.5

Code restructure: modulation, testing suite

Drift-diffusion transport for III-V material

nanoMOS spinFET

nanoMOS phonon scattering

nanoMOS HEMT

nanoMOS 4.0

Unification of branches

Parallel support, Rappture interface

Zhibin Ren S. Clark, S. Ahmed Kurtis Cantley

Himadri Pal

Yang Liu

Himadri Pal

Yunfei Gao

2000

Today

2008





Xufeng Wang

Inside thesis

GOAL: Deliver a comprehensive documentation and understanding of nanoMOS, physics and software wise.

Code restructure: modulation, testing suite Unification of branches



Models and techniques Software development

• Various transport models• Non-linear damping• Boundary conditions• Recursive Green’s function• Scharfetter and Gummel method• nanoMOS applications……

• Rappture interface• nanoMOS parallelization• Benchmark and testing suite





Xufeng Wang

Numerical Approach

Gummel’s Method

Solve Poisson’s Equation

Solve Transport Equations

Initial Guess for carrier density

€

n(old)⇒ ϕ

€

ϕ ⇒ n(new)

No

€

n(old) − n(new) < tolerance?Converge!Yes

€

∇2ϕ = −∇E = −1ε

(p− n + Nd+ − Na

−)

€

Jn = −qn(x)μ∇ϕ + qDn∇n

€

∂n∂t

= 1q∇Jn + Gn

€

n(new) ⇒ n(old)

Regardless how we start, all equations must be self-consistently satisfied at the same time

Xufeng Wang

“Straight-forward” Method of Solving Transport Equations

Solving the Transport Equations

Jn n(x)kT

( n(x)ddx

(x)ddx

n(x))

• In order to solve this equation, we first need to find a linear approximation to turn the differential equation into a discretized linear equation.

x i 1

x i

x i1

xi 1

2

xi1

2

First step is to use the mesh point variables to interpolate the midpoint variables

Xufeng Wang

“Straight-forward” Method of Solving Transport Equations


Substitute the approximated variables back to transport equations

x i 1

x i

x i1

xi 1

2

xi1

2

Continuity Equation tells us:

x i 1

x i

x i1

xi 1

2

xi1

2

Ji-1/2 Ji+1/2

Xufeng Wang

Stability Problem of “Straight-forward” Method


• Observe the equation:

ni1( i1 i 2) ni(i i 1

i i1

( i i 1 2) ( i1 i 2)) ni 1(i i 1

i i1

( i i 1 2)) 0

If both > 2

i1 i

i i 1Then, at least 1 carrier density is forced to

be negative

• This means if electric potential difference between any two neighboring nodes is greater than 2kT/q, the “straight-forward” method might get negative non-physical carrier density solutions.

• Therefore, a finer grid is required at regions that the rate of change of electric potential is high. This may lead to a huge number of grid nodes, thus increasing the computational cost dramatically.

Xufeng Wang

Scharfetter and Gummel Method


• We will attempt a direct integration by introducing the following factor:

n(x) e (x )u(x)• Carrier density• Exponential of electric potential• An unknown function of x

Find the derivative of carrier density (n)

Ji 1

2

i i 1

2kT( e (x )u(x) d

dx(x) e (x ) d

dxu(x) e (x )u(x) d

dx(x))

Substitute the introduced factor into transport equation

ddx

n(x) e (x ) ddx

u(x) e (x )u(x) ddx

(x)

Ji 1

2

i i 1

2kTe(x ) d

dxu(x)

Xufeng Wang



Recast the equation

Attempt a direct integration on both sides of the equation

Ji 1

2

i i 1

2kTe (x ) d

dxu(x)

Ji 1

2

e (x ) i i 1

2kTddx

u(x)

(Ji 1

2

e (x ))xi 1

xi

dx (i i 1

2kTddx

u(x))xi 1

xi

dx

Now, let’s look at this equation’s left and right hand side separately.

Xufeng Wang



Join the left and right hand side together

(Ji 1

2

e (x ))xi 1

xi

dx (i i 1

2kTddx

u(x))xi 1

xi

dx

Ji 1

2

x i i 1

(e i 1 e i ) i i 1

2kT(nie

i ni 1e i 1 )

Ji 1

2

i i 1

2kTB( i i 1)e

ini ni 1e

i i 1

x

B(z) zez 1

is the Bernoulli Function

Xufeng Wang



This is the 1-D electron Transport Equation via finite difference with Scharfetter and Gummel Method at node xi-1/2

• Similarly, one can write down the transport equation at node xi+1/2

Ji 1

2

i i 1

2kTB( i i 1)e

ini ni 1e

i i 1

x at node xi-1/2

Ji1

2

i i1

2kTB( i1 i)e

i1ni1 nie

i1 i

x at node xi+1/2

• Now, just as what we did in “straight forward” method, we can use the relationship establish by Continuity Equations to solve the problem

Xufeng Wang

0)()()))(()(()()( 11111111111

iiiiiiiiiiiiiiiii neBnBBneB iiii



0)()()))(()(()()( 11111111111


• How can the stability of transport equation be guaranteed by Scharfetter and Gummerl Method?

• Notice that the Bernoulli Function is ALWAYS positive.

B(z) zez 1

• One coefficient is always negative, so the carrier densities are no longer forced to be negative.

Xufeng Wang

Numerical Approach

Gummel’s Method




€

n(old)⇒ ϕ

€

ϕ ⇒ n(new)

No

€


€

∇2ϕ = −∇E = −1ε

(p− n + Nd+ − Na

−)

€


€

∂n∂t

= 1q∇Jn + Gn

€

n(new) ⇒ n(old)


Xufeng Wang

Solving the Poisson Equation

Boundary Conditions for Poisson Equation

• Although source and drain bias are given as inputs, we still use Neumann boundary for source and drain ends to avoid convergence problem.• Source and drain bias are used to calculate electron density, thus indirectly influence the potential at ends.

Xufeng Wang

Between 2D Poisson solver and 1D transport

• Effective mass Schrodinger equation is solved in confinement direction

Xufeng Wang

Complete Scheme of Drift-Diffusion Modeling





€

n(old)⇒ ϕ

€

ϕ ⇒ n(new)

No

€


€

∇2ϕ = −∇E = −1ε

(p− n + Nd+ − Na

−)

€


€

∂n∂t

= 1q∇Jn + Gn

€

n(new) ⇒ n(old)

Newton Iteration Converge?

Yes

No

Schrodinger Equation Solver

€

ϕ _ 2D(poisson)⇒ ϕ _1D(subbands)φ(subband,x)

Xufeng Wang

Other available transport models

Drift-diffusion

Semiclassical ballistic

Quantum ballistic

Quantum dissipative with phonon scattering

computationally efficient mobility difficult to determine

evaluates device ballistic limit may be too optimistic

RGF based; quantum effects no scattering; longer run time

Phonon scattering longest run time

Xufeng Wang

Software development: Overview

Developer UserSVN

test & benchmark

parallel job submitterRappture on nanoHUB

Xufeng Wang

Software development: Rappture interface

Xufeng Wang

Conclusion

• Overviewed nanoMOS development history• Demonstrated Scharfetter and Gummel method as numerical

sample• Demonstrated Rappture interface as software sample

GOAL: Deliver a comprehensive documentation and understanding of nanoMOS, physics and software wise.

Xufeng Wang

Acknowledgement

• Committee members: Professor Klimeck, Professor Lundstrom, and Professor Strachan.

• Funding and support from my advisors.• Encouragement and help when needed from my colleagues.• Mrs. Cheryl Haines and Mrs. Vicki Johnson for scheduling the

examination and being the most helpful secretaries.• As always, thank and love to my entire family.

Xufeng Wang

Now, welcome the questions……

Xufeng Wang

Xufeng Wang

Device geometry #1: Si/III-V double gate MOSFETs

• Si/III-V as channel material• Thin body (< 10nm). Single channel conduction if thin enough.• Double gates can be biased separately• Source/drain can be metallic and turn into Schottky barrier FET

Sample double gate MOSFET geometry

3D electron density

Conduction band profile

Xufeng Wang

Device geometry #2: SOI MOSFET

• Si/III-V as channel material.• Similar to previous structure, except the bottom oxide layer is thick.• Back gate can be biased to push channel electron toward front gate.

Sample SOI geometry

3D conduction band near front gate

Conduction band in transverse direction

Xufeng Wang

Device geometry #3: HEMT

• Intrinsic III-V material as channel = high mobility.• Delta-doped layer controls threshold voltage.* Y. Liu, M. Lundstrom, “Simulation-Based Study of III-V HEMTs Device Physics for High-Speed Low-Power Logic Applications”, ECS meeting, 2009

Sample HEMT geometry

Charge and conduction band profile from Yang Liu *

Xufeng Wang

Device geometry #4: spinFET

• Device structure suggested by Sugahara & Tanaka• Controls current by manipulating electron spin

Sample spinFET geometry

Xufeng Wang

What are we trying to solve?

Problem Statement and the Semiconductor Equations

Source DrainTop Gate

Buttom Gate

• Given device geometry and material parameters (such as gate length, dielectric constant, mobility)• Look for solution for:

carrier densityelectric potential

• Both carrier density and electric potential solutions must satisfy all the equations.


Xufeng Wang

Transport model #1: drift-diffusion

• Computationally efficient• Account scattering via mobility, thus suitable for long channel

devices• Do not consider quantum effects such as tunneling and

interference.

Xufeng Wang


SG method ensures stability of carrier density solutions.

• If apply finite difference method directly:

ni1( i1 i 2) ni(i i 1

i i1

( i i 1 2) ( i1 i 2)) ni 1(i i 1

i i1

( i i 1 2)) 0

If both > 2

i1 i

i i 1

Then, at least 1 carrier density is forced to be negative

n(x) e (x )u(x) • Carrier density• Exponential of electric potential• An unknown function of x

0)()()))(()(()()( 11111111111


• Introduce Scharfetter and Gummel method

Xufeng Wang

Transport model #2: Semiclassical ballistic

• Simple model exploring device behavior at ballistic limit• Do not consider quantum effects such as tunneling and

interferences.

€

v =1hdEdk

=2Em*

€

J = qvn

€

n(E) = D(E) f (E)

Xufeng Wang

Transport model #3 & #4 : Quantum ballistic & dissipative

€

G(E) = [E lI − H[E i(x)] − Σ]−1

€

Σ ΣS L 0M O M0 L ΣD

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

€

H =

H(E1(x)) 0 L 00 H(E2(x)) L 0M M O M0 0 L H(E i(x))

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

H(E i(x)) =

2t − E i(1) 0 L 00 2t − E i(2) L 0M M O M0 0 L 2t − E i(Nx )

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

€

t =h

2mx*a2

€

ΣS (E) = −te ikl a

€

E = E i(1) + 2t(1− coskla)

Xufeng Wang

Transport model #3 & #4 : Quantum ballistic & dissipative

For dissipative transport, nanoMOS can treat phonon scattering, or general scattering via Buttiker probe approach (now obsolete).

An GGnG

€

GS = i(ΣS − ΣS+)

€

GD = i(ΣD − ΣD+ )

€

n(E l ) =1

2πa(ℑ −1/ 2(μS − E l )AS +ℑ −1/ 2(μD − E l )AD )

€

TSD (E l ) = Trace(ΓSGΓDG+)

€

ISD (E l ) =qh

(ℑ −1/ 2(μS − E l ) −ℑ −1/ 2(μD − E l ))TSD (E l )

Xufeng Wang

Development historynanoMOS 1.0 (Published in 2000)• Developer: Zhibin Ren• Original nanoMOS code for silicon MOSFETs is written in MATLAB.nanoMOS 2.0 (Published in 2005)• Developer: Steve Clark, Shaikh S. Ahmed• Rappture interface is added to nanoMOS, and the code becomes

avaliable on nanoHUB.org.nanoMOS 3.0 (Published in 2007)• Developer: Kurtis Cantley• Support for III-V materials in semi-classical ballistic and quantum

ballistic transport models is added. Rappture interface is updated to reflect the III-V implementation.

nanoMOS 3.0 (Published in 2007)• Developer: Himadri Pal• Top and bottom gate can now have asymmetric configurations with

different gate dielectrics and capping layers.nanoMOS 3.5 (Published in 2008)• Developer: Xufeng Wang• Support for III-V materials in drift-diffusion transport is added.

Additional mobilities models are added.nanoMOS 3.5 (Published in 2009)• Developer: Xufeng Wang, Dmitri Nikonov• nanoMOS source code is restructured and modularized. Material

parameters are separated out as a mini-library. Debugging functions are planted within source code to assist code developments. Benchmark and testing suite is created based on a script from Dmitri Nikonov.

nanoMOS 4.0 (Developed in 2009)• Developer: Himadri Pal• Support for Schottky FET is added. NanoMOS now has the

ability to simulate a double gate MOSFETs structure with metallic source/drain via NEGF\ formalism.

nanoMOS 4.0 (Developed in 2009)• Developer: Yang Liu• Support for HEMT is added. NanoMOS now has the ability to

simulate a III-V HEMT structure via NEGF formalism.nanoMOS 4.0 (Developed in 2009)• Developer: Xufeng Wang• Parallel Jobs Submitter (PJS) is added. PJS allows nanoMOS

to sweep gate/source bias and run each bias on a cluster node. It supports only clusters with Portable Batch System (PBS) installed such at steele (steele.rcac.purdue.edu) or coates (coates.rcac.purdue.edu).

nanoMOS 4.0 (Developed in 2009)• Developer: Yunfei Gao• Support for SpinFET is added. NanoMOS now has the ability

to simulate a SpinFET structure via NEGF formalism.nanoMOS 4.0 (To be published in 2010)• Developer: Xufeng Wang• Merge working branches of Schottky FET, HEMT, and

SpinFET modules. Code is restructrued. Rappture interface is updated to accommodate the newly published features.

Documents

nanoMOS 4.0: A Tool to Explore Ultimate Si Transistors and Beyond