Transcript defense - Purdue University

Network for Computational Nanotechnology (NCN)
Purdue, Norfolk State, Northwestern, UC Berkeley, Univ. of Illinois, UTEP
nanoMOS 4.0: A Tool to Explore
Ultimate Si Transistors and Beyond
Xufeng Wang
School of Electrical and Computer Engineering
Purdue University
West Lafayette, IN 47906
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
Si/III-V double gate MOSFET
SOI MOSFET
HEMT
spinFET
Flexible and efficient modeling is needed to explore these device proposals.
Xufeng Wang
Why nanoMOS?
• It studies a very general structure: double gate, thin body, nMOSFET 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
Development history
Zhibin Ren
Kurtis Cantley
S. Clark, S. Ahmed
Creation
Quantum transport for
2008
III-V material
nanoMOS 2.0
nanoMOS 3.0
Asymmetrical gate
Code restructure:
configuration
modulation, testing suite
Rappture interface
nanoMOS 1.0
2000
Himadri Pal
nanoMOS HEMT
Yang Liu
X. Wang, D. Nikonov
Xufeng Wang
Unification of branches
nanoMOS phonon scattering
Himadri Pal
Xufeng Wang
Xufeng Wang
Drift-diffusion transport
for III-V material
Parallel support,
Rappture interface
nanoMOS 3.5
nanoMOS spinFET
nanoMOS 4.0
Yunfei Gao
Today
Xufeng Wang
Inside thesis
Unification of branches
Drift-diffusion transport
for III-V material
Models and techniques
• Various transport models
• Non-linear damping
• Boundary conditions
• Recursive Green’s function
• Scharfetter and Gummel method
• nanoMOS applications
Drift-diffusion transport
……
for III-V material
Code restructure:
modulation, testing suite
Parallel support,
Rappture interface
Code restructure:
modulation, testing suite
Software development
• Rappture interface
• nanoMOS
Unification parallelization
of branches
• Benchmark and testing suite
Parallel support,
Rappture interface
GOAL: Deliver a comprehensive documentation and
understanding of nanoMOS, physics and software wise.
Xufeng Wang
Numerical Approach
Gummel’s Method
Initial Guess for
carrier density
No
n(new)  n(old)
Yes
Converge!
n(old)  n(new)  tolerance?

Solve Poisson’s
Equation
1
 2  E   ( p  n  N d  N a )

  
n(old)

Solve Transport

Equations
Jn  qn(x)  qDnn
n 1
 J  Gn
t q n
 n(new)


Regardless how we start, all equations must be self-consistently satisfied at
the same time
Xufeng Wang
Solving the Transport Equations
“Straight-forward” Method of Solving Transport Equations
 n(x)
d
d
Jn 
(n(x)  (x) n(x))
kT
dx
dx
• 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
1
i
2
xi
x
x i1
1
i
2
First step is to use the mesh point variables to interpolate the midpoint variables


Xufeng Wang



Solving the Transport Equations
“Straight-forward” Method of Solving Transport Equations
 Substitute the approximated variables back to transport equations
x i1
x
i
1
2
xi

Continuity Equation tells
 us:

Xufeng Wang
x
i
i
x i1
1
2


Ji-1/2
x i1
x
Ji+1/2
1
2
xi
x
i
1
2
x i1
Solving the Transport Equations
Stability Problem of “Straight-forward” Method
• Observe the equation:
i i1
 
(i i12)(i1i 2) n
) i1( i i1(i i1 2) 0
)
i i1
i i1
ni1(i1i 2)ni(
i1i
If both
i i1> 2


Then, 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
Solving the Transport Equations
Scharfetter and Gummel Method
• We will attempt a direct integration by introducing the following factor:
(x)
n(x)e u(x)
• Carrier density
• Exponential of electric potential
• An unknown function of x
 Find the derivative of carrier density (n)
d
d
d
n(x)  e (x )
u(x)  e (x)u(x)
 (x)
d x
d x
d x

 Substitute the introduced factor into transport equation
i  i1 (x)
d
d
 (x) d
(x)
J

(e
u(x)

(x)e
u(x)e
u(x)
(x) )
  i1
2k T
d x
d x
d x
2
J
i
 
Xufeng Wang
1 
2
i  i1
2k T
e(x)
d
u(x)
dx
Solving the Transport Equations
Scharfetter and Gummel Method
 Recast the equation
J
i
1 
2
i  i1
2k T
e(x)
d
u(x)
dx
(x)
J 1e
i
2

i  i1 d
u(x)
2k T dx
 Attempt a direct integration on both sides of the equation

xi
 (J
xi1
e
1
i
2
i i1 d

)d x (
u(x) d) x
(x)
xi

xi1
2k T d x
Now, let’s look at this equation’s left and right hand side separately.

Xufeng Wang
Solving the Transport Equations
Scharfetter and Gummel Method
 Join the left and right hand side together
xi
 (J
xi1
J

i
1
2
xi
e
1
i
2
xi1
u(x) d) x
2k T d x
x
  i1
(e i 1  e i )  i
(n ie i  n i1e i1 )
 i   i1
2k T
J
 
1
i
2

i  i1
2k T
B(i i1)e
B(z) 

i i1 d
)d x  (
(x)
Xufeng Wang
i
ni ni1ei i1
x
z is the Bernoulli
e z 1
Function
Solving the Transport Equations
Scharfetter and Gummel Method
J
1 
i
2
i  i1
2k T
B(i i1)ei
ni ni1ei i1
x
at node xi-1/2
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
i1i
n
n
e
J 1
B(i1 i)ei1 i1 i
i
2k T
x
2
i i1
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
Solving the Transport Equations
Scharfetter and Gummel Method
• How can the stability of transport equation be guaranteed by Scharfetter and
Gummerl Method?
( i  i 1 ) B(i  i 1 )ei i1 ni 1  ( B(i  i 1 )  B(i 1  i ))( i  i 1 )ni  ( i  i 1 ) B(i 1  i )ei i1 ni 1  0
( i  i 1 ) B(i  i 1 )ei i1 ni 1  ( B(i  i 1 )  B(i 1  i ))( i  i 1 )ni  ( i  i 1 ) B(i 1  i )ei i1 ni 1  0
• Notice that the Bernoulli Function is ALWAYS positive.
B(z) 
z
e z 1
• One coefficient is always negative, so the carrier densities are no longer
forced to be negative.

Xufeng Wang
Numerical Approach
Gummel’s Method
Initial Guess for
carrier density
No
n(new)  n(old)
Yes
Converge!
n(old)  n(new)  tolerance?

Solve Poisson’s
Equation
1
 2  E   ( p  n  N d  N a )

  
n(old)

Solve Transport

Equations
Jn  qn(x)  qDnn
n 1
 J  Gn
t q n
 n(new)


Regardless how we start, all equations must be self-consistently satisfied at
the same time
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
Solving the Transport Equations
Complete Scheme of Drift-Diffusion Modeling
Initial Guess for
carrier density
No

Yes
Converge!
n(new)  n(old)
Solve Poisson’s
Equation

n(old)  n(new)  tolerance?
  n(new)

Solve Transport
Equations
n 1
 J  Gn
t q n
Jn  qn(x)  qDnn
Xufeng Wang
Newton Iteration
Converge?
Yes
n(old)  
Schrodinger

Equation Solver
 _ 2D( poisson)   _1D(subbands)
 (subband, x)
1
 2  E   ( p  n  N d  N a )

No
Other available transport models
Drift-diffusion
computationally efficient
mobility difficult to determine
Semiclassical ballistic
evaluates device ballistic limit
may be too optimistic
Quantum ballistic
RGF based; quantum effects
no scattering; longer run time
Quantum dissipative with phonon scattering
Phonon scattering
Xufeng Wang
longest run time
Software development: Overview
test &
benchmark
Developer
SVN
User
Rappture on nanoHUB
parallel job submitter
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
Sample double gate MOSFET geometry
3D electron density
Conduction band profile
• 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
Xufeng Wang
Device geometry #2: SOI MOSFET
3D conduction band
near front gate
Sample SOI geometry
Conduction band in
transverse direction
• 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.
Xufeng Wang
Device geometry #3: HEMT
Charge and conduction band
profile from Yang Liu *
Sample HEMT geometry
• 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
Xufeng Wang
Device geometry #4: spinFET
Sample spinFET geometry
• Device structure suggested by Sugahara & Tanaka
• Controls current by manipulating electron spin
Xufeng Wang
Problem Statement and the Semiconductor Equations
What are we trying to solve?
Top Gate
Source
Drain
Buttom Gate
• Given device geometry and material parameters
(such as gate length, dielectric constant, mobility)
• Look for solution for:
carrier density
electric potential
• Both carrier density and electric potential
solutions must satisfy all the equations.
Regardless how we start, all equations must be self-consistently satisfied at
the same time
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
Scharfetter and Gummel Method
• If apply finite difference method directly:
i i1
 
(i i12)(i1i 2) n
) i1( i i1(i i1 2) 0
)
i i1
i i1
ni1(i1i 2)ni(
If both
i1i
i i1
>2



Then, at least 1 carrier density is forced to
be negative
• Introduce Scharfetter and Gummel method
n(x)e(x)u(x)
• Carrier density
• Exponential of electric potential
• An unknown function of x
( i  i 1 ) B(i  i 1 )ei i1 ni 1  ( B(i  i 1 )  B(i 1  i ))( i  i 1 )ni  ( i  i 1 ) B(i 1  i )ei i1 ni 1  0
SG method ensures stability of carrier density solutions.

Xufeng Wang
Transport model #2: Semiclassical ballistic


n(E)  D(E) f (E)
J  qvn
1 dE
2E
v

dk
m*

• Simple model exploring device behavior at ballistic limit
• Do not consider quantum effects such as tunneling and
interferences.
Xufeng Wang
Transport model #3 & #4 : Quantum ballistic & dissipative
S

  

0
G(E)  [E l I  H[E i (x)]  ]1
H(E1 (x))
0

0
H(E 2 (x))
H  


0
 0





H(E i (x)) 
0
0

2t  E i (1)
0

0
2t  E i (2)

H(E i (x)) 


0
 0




 t  2m * a 2
2t  E i (N x ) 
x
0
0

Xufeng Wang
S (E)  teikl a

0 


D 

E  Ei (1)  2t(1 coskl a)
Transport model #3 & #4 : Quantum ballistic & dissipative
An  GnG
n(E l ) 
S  i( S  S )
D  i( D   D )
1
( (  E l )AS  1/ 2 ( D  E l )AD )

2
a 1/ 2 S
TSD (E l )  Trace(S GD G  )
q
ISD (E l )  (1/ 2 ( S  E l )  1/ 2 (D  E l ))TSD (E l )
For dissipative transport, nanoMOS can treat phonon scattering, or
general scattering via Buttiker probe approach (now obsolete).
Xufeng Wang
Development history
nanoMOS 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.
Xufeng Wang
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.