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Jesse Maassen
(Supervisor : Prof. Hong Guo)Department of Physics, McGill University, Montreal, QC Canada
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Towards parameter-free device modeling
The first electronic computer: ENIAC --- large sizes
This computer is made of vacuum tubes, 17,000 of them.
People work inside the CPU of this computer.
1800 square feet
ENIAC: Electronic Numerical Integrator and Computer. It was 2400 times faster than human computing.
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Today: transistors are very small
200 million transistors can fit on each of these pin head.
€
1960 : L ≈10μm
2000 : L ≈100nm
2010 : L = 22nm
How to compute charge conduction in these atomic systems?
Line of ~ 50 atoms
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
As the size of a device goes down, physics change
Channel Length, L
1 mm
0.1 mm
10 µm
1 µ m
0.1 µm
10 nm
1 nm
0.1 nm
Transistor
2000
Atomic dimensions
1975
2016
Top
Bottom
Macroscopicdimensions
L
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
V = I R or I = V G
Conductance G = 1/R L
AG σ=
Conductivity
“Not obvious”
Conduction is usually studied “top down”
μσ nq= Mobility
?=τm = ? n = ?
mqτμ = Scattering timeChannel
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
What device parameters?
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
VVV
III
G
S
D
G
S
DDevice
parameters
These parameters specify properties of each individual device.
How to obtain device parameters? --- by experimental measurements - now; --- by computational modeling;
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Practical modeling method: need for many parameters
capacitanceTransconductance
Geometry scalingdiodes
More than a 400 parameters are needed.May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Moore’s law for model parameters
Number of parameter double every 18 months
Reflects the complexity in modern technology
103
Year of introduction
Implem
ented feature(arbitrary unit in log scale)
1
102
10
1965 1980 1990 2000
Level 1
Level 2
Level 3
BSIM1
BSIM2
BSIM3v3
BSIM4
Num
ber
of p
aram
eter
s
parameter per feature
PSP
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Different modeling method: includes quantum and discrete material properties (all parameter free, no m, n or τ)
Quantum:
Tunneling – cannot turn off transistor; Size quantization ; electron-phonon scattering during current flow; Quantum dissipation; Spin transport; Spin-orbital effects …
Atomistic structures:
Materials are no longer a continuous medium. Atomic simulations are useful when: more atomic species are used in nano-systems; charge transfer; interfaces, surfaces, domain boundaries; external potential drop; disorder …
It is highly desirable to develop parameter-free theory and modeling method.
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
large scale device
modeling
device parametersatomic simulations
materials, chemistry, physics
quantum mechanics Physics
device modeling < 50nm (1000 atoms)
Nanoelectronic device physics
crash
science engineering
Goal of nanoelectronics theory and modeling
This is largely applied physics: it is absolutely important that our theory is not only fundamentally correct, but also practical.
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Theoretical transport model
A scattering region; semi-infinite leads; coherence; external potentials; coupling to other bath (the X-probe), etc.. We build an atomic model for this picture (for material specific properties).
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Theoretical transport model (cont.): Landauer theory
Under a voltage bias, electrons elastically (coherent) traverse the device from left to the right. They are “hot” electrons on the right, and some dissipation occurs and electrons end up inside the right reservoir.
We compute the transmission process from left to the right.
Left reservoir
Lμ
Rμ
empty
Right reservoir
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Device Hamiltonian
The Hamiltonian determines the energy levels of the device. (How to fill these levels non-equilibrium statistics.)
What kind of H to use is an issue of accuracy (tight-binding, DFT, GW, …).
In the end, we want to compare our results with experimental data without adjusting theoretical parameters.
DFT offers a good trade-off between accuracy and speed.
H = Hleads + Hdevice + Hcoupling
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Density functional theory : Kohn-Sham Hamiltonian
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Hamiltonian
Potential of ions
Potential of electrons
(Poisson equation)
Quantum/ many-body effects
Assumption : All electrons are independent
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
DFT approximately solves how atoms interact :
DFT for materials: put atoms in a simulation box, compute interactions between electrons and nucleus.
But, DFT solves only 2 kinds of problems: finite or periodic systems.
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
A device is neither finite nor periodic
For a device:
• There is no periodicity.
• There are infinite number of atoms because the device is hooked up to external leads…
These difficulties must be overcome in first principles modeling of transport.
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Essentially, must solve two problems:
Effective scattering regionLeft lead Right lead
How to reduce the infinitely large system to something calculable on a computer?
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Screening approximation --- reducing the infinitely large problem:
Within DFT, once the potential is matched at the boundary, charge density automatically goes to the bulk-electrode values at the boundaries:
Within screen approx., we only have to worry about a finite scattering region.
Charge density
Left lead Right leadScattering region
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Another example
Using the screening approximation and solving Poisson Equation in real space, we can deal with systems with different leads.
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Keldysh non-equilibrium Green’s function (NEGF):
Book of Jauho; book of Datta; Wang, Wang, Guo PRL 82, 398(1999)
NEGF:
Correct non-equilibrium physics, correct transport boundary conditions, easiness of adding new physics (e-p).
Effective scattering region
Left lead
Right lead
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Transmission
][),( Ra
Lr GGTrVET ΓΓ=Δ
(This is one of several ways of getting T)
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
NEGF-DFT: Taylor, Guo and Wang, PRB 63, 245407 (2001).
Use density functional theory (DFT) to compute the electronic structure and all other materials properties of the open device structure;
Use Keldysh non-equilibrium Green’s function (NEGF) to populate the electronic states (non-equilibrium quantum statistics);
Use numerical techniques to deal with the open boundary conditions.
Molecular transport junctionsSolid state devices
May 4, 2011 Roberto Car’s group, Chemistry Department, Princeton
Wide range of research has been carried out by NEGF-DFT
• Leakage current in MOSFET;• Transport in semiconductor devices, photocells;• Transport in carbon nanostructures;• Resistivity of Cu interconnects; • Conductance, I-V curves of molecular transport junctions;• Computation of capacitance, diodes, inductance, current density;• TMR, spin currents, and spin injection in magnetic tunnel junctions;• Transport in nanowires, rods, films, clusters, nanotubes;• Resistance of surface, interface, grain boundaries;• STM image simulations;• Strongly correlated electrons in transport;• Transport through short peptides;• ….
it is a progressing field and not all is perfect yet.