Jesse Maassen (Supervisor : Prof. Hong Guo)

Jesse Maassen

(Supervisor : Prof. Hong Guo)Department of Physics, McGill University, Montreal, QC Canada

Semi-classical device

modeling

device parametersAtomic, materials,

chemistry modeling

quantum physics device modeling < 50nm (1000 atoms)

crash

science engineering

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.


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


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


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


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;


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


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.


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.


Basic ingredients of a theory:

Picture from: Nitzan & Ratner, Science, 300, 1384 (2003).

1. A transport model

2. Device Hamiltonian

3. Non-equilibrium Physics

4. Transmission

5. Fermi level alignment

6. Calculable!


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).


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


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


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


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.


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.


Essentially, must solve two problems:

Effective scattering regionLeft lead Right lead

How to reduce the infinitely large system to something calculable on a computer?


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


Another example

Using the screening approximation and solving Poisson Equation in real space, we can deal with systems with different leads.


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


Transmission

][),( Ra

Lr GGTrVET ΓΓ=Δ

(This is one of several ways of getting T)


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


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.

Recently developed modeling tool allows for:

Large-scale systems (~1000 atoms & ~10 nm).

Disorder: Surface/interface roughness, dopants, impurities.


An example:

• Graphene-metal interface



• Experimental studies:

Nature Nanotechnology 3, 486 (2008) Phys. Rev. B 79, 245430 (2009)

Photocurrentexperiments

Motivation (graphene-metal interface)


Parameter-free transport (NEGF-DFT*) calculation of a

graphene / metal interface

* Jeremy Taylor, Hong Guo and Jian Wang, PRB 63, 245407 (2001).

Our goal


Which metals? What configuration at the interface?

• Cu, Ni and Co (111) have in-place lattice constants that almost match that of graphene.

• Previous study found most stable configuration (PRL 101, 26803 (2008)).

Metal

Atomic structure

Graphene-Cu interface


Appl. Phys. Lett. 97, 142105 (2010)

Bandstructure of hybrid

graphene | Cu(111) system

Graphene states in black Weak hybridization n-type graphene

Metal


Transport properties: graphene-Cu(111) system

Appl. Phys. Lett. 97, 142105 (2010)


Double minimum T.

T almost perfectly described by pure graphene at TMIN.


Transport properties: graphene-Cu(111) system

• One Dirac point pinned, while other moves with V.

QuickTime™ and aTIFF (Uncompressed) decompressor


• Peak in conductance doping level of graphene

Appl. Phys. Lett. 97, 142105 (2010)


Graphene-Ni interface

Band structure : graphene-Ni(111) system

• Strong hybridization with metal

• Band gap opening

• Graphene is spin-polarized

: A-site C(pz): B-site C(pz)

: Ni(dZ2)

Nano. Lett. 11, 151 (2011)



Nano. Lett. 11, 151 (2011)


Transport properties :

graphene-Ni(111) system


Nano. Lett. 11, 151 (2011)





Nano. Lett. 11, 151 (2011)




Graphene-metal interface


Cu merely n-dopes the graphene resulting in:

• Peak in dI/dV provides doping level

• Can be simply modeled assuming a n-i junction

• Similar trends for Al, Ag, Au & Pt

Ni & Co create spin-dependent (pseudo-) band gaps in

graphene.

• Large spin injection efficiencies ~80%

SUMMARY

1. Quantum physics Eigler (IBM)

9nm

2. Materials physics Williams (HP)

3. Nonequilibrium statistical physics (picture from Ratner)

To make quantitative predictions without phenomenological parameters, a formalism has been developed that includes these ingredients.

Take home message:


Thank you !(and to my supervisor and colleagues)

$: NSERC, FQRNT, CIFAR, DRDC; Computers :

[email protected]

SRC, LuXin Energy. RQCHP,CLUMEQ


Another example:

• Graphene-metal interface

• Ultrathin Si films



The main motivation for our research was the experimental work by Pengpeng Zhang et al. with silicon-on-insulators.

QuickTime™ and aTIFF (LZW) decompressor




Nature 439, 703 (2006)

SiO2



SiSiO2Vacuum

Charge traps

Used STM to image 10 nm Si film on SiO2

Surfacestates

Motivation (Si nano-film)


are needed to see this picture.CurrentElectrode Electrode

Our goal




Length

Th

ickn

ess

Surface

CurrentElectrode Electrode

Doping level(lead or channel)

Orientation

Our goal


Hydrogenated surface vs. clean surface









H

Si (top)

Si

Si (top:1)

Si (top:2)

Si

H terminated [21:H] Clean [P(22)]


Atomic structure (surface)


Atomic structure & bandstructure

H terminated [21:H] Clean [P(22)]

|| dimers dimers || dimers dimers

•Large gap ~0.7 eV •Small gap ~0.1 eV





d

imer

s

|| dimersQuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.



|| dimers

d

imer

s

Electronic structure (surface)



n++ n++iUltrathin Si films




are needed to see this picture.H

Si (top)

Si

Two-probe system

Channel : intrinsic Si

Leads : n++ doped Si

21:H surface

Periodic to transport


are needed to see this picture.n++ n++i

L = 19.2 nm


are needed to see this picture.n++ n++i

L = 3.8 nm

T = 1.7 nm




n++ n++i

Conductance vs. k-points ( dimers)

Shows contribution from k-points to transport

Transport occurs near Γ point.

Conductance drops very rapidly



in+ n+



TOP VIEW

Ultrathin Si films




n++ n++i

Conductance vs. k-points ( || dimers)

Largest G near Γ point

Conductance drops rapidly, but slower than for transport to dimers.



in++ n++

TOP VIEW



Ultrathin Si films




n++ n++i



Conductance vs. Length

Conductance has exponential dependence on length, i.e. transport = tunneling.

Large difference due to orientation.

Better transport in the direction of the dimer rows.



Ultrathin Si films




n++ n++iUltrathin Si films


Electronic structure

• 21:H [~0.7 eV gap]

• p(22) [~0.1 eV gap]

Transport properties

• Large effect of orientation in G for 21:H

More complete study to come soon!

SUMMARY

Some (more) examples:

Effect of dephasing on electron transport

Scattering properties of a nano-electromechanical system [Published in Phys. Rev. Lett. 105, 217206 (2010)]

Raman spectra of graphene on Cu substrate [Submitted to Phys. Rev. Lett.]

Effect of disorder and vacancies on electronic transport through a Au conductor [in progress]

Electron transport through a Si/Ge interface [in progress]


• Phase breaking : Phenomenological model (Buttiker probe model)

• Self-energy : Simple implementation

J. Maassen, F. Zahid and H. Guo PRB 80, 125423 (2009)

I(μX) = 0


Al - BDT - Al

Left reservoir

Right reservoir

Lμ

Rμ



Narrow Al nanowire

• Ballistic transport independent of length

• Ohm’s law behavior

V = 0.05 V


Documents

Jesse Maassen (Supervisor : Prof. Hong Guo)