atk 方法的扩展及其应用王雪峰苏州大学物理系 2009.11.28 zjnu-jinhua

ATK 方法的扩展及其应用

王雪峰

苏州大学物理系

2009.11.28 ZJNU-JinHua

Introduction

Inelastic scattering

Gate effect: electrostatic potential profile

Optimization of tridiagonal matrix inverter

Thermoelectric effect

Improvement of functionals

Summary

Outline


SIESTA TranSIESTAKS Hamiltonian Matrix

Nonequilibrium Green’s Function

TranSIESTA-C

ATK+VNL

Other transport packages:Smeagol, OpenMX,WanT,PWSCF


Jose M. Soler, et al., J. Phys.: CM 14, 2745 (2002); J. Taylor, H. Guo and J. Wang, Phys. Rev. B 63, 245407 (2001); M. Brandbyge, et al., Phys. Rev. B 65, 165401 (2002);www.openmx-square.orgwww.smeagol.tcd.iewww.icmab.es/siesta/www.quantumwise.comwww.wannier-transport.orgwww.pwscf.org

http://www.openmx-square.org/

http://www.smeagol.tcd.ie/

http://www.icmab.es/siesta/

http://www.quantumwise.com/

http://www.wannier-transport.org/

http://www.pwscf.org/

The systemT1, T2,

vibration

http://nanohub.org/


Band-diagram of the system in equilibrium

Effect of gate voltage

Not just a shift in reality!

Effect of source-drain voltage

Escape time, contact coupling

Current

N

E ( ) EI dED E I

Occupation number

Broadening


2 2

/ 2for ( )

( ) ( / 2)D E

E

N is then obtained

More general expression1 2

1 2 1 2( ) [ ( ) ( )]qI dED E f E f E

Potential profilecapacitance

Real system: numbers to matrices

Ref. S. Datta, “Quantum Transport: Atom to Transistor”, Cambridge University Press (2005).


1† † †

, 1 1 11 1

[ ] [ ( )]N N

n n n n n n n nnn n

c c t c c c c

H

Tight-binding model

1,2

2,1 2,3

3,2 3,4

4

1

2

3

,3 4

0

0

0

0

0 0

t

t t

t t

t

H

Inflow and outflow

11 12

1 1

Trace[ ]

Tr[ ]

q in n

q in p out nh

I dE A G

dE G G

Coherent transport for one level model

2 2

2 2

1 2

1

/2( ) /4

1

†1 2

( ) /4

1 2 1 1 2 2

†1 1 2 2

assuming imaginary constant

electrode self energy

/ 2

[ ]

( ) ( / 2)

( )

2 Im( )

i i

i i i

E iE

i

E

in in in

n in

i

i

G E E i

A G G

A A A i G G

G

f f

G A f A f G G

2 2

1 1/2 /2

( ) /4/

in

inE i E i

in

EA

1 1 2 2

1 2

1 2 1 1 2 2( ) ( )

n in f fGA

nG f f A

1 1 2 2

1 2

2 1 2 2 1 2

1 2 1 2

11 1 12

1 1 1 1 1

1 1 1 1

1 1

1 1 2

[ ]

/

( / ) ( / )

( )

( )

q in n

in n in

in n

f f

f f

I dE Tr A G

A G f A A

A f A f G A

A f

A A f f

1 1 1 1 1 2 1 1 1 2 2

1 2 1 2 1 2 1 2

21 1 2 1 2

( ) ( )

( ) G G ( )

[ G G ]( )

in n

eh

A G f A A A f A f

A f f f f

I dETr f f

Electronic structures of two electrodes and equivalent bulk system: self-consistent Kohn-Sham potentials and Hamiltonian matrices

Kohn-Sham Hamiltonian:

Poisson equation:

Hamiltonian Matrix H:

The Green’s function of open system, G:

Formalism

2

ps H xc2ˆ ˆ ˆ ˆ( ) [ ( )] [ ( )]H V r V V r r

2H ( ) 4 ( )V r r

* ˆ( ') ( ) 'ij i jH d d H r r r r

1

†

†

0

( )

0

L L LC

LC C RC

RC R R

V

E V V

V

H

G H

H0( ) ( )( )C C C Cz g E z

S V S V0 1( ) ( ) ; ; {L,R}Cg E z z E i S H

M. Brandbyge, J.-L. Mozos, P. Ordejon, J. Taylor, and K. Stokbro, PRB 65, 165401 (2002).K. Stokbro, J. Taylor, M. Brandbyge, and P. ordejon, Ann. NY. Acad. Sci. 1006, 212 (2003).


Density matrix:

In equilibrium:

In nonequilibrium:

The electron density:

The current through the contact:

1Im ( ) ( )E i f E dE

D G

†

1Re

1Re ( ) ( )

n

L L R R

dE

dE f E f E

D G

A A

A GΓ G

( ) Tr ( ) ( )i ij jijD r Dρ r r

†

L R

2( ) ( ) ( ) ( )

( ) Tr ( ) ( ) ( ) ( )

L R

L R

eI V dE f E f E T E

h

T E E E E E

V

Γ G Γ G

Calculate Kohn-Sham Hamiltonian

Calculate , , and derive ,L R L RG G

Guess for the system

1Calculate Green's function ( ) [ ]L RG E E H

Initially define the system geometry

Bulk calculations of Veff for the left and right electrodes

Calculate from G

' ?

Calculate Current

Mixing density:

= '+(1- )

Self-consistent Loop


Phonon effect in the lowest order

D

G

Inelastic scattering

p.287

p.336

Elastic scattering for one level model

2 2

2 2

2 2

1 2 0

1

/2( ) /4

†

( ) /4

1 1 2 2 0

† 1 1/2 /2

( ) /4

( ) ( / 2)

( ) 2 Im( )

/in

E iE

E

in n

n in inE i E i

in

E

D A

G E E i

A i G G G

f f D G

G G G

A

1 1 2 2 0

1 2 0

1 1 2 2

1 2

0 1 2

1 1 2 2 0

1 2 1 1 2 2

( )

( )

( ) ( )

nn in

n

f f D GGA D A

n n

n

n

f fGA

D AG G

f f A D G A

G f f A

1 1 2 2

1 2

2 1 2 2 1 2

1 2 1 2

12

0 0

11 1 12

1 1 1 1 1

1 1 1 1

1 1

1 1 2

[ ]

0

0

[ ]

/

( / ) ( / )

( )

( )

q in ns s s

in n n ns s

s

q in n

in n in

in n

f f

f f

I dE Tr A G

A G D AG D G A

I

I dE Tr A G

A G f A A

A f A f G A

A f

A A f f

Inelastic scattering for one level model2 2

2 2

2 2

1 / 2( ) / 4

( ) / 4

1 2

1 1/ 2 / 2 ( ) / 4

1 1 2 2

( ) ( / 2)

( ) 2 Im( )

[ ( ) ( )]

/

( )

in

E iE

E

ph n p

n in in inE i E i E

p n

in ph n

G E E i

A i G G G

D G E G E

G G G A

G A G

f f D G E

1 1 2 2

1 2

( )

[ ( ) ( )]

( ) [( 1) ( ) ( )]

ph nn in

ph n p

f f D G EGA D G E G E

phi i i i i

i

D d D N N

12

11 1 12

[ ]

[ ]

q in ns s s

q in n

I dE Tr A G

I dE Tr A G

D

G

A7-atom gold wire with L=29.20 Å is coupled to semi-infinite electrodes. The vibrational region is taken to include the atoms in the pyramidal bases. The device region (describing the e-ph couplings) includes also the outermost surface layers.

T. Frederiksen, M. Paulsson, M. Brandbyge, A. P. Jauho, Phys. Rev. B 75, 205413 (2007)2009.11.28 ZJNU-JinHua

The measured (noisy black curves) are for different strain. The calculated (smooth colored lines) are for different damping.

T=4.2K

Gate effect: Si MOSFET devices

Equivalent capacitive circuit

Vg

MOSFETs are the most important building blocks Si nanostructures are still the fundamental units: Si cluster, nanowire, nanoslab, and so on


Top-down technologiesin traditional semiconductor industry --

Microelectronicsm 100nm 10nm nm 0.1nm

45 32 0.54Si latticeCPU wireITRS

Intl. Tech. Roadmap Semi.

Bottom-up technologiesMolecular electronics, Spintronics, Quantum computation

m0.1nm nm 10 nm 100nm

DNA

Si latticeH2O

Quantum dot

nanotube, nanowire


S D

(a) (b)

Atomistic model systems

Geometrically optimized Si-H bond length and Si-SiO2 interfaces


ATK Two-Probe method Multigrid Poisson solver Norm-conserving pseudopotential of Troullier-

Martins scheme LDA with Perdew-Zunger parameterization Standard SIESTA SZP basis set Mesh cutoff 4348 eV or 0.092 Å

Calculation Method


L. N. Zhao, et al., J. Comp. Electronics 7, 500 (2008); X. F. Wang, et al., Int. J. Nanoscience 8, 113 (2009).

Si-slab based MOSFET capacitor

Result:


Charge and electrostatic potential distribution

X

V

X


Total induced charge and surface potential versus gate voltage

Q Vs

Vs

Vg Vg Vg

Q


Transmission spectrum under gate voltageT

T


Thermoelectric effects

thermoelectric devices

Thermal conductance


1 1 1 2 2 2

1 1 1 2 2 2

( ) ( )[ ( , , ) ( , , )]

( ) ( )[ ( , , ) ( , , )]

qh

qQ h

I dED E T E f E T f E T

I dED E ET E f E T f E T

Charge current and heat current

2 1

2 1

0,I

VS

T T T

Q

I G V SG T

I TSG V K T

Here we do not include the phonon effect


silicon nanowiresA. I. Hochbaum et al., Nature 451, 163 (2008).

silicon nanowires array

Akram I. Boukai, ibid. 451, 168 (2008).

ZT=S2T/

ATK method is under development Optimize algorithm: faster, better

accuracy, larger system Inelastic scattering Multi-terminal systems: transistor Temperature bias: thermoelectric

effect Better density functional

Summary


atk 方法的扩展及其应用 王雪峰 苏州大学物理系 2009.11.28 zjnu-jinhua

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atk 方法的扩展及其应用王雪峰苏州大学物理系 2009.11.28 zjnu-jinhua