ANDREI MANOLESCU, ASSOCIATE PROFESSORSCOOL OF SCIENCE AND ENGINEERING | RU LECTURE MARATHON

COULOMB EFFECTS IN ULTRA-SMALL ELECTRONIC DEVICES

Nanoelectronic devices

GaAs/AlGaAs heterostructure

100 – 500 nm

Quantum dots Quantum wires

Quantum mechanics ( )

( ) ( )2

Wave functions: ,

Schrödinger equation: , ( ) ,

Hamiltonian (energy) operator: ( ) ( , )2

r t

i r t H t r tt

H t U r tm

Ψ

∂Ψ = Ψ

∂

= − Δ +

r

r rh

h r

Example: The quantum well

Open quantum dots in contact with leads

Unknown wave functions of the electrons in the sampleStatistical description: wave functions with probabilities Statistical operator | |

Expected values of physical observables:

n n

n n nn

n

pp

A

ψ

ρ ψ ψ= ><

< > =

∑

( ) ( )

n n n n nn

nnn

A A p A

A Tr A A

ψ ψ ψ ψ

ρ ρ

< > → < >= < >

⇒ < >= =

∑

∑

The leads-sample contacts Left lead Right lead

Sample

Overalp of wave functions

* ( ) ( , )́ ( )́ Generic forml l

qnl ql nT dr dr r g r r rψ ϕΩ Ω

′ ′ ′ →∫ ∫r r r r r r

The external field (bias)

Left lead Right leadSample

μL,R chemical potentials

Coulomb interaction and many-body effects

212

2* *

,

electrons in the sample

( ) ( ) ( ) ( )

Sample nn n m n m

nm pq n m p qn m

eN H Tr r

eV drdr r r r rr r

≠

⇒ = +−

′ ′ ′=−

∑ ∑

∫

r r

r r r r r rr rψ ψ ψ ψ

The equation for the statistical operator (generalized master equation)

[ ] { }

[ ]{ }

2

/ / ( ) /

/

0

/ ( ) /† †

1( ) , ( ) , ( ) . .

( ) ( )

( ) ( )

( ) 1 ( ( ) ( ) ( ( )

S S l

S

S l

S S S l ql qll

itH itH it qql ql

titH

ql l

isH is qql S l S ql l

it H dq t T t h c

t e t e e

t ds s e

T s f q s T f q e e

ε

ε

ρ ρ χ

χ

ρ ε ρ ε

− −

−

⎡ ⎤= − − Ω +⎣ ⎦

Ω = Π

Π = ×

× − −

∑ ∫

∫

h h h

h

h h

&h h

Charging of the sample

Currents

Left lead Right leadSample

Double system

Induced currents

Time (a.u.) Time (a.u.)

Electric pulses

The team

Viðar Guðmundsson,Professor HÍ

Valeriu Moldoveanu,Scientist, Romania

Andrei, speaker☺

Cosmin Gainar,PhD student HÍ

Kristinn Torfason,PhD student HR