SemOI transistors: from classical to quantum computing

SemOI transistors: from classical to quantum computing

A. Orlikovsky¹, S. Filippov¹², V. Vyurkov¹², and I. Semenikhin¹

¹Institute of Physics and Technology Russian Academy of Sciences Moscow, Russia

1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine

²Moscow Institute of Physics and Technology

Outline

Introduction: a brief review of the history of transistors

Simulation of fully depleted (FD) extremely thin (ET) SOI FET

Towards SemOI-based quantum computers


The end of Moore’s ‘law’?


Micrometer channel length

Nanometer channel length

Semiconductors

Metals

Where does nanoelectronics start from?


Evolution of models

Charged fluid:Hydrodynamic equations

Charged particles:Boltzmann kinetic equation

Charged waves:Schrödinger equation


ET FD SOI FET


IBM Gains Confidence in 22 nm ETSOI(IEDM Conf., Dec. 2009)


Fermi-Dirac statistics.

Transversal quantization in channel: Quantum longitudinal motion: a) interference on random impurities; b) quantum reflection; c) source-drain tunneling.

Quantum effects in nanotransistors


Silicon conduction band structure

Effective mass and transversal quantization energy

0 00.19 , 0.98t lm m m m

22

0 2

Sidm


Quantum descriptionCharged waves:Schrödinger equation

Transversal quantization Wave-guide modes in the channel Landauer-Buttiker formalism

)()()(2)(0

EfEfEdETheVI dsii

sd


High self-consistent barrier at S/D contacts

Few of incident particles surmounting the barrier is followed by equilibrium distribution for particles coming in the channel

-10 -5 0 5 10-1.0x100

-8.0x10-1

-6.0x10-1

-4.0x10-1

-2.0x10-1

0.0

2.0x10-1

Pot

entia

l ene

rgy,

[eV

]

x, [nm]

Potential energy for different drain voltage Fermi level in source contact


2

( , , ) ( , , ) ( , , ) ( , , )2

x y z V x y z x y z x y zm

V(x,y,z) is a potential.

The direct solution of the stationary 3D Schrödinger equation via a finite difference scheme comes across a well known instability caused by evanescent modes.

In fact, the exponential growth of upper modes makes a computation impossible.

Solution of 3D Schrödinger equation


D.K.Ferry et al. (2005) (USA, Arizona State University):results of simulation


D.K.Ferry et al. (2005) (USA, Arizona State University):results of simulation


Solution of Schrödinger equation: transverse mode representation + some mathematical means

1

( , , ) ( ) ( , )N

i ii

x y z a x y z

where ψi(y,z) is the i-th transverse mode wave function, N is a number of involved modes.

The space evolution of coefficients ai(x) is governed by matrix elements

( ) ( , ) | ( , , ) | ( , )ij i jM x y z V x y z y z


Calculated transmission coefficient vs. electron energy E

Transistor parameters are 10nm channel length and width, 5nm body thickness, 10^20 cm^-3 source/drain contact doping, 5nm spacers.

[100] and [010] valleys (small mass along the channel) [001] valleys

(big mass along the channel)

(4 random impurities in a channel)

0

0,2

0,4

0,6

0,8

1

1,2

0 0,1 0,2 0,3 0,4 0,5

Energy, eV

Tran

smis

sion

0

0,2

0,4

0,6

0,8

1

1,2

0 0,1 0,2 0,3 0,4 0,5

Energy, eV

Tran

smis

sion


Gate voltage characteristics

-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

1E-8

1E-7

1E-6

1E-5

Vd=0.9

Vd=0.05

Dra

in C

urre

nt, [

A]

Gate Voltage, [V]

Sub-threshold swing is 71 mV per decade of current.


Corrugated channel:

0.0 0.2 0.4 0.6 0.8 1.00.0

4.0x10-6

8.0x10-6

1.2x10-5

Dra

in C

urre

nt, [

A]

Drain Voltage, [V]

flat channel corrugated channel ( 0.5 nm step)

channel thickness 3 nm


Corrugated channel:

0.0 0.2 0.4 0.6 0.8 1.00.0

4.0x10-6

8.0x10-6

1.2x10-5

Dra

in C

urre

nt, [

A]

Drain Voltage, [V]

without narrow in channel Narrow Left 0.5 nm Narrow Right 0.5 nm



Impurities in channel:

0.0 0.2 0.4 0.6 0.8 1.00.0

4.0x10-6

8.0x10-6

1.2x10-5

Dra

in C

urre

nt, [

A]

Drain Voltage, [V]

without impurities in channel 1 positive impurity in channel 1 negative impurity in channel



Impurities in channel:

0.0 0.2 0.4 0.6 0.8 1.00.0

4.0x10-6

8.0x10-6

1.2x10-5

Dra

in C

urre

nt, [

A]

Drain Voltage, [V]

without impurities in channel 1 impurity near drain 1 impurity near source



Dispersion of characteristics

5-15% in calculated I-V curves

< 10% is an everlasting condition for large integrated circuits

More severe demands to technology may arise.


Semi-analytical models of FETs with low-dimensional channels A. Khomyakov (IPT RAS)

Poster P8 at 19-00!(Conference Abstracts, page 109)


Quantum Computers


Quantum bits


26

Charge qubits in double quantum dots (DQDs)


27

Solid state implementation


Gorman et al, PRL, 2005

28

Two phosphorus atoms in silicon



Hollenberg et al, PRB, 2004

29


Gate-engineered quantum dots


Hayashi et al, PRL, 2003

Qubits based on space states

Advantages:

quite simple read-out (measurement of final state)

explicit initialization scaling and integrity

with modern microelectronic technology

Disadvantages: strong decoherence

caused by uncontrollable Coulomb interaction between even far-distant qubit

decoherence caused by interaction with gates and phonons


Unavoidable obstacle

strong decoherence caused by uncontrollable Coulomb interaction between even far-distant qubit

independent of temperature quantum calculations seem impossible


Coulomb interaction Long range Coulomb interaction

d

D

eˉeˉ

eˉ

2 2 21

32 2

1 1~ ~2phase

e e dD DD d

For , , one obtains100D nm 10d nm 10 10~ 10 s


Qubit and its operationConsists of two double quantum dots

Electrode Е operates upon the strength of exchange interaction between electrons.

Electrode Т operates upon tunnel coupling between dots.

E

T1eˉ

1eˉ+—

—+


Vyurkov et al, PLA, 2010

Basic states in a DQD

Potential in a DQDSymmetric Antisymmetric

Electron wave-function in a DQD


Basic states of two DQDs

Potential in two DQDs Wave-function of two electrons in two DQDs

basis*


1 2 2 1102

1 2 2 1112

Basic states of a qubitSpin-polarized electrons:


1 2 2 1102

Qubit states

1r

2r

1r

2r


Qubit states

1r

2r

1r

2r

1 2 2 1112


Distribution of charge in a qubit Probability density

For region Ω:

Charge in a dot Ω:

1 (1) (2) (2) (1)2

1, 2 1 2 1 2( , ) 0P dr dr r r

31 2 1 21, 2

\

1( , )4R

P dr dr r r

31 2 1 21, 2

\

1( , )4R

P dr dr r r

1, 2 1, 2 1, 2122

q P e P e P e e


Distribution of charge in a qubit


Arbitrary qubit states Arbitrary qubit state

Hamiltonian in matrix representation

Evolution operator

2 2

0 1a b

a b

0 1 1 0ˆ1 0 0 1

H A P

0

ˆ ( )ˆ ˆ( )

ti H d

U t Te


Initialization Cooling in magnetic

field, positive potential on gate Т

Transformation

Pumping of electrons along the chain


Initialization

Pumping electrons from

a spin-polarized source,

for instance, ferromagnetic

Single-electron turnstile


Decoherence

Particular symmetry makes the qubit insensitive to voltage fluctuations

5~ ( ) 3~ ( )

x

y

Small energy gap between basic states in a DQD secures against the decoherence on phonons :deformation acoustic phononspiezoelectric acoustic phonons

–VT


Decoherence ‘Frozen’ qubit: only two-phonon processes are possible, Decoherence rate is independent of energy gap

2 2' '

'

2'

2

'

( )

2 | | | | ( ) 1 ( )

( )

q q q qq q

iq r iqr

z

z q

f q q

W f

F F n n

f e z z e


Read-out To read-out one must distinguish

from

An additional electrode by the DQD makes it possible tunneling of an electron into first or second dot depending on the initial state of DQD: or


Realistic structure


SemOI quantum register


Potential defined quantum dotsConfinement energy

20

1 1~l Si tm d m D

0 00.19 , 0.98t lm m m m ~ 2 , ~ 10Sid nm D nm

0 ~ 0.02eVCoulomb repulsion energy

~ 0.01C eV

=> one electron in a dot


How a read-out is possible?

Transistor currentdepends on position ofan electron in thechannel

0.0 0.2 0.4 0.6 0.8 1.00.0

4.0x10-6

8.0x10-6

1.2x10-5

Dra

in C

urre

nt, [

A]

Drain Voltage, [V]

without impurities in channel 1 impurity near drain 1 impurity near source



Compare with Tanamoto et al, PRA, 2000

Summary The efficient program for 3D all quantum

simulation of field effect nanotransistors is elaborated.

The results of simulation demonstrate the impact of realistic channel inhomogeneities on transistor characteristics.

SOI structure for quantum computation is proposed.


Эпилог

С

Light at the end of the tunnel

AcknowledgementsRussian Foundation for Basic Reasearch, grant # 08-07-00486-а

NIX Computer Company ([email protected]), grant # F793/8-05

grant # 14.740.11.0497



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