Download 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
²Moscow Institute of Physics and Technology
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
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

1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
The end of Moore’s ‘law’?
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Where does nanoelectronics start
from?
Micrometer channel length
Nanometer
channel length
Semiconductors
Metals
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Evolution of models
Charged waves:
Schrödinger equation
Charged particles:
Boltzmann kinetic equation
Charged fluid:
Hydrodynamic equations
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
ET FD SOI FET
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
IBM Gains Confidence in 22 nm ETSOI
(IEDM Conf., Dec. 2009)
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Quantum effects in nanotransistors

Fermi-Dirac statistics.

Transversal quantization in channel:

Quantum longitudinal motion:
a) interference on random impurities;
b) quantum reflection;
c) source-drain tunneling.
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Silicon conduction band structure

Effective mass and
transversal quantization
energy
mt  0.19m0 , ml  0.98m0
  
0 


2m  d Si 
2
2
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Quantum description
Charged waves:
Schrödinger equation
Transversal quantization
 Wave-guide modes in the channel
 Landauer-Buttiker formalism

2e 
I (Vsd )  
h i 0
 dET ( E ) f ( E )  f
i
s
d
( E )
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
High self-consistent barrier at S/D
contacts
-1
2.0x10
Potential energy, [eV]
0.0
-1
-2.0x10
-1
-4.0x10
-1
-6.0x10
-1
-8.0x10
Potential energy for different drain voltage
Fermi level in source contact
0
-1.0x10
-10
-5
0
5
10
x, [nm]

Few of incident particles surmounting the barrier is
followed by equilibrium distribution for particles coming in
the channel
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Solution of 3D Schrödinger
equation


2
2m
( x, y, z )  V ( x, y, z )( x, y, z )  ( x, y, z )
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.
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
D.K.Ferry et al. (2005)
(USA, Arizona State University):
results of simulation
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
D.K.Ferry et al. (2005)
(USA, Arizona State University):
results of simulation
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Solution of Schrödinger equation:
transverse mode representation + some mathematical
means

 ( x, y , z ) 
N
a
i 1
i
( x ) i ( 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
M ij ( x )   i ( y , z ) | V ( x, y, z ) | j ( y, z ) 
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Calculated transmission coefficient
vs. electron energy E
1,2
1,2
1
1
Transmission
Transmission
(4 random impurities in a channel)
0,8
0,6
0,4
0,2
0
0
0,1
0,2
0,3
0,4
Energy, eV
0,5
0,8
0,6
0,4
0,2
0
0
0,1
0,2
0,3
0,4
0,5
Energy, eV
[100] and [010] valleys
(small mass along the channel)
[001] valleys
(big mass along the channel)
Transistor parameters are 10nm channel length and width, 5nm body thickness,
10^20 cm^-3 source/drain contact doping, 5nm spacers.
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Gate voltage characteristics
Drain Current, [A]
1E-5
Vd=0.9
Vd=0.05
1E-6
1E-7
1E-8
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Gate Voltage, [V]
Sub-threshold swing is 71 mV per decade of current.
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Corrugated channel:
-5
Drain Current, [A]
1.2x10
-6
8.0x10
flat channel
corrugated channel (0.5 nm step)
channel thickness 3 nm
-6
4.0x10
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Drain Voltage, [V]
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Corrugated channel:
-5
Drain Current, [A]
1.2x10
-6
8.0x10
without narrow in channel
Narrow Left 0.5 nm
Narrow Right 0.5 nm
channel thickness 3 nm
-6
4.0x10
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Drain Voltage, [V]
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Impurities in channel:
-5
Drain Current, [A]
1.2x10
-6
8.0x10
without impurities in channel
1 positive impurity in channel
1 negative impurity in channel
channel thickness 3 nm
-6
4.0x10
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Drain Voltage, [V]
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Impurities in channel:
-5
Drain Current, [A]
1.2x10
-6
8.0x10
without impurities in channel
1 impurity near drain
1 impurity near source
channel thickness 3 nm
-6
4.0x10
0.0
0.0
1st
Ukrainian-French Seminar and
0.2
6th
0.4
0.6
Drain Voltage, [V]
0.8
1.0
International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
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.
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Semi-analytical models of FETs
with low-dimensional channels

A. Khomyakov (IPT RAS)
Poster P8 at 19-00!
(Conference Abstracts, page 109)
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Quantum Computers
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Quantum bits
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Charge qubits in double quantum dots
(DQDs)
26
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Solid state implementation
Gorman et al, PRL, 2005
27
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Solid state implementation

Two phosphorus atoms in silicon
Hollenberg et al, PRB, 2004
28
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Solid state implementation

Gate-engineered quantum dots
Hayashi et al, PRL, 2003
29
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
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 fardistant qubit
decoherence caused
by interaction with
gates and phonons
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Unavoidable obstacle
strong decoherence caused by
uncontrollable Coulomb interaction
between even far-distant qubit
 independent of temperature
 quantum calculations seem impossible
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Coulomb interaction

Long range Coulomb interaction
eˉ
eˉ
d
eˉ
D
1
 phase
 e2 d 2
e2  1
1
~
~
 

3
2
2
 D

2
D
D d 
For D  100nm , d  10nm ,   10 one obtains  ~ 1010 s
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Qubit and its operation
Consists of two double
quantum dots
—
+
1eˉ
Electrode Е operates
upon the strength of
exchange interaction
—
T+
between electrons.
1eˉ
E
Vyurkov et al, PLA, 2010
Electrode Т operates
upon tunnel coupling
between dots.
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Basic states in a DQD
Electron wave-function in a DQD


Symmetric
Antisymmetric
Potential in a DQD
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Basic states of two DQDs



basis*

Potential in two DQDs
Wave-function of two
electrons in two DQDs
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Basic states of a qubit
Spin-polarized electrons:
1
0 
 1 2  2 1
2

1
1 
 1 2  2 1
2

1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Qubit states
1
0 
 1 2  2 1
2
r1

r1

r2
r2
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Qubit states
1
1 
 1 2  2 1
2
r1

r1

r2
r2
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Distribution of charge in a qubit



Probability density
1
    (1)   (2)   (2)   (1) 
2

For region Ω:
P1,2   dr1  dr2  (r1 ,r2 )  0

P1, 2

P1,2


1
  dr1  dr2  (r1 ,r2 ) 
4

R3 \ 
1
  dr1  dr2  (r1 ,r2 ) 
4

R3 \ 
Charge in a dot Ω:
q  P1,2  2e  P1, 2  e  P1,2  e 
1
e
2
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Distribution of charge in a qubit
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Arbitrary qubit states

Arbitrary qubit state
 

a 0 b 1
a b
2
2
Hamiltonian in matrix representation
0 1
1 0 
ˆ
H  A
  P

1
0
0

1





Evolution operator
ˆ
Uˆ (t )  Te

i
t

Hˆ ( ) d
0
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Initialization
Cooling in magnetic
field, positive potential
on gate Т
 Transformation   


Pumping of electrons
along the chain
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Initialization

Pumping
electrons from
a spin-polarized
source,
for instance,
ferromagnetic
Single-electron turnstile
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Decoherence


y
Particular symmetry makes the
qubit insensitive to voltage
fluctuations
x
Small energy gap between basic states in a DQD
secures against the decoherence on phonons :
 ~ ( )5 deformation acoustic phonons
 ~ ( )3 piezoelectric acoustic phonons


–VT
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Decoherence


‘Frozen’ qubit: only two-phonon processes are
possible,
Decoherence rate is independent of energy gap
W(   f ) 

2
2
2
|
F
|
|
F
|
 q q '  n( q )  1 n( q ' )
q


z
q'
2
f eiq ' r z z e  iqr  

z
    q 
2
  ( f  q '     q )
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
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 
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Realistic structure
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SemOI quantum register
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Potential defined quantum dots
Confinement energy
 1
1 
0 ~ 2 


 ml d Si mt D 
mt  0.19m0 , ml  0.98m0
d Si ~ 2nm, D ~ 10nm
 0 ~ 0.02eV
Coulomb repulsion energy
 C ~ 0.01eV
=> one electron in a dot
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
How a read-out is possible?
-5
1.2x10
Drain Current, [A]
Transistor current
depends on position of
an electron in the
channel
-6
8.0x10
without impurities in channel
1 impurity near drain
1 impurity near source
channel thickness 3 nm
-6
4.0x10
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Drain Voltage, [V]
Compare with Tanamoto et al, PRA, 2000
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
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.

1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
Эпилог

С
Light at the end of the tunnel
Acknowledgements
Russian Foundation for Basic
Reasearch, grant # 08-07-00486-а
NIX Computer Company
([email protected]), grant # F793/8-05
grant # 14.740.11.0497
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine
1st Ukrainian-French Seminar and 6th International SemOI Workshop, October 25-29, 2010, Kyiv, Ukraine