Sourin Das
Dept. of Physics and Astrophysics, University of Delhi HRI, 10 Oct 2011 Plan of the talk •  Introduc;on to spin transistor •  Spin transistor as quantum bit – Quantum Dot •  Introduc;on to Quantum Dot •  Quantum dot as a spin qubit in the non-­‐transport regime: –  Theore;cal proposals –  Experimental realiza;on •  Quantum dot in the transport regime: Spin manipula;on via spin blockade •  Conclusion Transistors 1956 Nobel Prize in Physics John Bardeen William Shockley Walter BraQain ,-*.(()/01*2
Field Effect Transistors (FET) Current through device can be switched on or off by the gate, hence it can be used as a classical bit Spin Transistor Two Dimensional Electron Gas !"#$%&'()*&#)+,-.,(/01#)-2+*
asymmetric !
301+&)(4 GaInAs/InP heterostructure:
Ga0.47In0.53As
_\_
.
n+
2+5)7*
5)6
5)6
grown by MOVPE
z
2DEG
Ga0.23In0.77As
high mobility: 200000 - 450000 cm2/Vs
large indium content (77%)
low band gap
InP
Ec
Rashba E
ffect !"#$%&'(#)%*+,"-#$./#$/0/1%2#34$5#+$0!"#$%&'(#)%*+,"-#$./#$/0/1%2#34$5#+$0Asymmetric potential
electric field
Asymmetric potential
6-47)'+$/80545/9:05;(0/6<<47)=
6-47)'+$/80545/9:05;(0/6<<47)=
Rashba Hamiltonian:
electric field E
Rashba Hamiltonian:
: Rashba couplingparameter
Energy splitting:
Asymmetric potential
electric
field electric
Asymmetric
potential
field
Energy
splitting:
E
: Rashba
parame
E. I. Rashba E.
D can be controlled by applying
gate
voltage by applying a gate v
D canabe
controlled
Rashba Hamiltonian:
Rashba Hamiltonian:
E
!"#$%&'($)#)*+',%!"#$%-'./.))#+$
E
: Rashba coupling: Rashba couplingStarting condition:
parameter
parameter
§1·
¨¨ ¸¸
©1¹ ( x pol )
§ 1·
§ 0·
¨¨ ¸¸
¨¨ ¸¸
© 0 ¹ ( z pol ) © 1 ¹ ( z pol )
§ 1 · ik x 2 x § 0 · ik x1x Bad
§ 1 ·
2010
¸¸ e ¨¨ ¸¸ e ¨¨ i'k x Honnef,
¸
e ik x1x ¨¨ Nano-Spintronic
x¸
0
1
© ¹
© ¹
©e
¹
Institut für Bio- und Nanosysteme (IBN-1)
Nano-Spintronic Bad Honnef, 2010
Datta-Das Transistor
Gate Energy
Energy
splitting: FM FM splitting:
D can be controlled
a by
gate
voltagea gate voltage
D can by
be applying
controlled
applying
x
0
§1·
¨ ¸
S
S
2'k x
'k x
§1·
e iI ¨ ¸
i 2I
3S
2 'k x
§ 1 · i 3I § 1 ·
¨ ¸e ¨ ¸
EF
k
I
13. M. Wegener, S. Linden, Physics 2, 3 (2009).
14. E. Hecht, Optics (Addison-Wesley, San Francisco, ed. 4,
2002), chap. 8.
15. M. S. Rill et al., Nat. Mater. 7, 543 (2008).
16. N. Liu, H. Liu, S. Zhu, H. Giessen, Nat. Photon. 3, 157
(2009).
17. M. G. Silveirinha, IEEE Trans. Antenn. Propag. 56, 390 (2008).
18. Materials and methods are available as supporting
material on Science Online.
19. J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Steward,
IEEE Trans. Microw. Theory Tech. 47, 2075 (1999).
J.K.G., M.T., M.S.R., and M.D. is embedded in the
Karlsruhe School of Optics and Photonics (KSOP).
Supporting Online Material
www.sciencemag.org/cgi/content/full/1177031/DC1
Materials and Methods
Experimental Status Control
of Spin Precession in a
Spin-Injected Field Effect Transistor
M
Hyun Cheol Koo,1 Jae Hyun Kwon,1 Jonghwa Eom,1,2 Joonyeon Chang,1*
Suk Hee Han,1 Mark Johnson3
Spintronics increases the functionality of information processing while seeking to overcome some
REPORTS
of the limitations of conventional electronics. The
spin-injected field effect transistor, a lateral
semiconducting channel with two ferromagnetic electrodes, lies at the foundation of spintronics
magnetization states along the y axis, TMy. A
research. We demonstrated a spin-injected field effect transistor in a high-mobility InAs
small external magnetic field applied along the y
heterostructure with empirically calibrated electrical injection
andcreate
detection
of ballistic
spinaxis (Ba) can
conditions
with the
injector
polarized electrons. We observed and fit to theory an (source)
oscillatory
channel
conductance
as
a
function
and detector (drain) magnetizations
parof monotonically increasing gate voltage.
allel or antiparallel, resulting in relatively high
29 May 2009; accepted 4 August 2009
Published online 20 August 2009;
10.1126/science.1177031
Include this information when citing this paper.
materials with relatively small spin-orbit interaction such as GaAs and Si (3–5). However,
modulating the channel conductance by using
an electric field to induce spin precession has
remained elusive. A material with large spin-orbit
interaction will not permit the observation of the
Hanle effect, yet this type of material is necessary
for gate voltage–induced spin precession. These
two phenomena are mutually exclusive within
any single material. We used a high-mobility
InAs heterostructure with strong intrinsic spinorbit interaction a, and we measured the nonlocal
channel conductance (5, 6) rather than the direct
source-drain conductance suggested by Datta and
Das. Conventional lateral spin valve techniques
measured the population of ballistic spins.
Shubnikov–de Haas (SdH) experiments provided
an independent measurement of the dependence
of the spin-orbit interaction on gate voltage.
Apart from a small phase shift, the oscillatory
conductance that we measured fits to theory (1)
with no adjustable parameters. The temperature
dependence indicates that the modulation is only
observed when the injected electrons have ballistic trajectories to the detector.
A conventional lateral spin valve device
(Fig. 1A) is a convenient structure to investigate
spin injection and detection for several reasons.
First, the ferromagnetic (FM) electrodes have a
uniaxial shape anisotropy that can create binary
Downloaded from www.sciencemag.org on Oct
tinuously increases along the helix axis—rather
than for the constant helix diameter discussed
here—antenna theory (28, 29) promises bandwidths considerably exceeding one octave. This
approach could lead to a further increase of the
circular-polarizer operation bandwidth.
Metallic wire-grid linear polarizers (“onedimensional metamaterials”) have been known
since the pioneering experiments on electromag-
or low spin-dependent voltages at the detector
(2, 7, 8). Second, a small channel length, L,
Dattaand
anddetector
Das (1),
any types of spintronic devices have Proposed
between by
injector
canthe
be demondefined
lithographically.
in the “nonlocal”
configof a spinThird,
FET involves
spin injection
been proposed, investigated, and de- stration
(5–12),
theabias
current is grounded
at
detection
using
ferromagnetic
source and
veloped. However, the spin-injected anduration
one However,
end of the asample,
is no
current
specialthere
feature
ofcharge
the spin
FET
field effect transistor (spin FET), which lies at drain.
vicinitymodulation
of the spin of
detector,
background
thethe
periodic
source-drain
conthe heart of spintronics, has yet to be realized. is in
effects are minimized, and the signal-to-noise
ductance as controlled by gate voltage–induced
ratio is maximized. Spin-polarized carriers with
1
precession
of the injected
spin
Center for Spintronics Research, Korea Institute of Science
ballistic trajectories
along spins.
the +xElectrical
and −x direcand Technology (KIST), 39-1 Hawolgok-dong, Seongbukinjection
and
detection
have
been
demonstrated
tions
are
injected
with
equal
probability.
gu, Seoul, 136-791, Korea. 2Department of Physics, Sejong
in a variety
of semiconductors
(2–6). gas
Carrier
spin
In a two-dimensional
electron
(2DEG)
University, 98 Gunja-dong, Gwangjin-gu, Seoul, 143-747,
3
precession
has
been
induced
by
using
an
external
channel
with
strong
spin-orbit
interaction,
the
Korea. Naval Research Laboratory, 4555 Overlook Avenue
structural
asymmetry
provides
intrinsic
elec-a
magnetic
field
and detecting
theanHanle
effect,
SW, Washington, DC 20375, USA.
tric field along the
z axis, Ez,0, where
the
Lorentzian-shaped
magnetoresistance
caused
*To whom correspondence should be addressed. E-mail:
subscripts
denote
the
z
direction
and
zero
gate
presto@kist.re.kr
by precessional dephasing of diffusing spins, in
voltage. In the rest frame of a carrier moving with
a weakly relativistic Fermi velocity, vFx ~ c/300,
with c the SCIENCE
speed of light,VOL
electric
field E
transz,0 SEPTEMBER
www.sciencemag.org
325
18
2009
1515
forms as an effective magnetic field BRy,0, which
is called the Rashba field (13). The Rashba field
is perpendicular to the directions of the carrier Fig. 1. Lateral gated spin valve device with an external magnetic field (Ba = 0.5 T) applied along
velocity and the electric field. In Fig. 1, BRy,0 is the y axis (A) and x axis (B). In (A), the magnetizations of the FM electrodes are shown oriented
along the y axis and has no effect on carriers that along the y axis. The injected spin-polarized electrons are oriented along the y axis and do not
precess under the influence of the Rashba field BRy. (B) shows the electrons injected with spin
Koo et al., Science 325, 1515-1518 (2009)
Spin-­‐Transistor as Quantum Bit •  Can we use this kind of spin-­‐transistor as a quantum bit ? •  Not possible with an open quantum system like the DaQa-­‐Das transistor •  Fundamental need : two level system with long decoherence ;me One possibility could be -­‐ a single electron Quantum Dot -­‐I Quantum Dot : The Constant Interac;on Model Assumption
•  Inter-­‐dot Coulomb interac;on + Interac;on of dot electron with environment : can be parameterized in terms of a single capacitance “C” •  Single parity energy spectrum is independent of these interac;ons 2
!
e
N
+
C
V
+
C
V
+
C
V
(
S S
D D
g g )
E(N)
=
2C
C = CS + C D + Cg
Quantum Dot: A leaky capacitor Electrochemical poten;al of the dot: 1
µ N = E(N ) ! E(N !1) = U(N + ) ! " !i e Vi
2 i=L,R,g
Where: !i = ci / C < 1, i = L, R, g
µ N ! µ N !1 = U = e 2 / C
Linear response: Liaing of coulomb blockade It is not a quantum level Quantum Dot: A leaky capacitor Nonlinear response: Quantum Dot: A leaky capacitor Nonlinear response: Quantum Dot: A leakey capacitor Few electron quantum dot (account for Pauli principle): µN = $ N
free
+ E ( N ) " E ( N " 1)
1
= $ N free + U ( N + ) " ! # i eVi
2 i= L,R, g
Odd – Even diamonds δ = ε1 − ε 2
Quantum Dot: A leaky capacitor Few electron quantum dot (account for Pauli principle): µN = $ N
free
+ E ( N ) " E ( N " 1)
1
= $ N free + U ( N + ) " ! # i eVi
2 i= L,R, g
Odd – Even diamonds δ = ε1 − ε 2
Excita'on Spectra of Circular, Few-­‐Electron Quantum Dots Kouwenhoven et al
Quantum Dot -­‐II through a tunnel
barrier represented
by a tunnel resistor
&#
are negligible, the double dot electrostatic
energy reads by
Theaelectrochemical
potential ( 1(2) (N
of dot a capacit
resented
tunnel
resistor
R
1 ,N
2 ) and
m
U " N 1 ,N 2 # !
full derivation is given
in the Appendix)
Rtransport
a(acapacitor
Cdots
connected in1(2)parallel.
is defined 3as theThe
energy needed to add the N 1(2) th
L(R) and
L(R)
el et al.: Electron
through
double quantum
parallel.
The
bias
voltage
iselectrons
applied
on dot to the so
electron
to dot 1(2),
while havingV
N 2(1)
1 2
1 2
dots
areet coupled
to
each
other
by
a
tunnel
barrier
rep3
van
der Wiel
al.: Electron
transport
through
double
quantum
dots
U " N 1 ,N 2 # ! N 1 E C1
" N
"N 1tact
Net
2(1). Using
the expression
for the
total
energy
Eq. (1),(asymme
2 E Cm
2 E C2Wiel
van
al.with
: Electron
transport
through
double
quantum
dots
drain
contact
grounded
2
2 der
van
derthe
Wiel
et al.C
: Electron
transport
through
double quan
2
resented by a tunnel
resistor
R
and
a
capacitor
in
the
electrochemical
potentials
of
the
two
dots
are
m
m
1
e
In thisto
we consider
the# linear transpor
"f " V g1
(1)section
E C2 !
g2 # , ,is applied
2 ,VV
(source
N 1 ,N 2 # #U " N 1 #1,N
2
parallel. The
bias
the
" N 1 ,N 2 # )U "conThis
is the energy
1
C 2 voltage
Cm
1
e
2 2
1
e
i.e.,
V!0.
If
cross
capacitances
(such
as
2 betw
1
3bias).
l et al.: Electron
transport
double 1#
quantum
dots
Egrounded
! "N E # (asymmetric
,
e
tact
withthrough
the drain
1
C2
2
E
!
,
! C 1$C
C g1
V
N
E
f " V g1 ,V g2 #contact
"
"N
and
a
capacit
2
C m2), other !voltage
2 and
N 1 # EC2
2 2 Cm dot
# ! e ! 2 g1 1 C1Cand
C1 "N
2E2
Cm
EC
!stray cap
C
sources,
C2
2
m
In this section we
consider the linear
1# transport regime,
2
#C1#
theCcapacit
2
m is
C 1C 2
e
"C1g2 V g2 " N 1 E Cm "N 2 Eare
#
%
2
C2
C
C
negligible,
the
double
dot
electrostatic
ene
1#
1 2
1
e
i.e., V!0.E Cm
If !cross
capacitances
(such as between# 1V" Cg1
.
Thus
a
large
interd
V
E
"C
V
E
,
(5)
#
g1
g1
C1
g2
g2
Cm
3
E ! Ctransport
,
van der Wiel et al.: Electron
dots
2quantum
C 1 Cthrough
12 21 2 2double
1e(a
!
is ! egiven
ine 2the Appendix)
2 full
2
1derivation
C 2m voltage
and dot 2), C2
other
capacitances
Csources,
"2 C
V g2
E C2 stray
m
2 #1
g1 V g1 E C1 " C g2and
one
big 1dot. e 2.
2
.2" N 1 ,N 2 # )U " N 1 ,N
apacitors represent!
E
C me 2 E Cm ! C
1#
(
#U
N
,N
#1
#
#
"
Cm
2
1
2
C 1C 2
are negligible, the double
energy1 reads
C1m The
C 1E
Celectrochem
m
C 1 C 2 dot 2electrostatic
2 !
e different elements
Cm
#1
1
e
2
2
#1C m C 1
2
1
2
Here
C 1(2)
is2 the
sum
attached
to
dot! N
tunnel resistors
represent"C g1 of
V g1 Call
Vcapacitances
E Cmcapacitors
, Appendix)
FIG.
1. capacitors
Network
of tunnel
resistors
and
representbarriers
are characg2!
g2the
U
N
N
,N
!
"
E
"N
(aand
full
derivation
is
given
in
Cm
#
"
C
N 2 #1 EEC1
"N
E1(2)
E
,
1
2
C2
1 Nas
2 EtC
2
C2
1
Cm
m
is
defined
C2
2
ecoupled
1Cseries.
2
2
2
!C
"C
"C
.
Note
1(2)
including
C m : in
CThe
C
sas
coupled
in series.
The
different
elements
ing
two
quantum
dots
different
elements
indicated
in the
FIG.
1.
Network
of
tunnel
resistors
and
capacitors
represent1(2)
L(R)
g1(2)
m
2
m
C
3
! E C1(2) is the charging
E Cm
vanenergy
der.Wiel
etthe
al.:individual
Electron transport
double quantum dots
where
ofas
dot ofthrough
1#
Here
C
is
the
sum
all
capacitances
attached
to
dot
electron
to
dot
1(2)
Here
C
is
the
sum
of
all
capac
can
be
interpreted
the
charging
energy
of
that in
E C1(2)
1(2)
text. Note that
barriers
are
charac1(2)
C
C
C
aretunnel
explained
the
text.
Note
that
tunnel
barriers
are
charac1
1
1
ingEtwo
dots
coupled
in
series.
The different elements
m quantum
C#!e!
coupling
energy,
and
1(2),
21 2 #1
2
1 Cis2
Cm is the electrostatic
#
V g1,V
E Cm "C g2
V g2
E C2 # .:C(6)
" C g1including
C
1(2)
C
:
C
!C
"C
Note
1(2)
including
C
the
single,
uncoupled
dot
1(2)
multiplied
by
a
correction
U
N
N
N
,N
!
E
"
E
"N
N
E
"f
#
is!C
theexp
su
"
2
terized
by
a
tunnel
resistor
and
a
capacitor,
as
indicated
in
esistor
and
a
capacitor,
as
indicated
in
the
m
1(2)
L(R
2(1).
Using
the
m
1(2)
g1(2)
m .# ,Here
! e"! V
1(2)
2 L(R)
1
2 electron
C1the
1 2 the
Cm
are
explained
in
text.
barriers
are
characg1 "C
g2
pacitors represent1 The
2Note
is thetunnel
the
charge.
coupling
energy
EC2
Cmthat
C
1
e
m that E 2 2 can be interpreted as the
2
can
be
asm
that
charging
ofinterpreted
factor that
accounts
for
theresistor
coupling.
When
!0,
inset.
E C2
! asand
,Eif,C1(2)
e anand
change
in the
of one
dot
when
electron
is CThe
1(2)
including
C
m
C1(2)
different elements
terized
byenergy
a tunnel
a1 capacitor,
atthe
fixedenergy
gate voltages,
N
change
in ( 1 (N
the
electrochemica
1 is
C 2 indicated
C m2 1 ,N32 ) in
!
.
E
van
der
Wiel
et
al.
:
Electron
transport
through
double
quantum
dots
added
to
the
other
dot.
These
energies
can
be
expressed
uncoupled
dot
1(2)
mu
Cm
Eq.
(1)
to
henceCE1(2)
Here
is the
sum
ofreduces
all capacitances
dot
the
single,
uncoupled
dotchanged
1(2)toby
multiplied
by2 )#a (correction
( 1(Nsingle,
is
1, the
1#
Cm !0,
can be
that
E, C1(2)
barriers are charac1 "1,N
1 (N 1 ,N
2 )!E C1
inset.
C
C 1 C 2 attached
m
C 1 C1 2 (1)
in terms of"f
the capacitances
as
follows:
V
,V
,
#
"
called
addition
energy
dotg1
1 and
equals
the1"
chargaccounts
for
the
coupl
,Vmg2
! factor
C
V
N
EN
"N
EC
#2Note
"V
"(
$ofthat
C 1(2)g2
!C
"C
"C
.the
1(2)
including
C m : g1
#1
factor
thatL(R)
accounts
for
the
coupling.
When
C
and
2 fg1(2)
g1
g1the
C1
as
indicated
in the
m !0,
uncouple
A.
Linear
transport
regime
1single,
1 ,N
2 #2)U
regime
nel
resistors and
capacitors
represent2g1 # C 2
2 " #N 1 ! e ! "C g1 V
#
!
e
!
ing
energy
of
dot
in
this
classical
regime.
Similarly,
the
1
e
e
1
e
1 hence E as!0,
m (1) reduces
Eq. (1)
reduces
to
where, can
charging
that U
Eelements
toofhence E Cm !0,factor
N 1 E,N
"C1(2)
2 # !be interpreted
E C2Cm
! theEq.
Eenergy
that
accounts
,2C
upled in series.
The different
(
(N
,N
Cm ! energy of dot .2 equals E C2 , and
2 (2), ; addition
; C1 !
2
cription
in which
1
1
2
1
C
C
C
C
A.
Linear
transport
regime
Cm
C1
C mdot 1(2)
1 multiplied
2
m
1 2
1. Classical
the theory
single,
uncoupled
byall
a"1)#
correction
Here
C 1(2)
is" the
sum
of
capacitances
attached
( 1 (N
( 2 (N
(Nhence
,N "2 )!E
1#
1#,N 22)!2#1
1 "1,N 2 )# ( 2to
1dot
Cm
tunnel
are
charac1#
#N
! e ! "C
!0,
Eq.
E1. Cm
,V
!
C
V
N
E
"N
E
f" V
s. Note
is notthat
taken
intobarriers
#
$
FIG.
1.
Network
of
tunnel
resistors
and
capacitors
representg1
#N
!
e
!
"C
V
#
"C
V
N
E
"N
E C2
"
#V% g
"
g1
g2
g1
g1
1
C1
2
Cm
C
C
C
C
C
1
g1
g1
2
g2
g2
1
Cm
2
m
1
2
1for
2 including
In
the
next
section
we
will
discuss
the
addition
energy
in
!
U
N
,N
!
factor
that
accounts
the
coupling.
When
C
!0,
and
#
"
#
!
e
!
:
C
!C
"C
"C
.
Note
1(2)
C
! e !""C
" #N
1m 2
or andetaal.,
capacitor,
inquantum
the dots coupled
ing two
in series.
The
elements
1(2) m L(R)
g1(2)
2U
g2 V#g2
N different
,N
!m#
uzin
1992; as indicated
Double quantum dot: An ar;ficial molecule $ % $ %
$ "% $ %
#
$
%
$ %
$ %
$ %
$ %
$ %
$ %
$ %
$ %
$ %
"
$
$
%$
%$
$
Wedescription
start with1.
purely
classical
inHere
which
1 22
Classical
theory description
2C 1 " #
purely classical
ina which
"
(3)
C 1(2)
is the
sumcharging
of all capacitances
attached
to dot
2C
eto be
1.
areEexplained
in the
text.
Note
that
tunnel barriers
are charac1
!0,
Eq.
(1)
reduces
hence
can
interpreted
as
the
energy
of
that
E
Cm
C1(2)
2C
led as a network
the states
influence
discrete
quantum
states
is not
taken
into C m : C 1(2)
! indicated
. 1(2) including
E
!C L(R)
"C g1(2) "C m . Note
1a correction
1by
1 2 # 2!
Cm2as
U " N 1 ,N
terized
a tunnel
resistor
a capacitor,
Rev. by
Mod.
Phys.,
Vol.
75, No.and
1, January
2003
screte
quantum
is notof
taken
into
C m Cclassical
Cin2 thedotE
2
2energy
"C
V
N
E
"N
#
1
"
%
the
single,
uncoupled
1(2)
multiplied
We
start
with
a
purely
description
in
2 which
#N
! eC
! "CV
"
2
g2
g2
1
Cm
2
C2
. 1). The number
2
g2
can
be
interpreted
as
the
charging
of
that
E
inset.
account
yet
(Pothier
et
al.,
1992;
Ruzin
et
al.,
1992;
#1
C
"
V
E
"
! "C
" #N 2 ! eC1(2)
V g1
#C 2 independent
" #N
g2 V g2 # 2
C1
This
is the
sum
ofrepresentthe
energies
two
dots.
g1
g1
g2
1 ! e ! "C g1of
FIG. 1.1992;
Network of
tunnel
resistors
and capacitors
hier et al.,
Ruzin
et
al.,
1992;
"
m
factor
that
accounts
for
the
coupling.
When
C
!0,
and
the
single,
uncoupled
dot
1(2)
multiplied
by
a
correction
e
2
2
influence
ofisdiscrete
quantum
states is not taken
. ; into
(3)
m
dot
is capacitively
U
Nthe
,N
"The
2C 2 #
Dixon,
double
dot
modeled
as" a network
1The
2#! C
ime
ing
two
quantum dots1998).
coupled
in series.
different
elements
becomes
the
dominant
capacitance
In the
case
when
cription
in
which
2C
double
dot
is
modeled
as
a
network
2C
m
factor
that
accounts
for
the
coupling.
When
C
!0,
and
1
1
1
2
1
m
Here
C
is
the
sum
of
all
capacitances
attached
to
dot
!0,
Eq.
(1)
reduces
to
hence
E
A. Linear
transport
regime
1(2)2 et1).
are explained
in the text. resistors
Note that
tunnel
barriers are
charac- (Pothier
account
yet
al.,The
1992;
1992;
ough a capacitor
2Cm
2Ruzin
2 Eq.et(1)al.,
ofinto
tunnel
and
capacitors
(Fig.
number
!0,
reduces
to
hence
EV
/C
→1),
the
electrostatic
energy
is!C
given
by
(C
"
is
notcapacitors
taken
Cm
This
is
the
sum of the energies of
C
C
"
V
E
"
E
and
(Fig.
1).
number
mThe
1(2)
:
C
"C
"C
.
Note
1(2)
including
C
terized
by
a
tunnel
resistor
and
a
capacitor,
as
indicated
in
the
m
1(2)
L(R)
g1(2)
m
C1
C2
2
g1the
g1
g2 g2ofas2two
2capacitively
rain (D) contact
This
is
sum
of
the
energies
independent
dots.
Dixon,
1998).
The
double
dot
is
modeled
a
network
of
electrons
on
dot
1(2)
is
N
.
Each
dot
is
#N
!
e
!
"C
V
#
"
1.
Classical
theory
e
2
2
2
1(2) 2that E C1(2) g2
V2g1 #"energy
"g2#N
can be
interpreted
as the g1
charging
of
inset. 1992;
uzin
et
1 ! e ! "C
#N
! "C g1V
V g1 # C
the
In1! ethe
case
when
CE
is al.,
N
. Each dot is capacitively&"
" Nbecomes
"C
, 2 # )U
1(2)
y 1(2)
a tunnel
resistor
.
(3)
1 ,N
g1
g1
g2 V(
g22m
Cm
#
N
"N
!
e
!
"C
V
"C
V
#
"
'
U
N
,N
!
#
"
becomes
the
dominant
capacitance
In
the
case
when
C
U
N
,N
!
#
"
of
tunnel
resistors
and
capacitors
(Fig.
1).
The
number
through
a
capacitor
coupled
to
a
gate
voltage
V
the
single,
uncoupled
dot
1(2)
multiplied
by
a
correction
1
2
g1
g1
g2
g2
1
2
1
2
m
g1(2)
We
start
with
a
purely
classical
description
in
which
2C
rely
classical
description
in
which
2C/C
led
a network
theiselectrostatic
(Cand
This
the sum of t
2C 1 When .C ; !0,
1
throughUa" Ncapacitor
ed
voltage
V g1(2)
2 accounts for the coupling.
#!
m
1(2) →1),
1 ,N
2source
inas
parallel.
The
factor
that
menergy
/C
→1),
the
electrostatic
is
given
by
(C
and
to
the
(S)
or
drain
(D)
contact
C
of
electrons
on
dot
1(2)
is
N
.
Each
dot
is
capacitively
the
influence
of
discrete
quantum
states
is
not
taken
into
A.
Linear
transport
regime
m
1(2)
g1(2)
1(2)
2
te
states
not taken
into
The
number
C̃
"C̃
#reduces to
"of
In the case when C
e1).quantum
source
(S)
or is
drain
(D)
contact
!0,
Eq.
(1)
hence
E2Cm
" #N
1two
2independent
unnel
barrier
rep2 2 ! e ! "C g2 V g2 #
account
yet (Pothier
etaV
al.,
1992;
Ruzin
et
al.,
1992;
This
is
the
sum
ofrepresented
the
energies
dots.
"C
C
V
E
,
through
a
tunnel
barrier
by
a
tunnel
resistor
#N
!
e
!
"C
V
N 1 "N
!e!"
#
#!
"
" of
through
a
capacitor
coupled
to
gate
voltage
V
g1
g1
g2
g2
Cm
where
E
is
the
charging
energy
the
"
.
(3)
2
g2
g2
2indiv
g1(2)C1(2)
2& #/C
etisal.,
Ruzin
al.,
1992;
(4)
dot
capacitively
1.1992;
Classical
theoryby aet
2
2C
→1),
th
(C
represented
resistor
Dixon,
1998).
The C
doublebecomes
dot is modeled
as adominant
network
#
N
"N
!
e
!
"C
V
V
aarrier
capacitor
C
in
#
2"C
&
"
'
#N
!
e
!
"C
V
#
"
m
1(2)
U
N
,N
!
"
.
(3)
#
"
the
capacitance
In tunnel
the
case
when
mL(R)
1
g1
g1
1
2
g1
g1
g2
g2
1
2
R
and
a
capacitor
C
connected
in
parallel.
The
m to the
L(R)
and
source
(S)
or
drain
(D)
contact
C
U
N
,N
!
#
"
2C
uble
dot
is
modeled
as
a
network
of
tunnel
resistors
and
capacitors
(Fig.
1).
The
number
g1(2)
is
the
electrostatic
coupling
energy,
a
1(2),
E
U
N
,N
!
.
1
2
#
"
ough
a
capacitor
1 dot
2energy
itor
in classical
parallel.
Theinthe
Cm
Weconnected
start
with
a purely
description
which
o
theCsource
con2 " C̃
isrepthe2by
sumN
of1the energies of two independent dots.
2C
This
iselectrons
the
energy
of
a by
single
with
a1 This
charge
L(R)
/C
→1),
electrostatic
is
given
(C
dots
are
coupled
to
each
other
a
tunnel
barrier
m
1(2)
of
on
dot
1(2)
is
N
.
Each
dot
is
capacitively
through
a tunnel
represented
by
awhen
resistor
where
EThe
is the
charging
energy
of
the
individual
dot
d
(Fig.
1).
number
2 "tunnel
C̃C1m"C̃
&#
# the dominant
thecontact
influence
quantum
states
isrepnot taken
into 1(2) barrier
ain
(D)other
becomes
capacitance energy E
In the
case
2The
C1(2)
asymmetric
bias).
o capacitors
each
by ofa discrete
tunnel
barrier
electron
charge.
coupling
2 of
This
isofthe
of
the
energies
two
independent
dots.
C
"N
and a capacitance
C̃ sum
"C̃the
, where
C̃
!C
"
#
#
$
when electrons
tunnel
through
both dots. This conwell-defined electron
number can
on each
dot.
For double
wheneverresonance
three charge
states become dedots coupled indition
series,isa met
conductance
is found
i.e.,
whenever
when electronsgenerate,
can tunnel
through
boththree
dots.boundaries
This con- in the honey4
van
der
Wiel
et
al.
: Electron
through
double q
comb diagram
indouble
one quantum
point.
In
Fig.transport
2(d) two
kinds
isWiel
met
whenever
three meet
charge
states
become
4
tic stability
diagram of the doubledition
dotder
system
van
et al.
: Electron transport
through
dots deof such triple
points
are distinguished,
(!) and ("), corgenerate,
three
boundaries
in the honey) intermediate, and (c) large interdot
coupling.i.e., whenever
responding
different
charge
processes. At the
comb
diagram meet
in onetopoint.
In Fig.
2(d)transfer
two kinds
charge
on double
each dotdot
in system
each domain
is denoted
am
of the
Here
V is the
electrostatic
potential
of
node through
j ("),
and
triple are
point
(!),
the
dots
cycle
of suchwith
triple points
distinguished,
(!)
and
cor-the sequence
e two
kinds interdot
of triple coupling.
points corresponding
ground is defined to be at zero potential, V !0. The
nd
(c) large
We write the total charge Q on the dot as the sum of
charges on transfer
the nodes are linearprocesses.
functions of the poten- At
the charges
responding
to different charge
theon all the capacitors connected to the dot
of the nodes
so
this can be expressed
more
comnsferinprocess
(!) and
the hole transfer
process
(see
Fig. 29)
dot
each domain
is denoted
→
N
"1,N
→
N
,N
"1
4
$
$
$→
# N 1 ,N 2tials
#
#
# N 1 ,N
van
der
Wiel
et
al.
:
Electron
transport
through
double
1
2
1
2
2$,
pactly in matrix form
triple
point
dotstransport
cycle
through
the quantum
sequence
Q !C V "V #C V "V #C V "V
4 corresponding
van
der
Wiel
et al.: the
Electron
through
double
dots
ple
points
ed in
(d).
The region in with
the dotted
square
in (!),
! !CV
!,
Q
(A2)
⇒Q #C V #C V #C V !C V ,
which
shuttles
the
system. This (A7)
where C isone
called the electron
capacitance matrix. Athrough
diagonal
and
the
hole
transfer
process
where
C is the total capacitance coupled to the dot,
n more detail in Fig. 5.
→ #ofNthe1capacitance
,N 2 "1
→
N
,N
,
$
$
# N 1 ,N 2 $ → # N 1 "1,N 2 $element
#
C is the total capaci1
2C !C #C #C . The capacitance matrix C only has
process is illustrated
bymatrix
the
counterclockwise
e and
tance of node j,
one element. Using path
Eq. (A4) and
substituting Q
egion in the dotted square in
!"(N "N ) # e # , we find
which
shuttles
one
electron
through
the
system.
This
the diagram Cof! !anc electron
sequentially
tunneling
from
.
(A3)
Fig. 5.
" N "N # e # #C V #C V #C V
U N !
,
What a
bout t
he S
pin ?
2C
process
is
illustrated
by
the
counterclockwise
path
e
and
off-diagonal
element
of the capacitance
matrix 2(d).
is mithe left leadAn
to
the
right
in
Fig.
At
the
other
triple
regime, where also the spacing between
(A8)
nus the capacitance between node j and node k, C
where
N is the number of electrons on the dot when all
the diagram ofpoint
an electron
from
!C sequentially
!"c . The electrostatic energy
of this system of
("),
the
sequence istunneling
voltage sources are zero, which compensates the positive
y levels plays a role.
conductors is the sum of the electrostatic energy stored
background
charge originating from donors in the hetthe N(N#1)/2
capacitors
and can
be conveniently
the left lead to the right inonexpressed
Fig.
2(d).
At
the
other triple
also the spacing
betweenin Eqs.
erostructure.
using the capacitance matrix
lectrochemical
potentials
(5)
and
N 1 "1,N
→
"1,N
N 1 ,N 2 "1
$stability
# 3.
#N
Schematic
diagram
the$ Coulomb
point ("), the FIG.
sequence
is U!2 "1
2 $ → #showing
1
1 1
1
role.
! •CV
!! V
! •Q
!! Q
! •C Q
!.
(A4)
V
2
2
uct a charge stability diagram, giving the
peak spacingsVoltage
given
in2 Eqs.
(8) and (10). These spacings can be
potentials in Eqs. (5) and
sources can be included in the network by
write
the total charge
Q $ on
as the sum
N
"1,N
"1
→
N
"1,N
→
N
"1to ground
$
$
$ → #ofNWe
#
#
#
"1,N
,connected
FIG.
3.dot 1(2)
Schemat
1
2determined1 experimentally
2 as nodes with large
1 ,N
2connecting
treating them
capacitances
2 "1
ectron numbers N 1 and N 2 as a function
the1the
charges on
all the
capacitors
to dot 1(2)
by
triple
points.
and
large
charges
on
them
such
that
V!Q/C.
In
this
Fig. 30),
FIG. 3. Schematic stability(seediagram
showing
the Coul
ability diagram, giving the
peak spacings
give
case, it is numerically difficult to compute the inverse of
. We define the electrochemical potenQ !C
VFig.
"V #C2(d).
V "V This
#C V "V ,
the capacitance
matrix
since
it "1,N
contains
largein
elements.
g2
corresponding
to
the
clockwise
path
h
in
→
N
"1
,
$
#
peak
spacings
given
Eqs.
(8)
and
(10).
These
spacings
ca
determined exper
1 invert the entire
2 capacis N 1 and N 2 as a function
However, it is not necessary to
Q !C V "V #C V "V #C V "V .
well-defined
electron
number
on
each
dot.
For
double
tance determined
matrix since the voltages
on the voltage sources
ft
right leads to
be zero if no bias
experimentally
connectingof
thea triple
(A9)
can be interpreted
as the
sequentialbytunneling
holepoints
are already known. Only the voltages on the other nodes
he and
electrochemical
potencorresponding
to
the
clockwise
path
h
in
Fig.
2(d).
WeThis
can write this as
need
to
be
determined.
These
voltages
can
be
deterdots
coupled
in series,
a conductance
resonance
isC found
well-defined
lied,to! Lbe
!!
in the
direction
opposite
to the
electron.
The energy
dif-V ele
R !0.
mined by writing
the relation between
the charges
and
"C
Q #C V #C V
ads
zero
if Hence
no biasthe equilibrium
can be interpreted
as
the
sequential
tunneling
of
a
hole
! ! dot. For
the voltages
as
!
"
" ! V " , dou
well-defined
electron
number
on
each
in
Q #C dots.
V #C Vdots
"Ccoupled
C
when
electrons
can
tunnel
through both
This
conanddirection
N2
e0.dots
are the
values of in
N 1the
ference
between
!
! processes determines the separa- (A10)
C both
C
Q
V
Hence
thelargest
equilibrium
FIG.
3.
Schema
opposite
to
the
electron.
The
energy
difdots
coupled
in
series,
a
conductance
resonance
is
fo
!
.
(A5)
when
electrons
" ! ! Schematic
" points
! " !C
dition
is met ! Qwhenever
three (!)
charge
become
de. The above expression
in the
wherestates
C !C #C #C
C 3. V
FIG.
stability
diagram
showing
the
Co
!
(N
,N
)
and
!
(N
,N
)
are
less
than
tion
between
the
triple
and
("),
and
is
given
1
1
2
2
1
2
form
of
Eq.
(A6)
reads
peak
spacings
giv
rgest values of N 1 and N 2
when
electrons can
tunnel
through both
dots.
This
ference between
both processes
the
dition
is met
whc
! and V
! determines
are the charges and the voltages
on theseparaHere Q
generate,
i.e.,
whenever
three
boundaries
the
peak
spacings
given
in
Eqs. (8)
and
(10).
spacings
C These
C honey1in
V
! and
! met
charge
nodes, Q
V
areEq.
the charges
and the volt- three
determined
expe
by
E
,
as
defined
in
(2).
is
larger
than
zero,
electrons
escape
to
the
dition
is
whenever
charge
states
become
!
! V " connecting
!generate,
" triple
i.e., w
Cm points
d ! 2 (N 1 ,N 2 ) are less than
tion between the
triple
(!)
andand
("),
and
is given
C C "C C
C
ages on the
voltage sources,
the
capacitance
matrix
determined
experimentally
by
thekinds
poi
comb
diagram
meet
one
point.
Fig.
2(d)
two
has been
expressed
inin
terms
of honeycomb
fourwhenever
submatrices.
The In
generate,
i.e.,
three
boundaries
in
the
hon
The
dimensions
of
the
cell
(see
Fig.
3)
can
comb
diagram
m
c stability
diagram
of the
double
dotN
nstraint,
plus
the
fact
that
N 1 and
must
Q #C V #C V
by
Esystem
,
as
defined
in
Eq.
(2).
ero,
electrons
escape
to
the
2Cm
voltages
on
the
charge
nodes
are
then
FIG. 2. Schematic
stabilitypoints
diagramare
of the
double dot system
$!
.
(A11)
Q #C
V #C
V "
of
such
triple
distinguished,
(!)
and
("),
corwell-defined
el
diagram
meet in
one point.
InofFig.
2(d)
twopo
k
such
triple
! "C
!
!C
Q
V
(A6)
and (c) large
interdot
coupling.
be
related
toV! comb
the
capacitances
using
Eqs. on
(5)
(6).
FIG.
Schematic
stability
diagram
the
double
dot
creates
hexagonal
domains
inof dimensions
the
for
(a)
small,
(b)
intermediate,
and
(c)cell
large
interdot
coupling.
The can
electrostatic
energy
of and
the double
dot system
can d
The
ofsystem
the honeycomb
(see
Fig.
3)
entermediate,
fact
that 2.N
well-defined
electron
number
each
dot.
For
1 and N 2 must
dots
coupled
in
now be processes.
calculated using
Eq.
(A4).
For
the
case
V
of
such
triple
are
(!)
and
("),
and
the
electrostatic
energy
can
be points
calculated with
Eq. distinguished,
responding
to
different
charge
transfer
At
the
responding
to di
for
(a)
intermediate,
(c)
large
interdot
coupling.
harge
on domains
each
dot(b)
in in
each
domainand
is
denoted
The equilibrium
charge (A4).
on each
dot
in each
domain
is adenoted
!Vconductance
!0 and Q
!"N
# e #resonance
this becomes
dots
coupled
in
series,
is
From
ase
space
insmall,
which
the the
charge
configurabe
related
to triple
the
capacitances
using
Eqs.
(5)
and
(6).
gonal
when
electrons
responding
to
different
charge
transfer
processes.
At
triple
point
(!),
point
(!),
the
dots
cycle
through
the
sequence
The
equilibrium
charge
on
each
dot
in
each
domain
is
denoted
by (N 1 ,N 2 ). The two kinds of
tripleelectrons
points corresponding
with
when
candots
tunnel
through
both
Thiw
two kinds
of tripleconfigurapoints corresponding
with
dition
is
met
From
hich
the
charge
triple
point
(!),
the
cycle
through
thedots.
sequenc
by
(N
,N
).
The
two
kinds
of
triple
points
corresponding
with
the
electron
transfer
process
(!)
and
the
hole
transfer
process
!
N
,N
;V
,V
!
!
N
"1,N
;V
"%V
,V
$
$
#
#
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is met
whenever
charge
states
#N
1 #N
1 ,N
2 $→
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g2"1,N
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2 three
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g2$1,,Nbecom
fer process
2 $ → #w
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,N
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tely
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(Cprocess
the
diagram
1 in (d).
2 The
1N ,Nin$2→
1 square
2$"1
2 N i.e.,
the electrondots
transfer
and
the
hole
transfer
process
m !0) (!)
(")
are
illustrated
the
dotted
in
N
"1,N
→
N
,N
"1
→
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$
#
#
#
#
generate,
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boundaries
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(7)1the
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1 g1 ,V g2
2 $
1
2 comb diagram
ddots
in (d).
The
region
in inthe
dotted
which
shuttles
o
FIG.
2.
Schematic
stability
diagram
of
the
double
dot
system
(C
!0)
the
diagram
(")
are
illustrated
(d).
The
region
in
the
dotted
square
in
theis depicted
. 2(a). The
in more
detailcomb
in Fig.
5.
m gate voltage V g1(2) changes(b)
diagram
meet
in
one
point.
In
Fig.
2(d)
two
which
shuttles
one
electron
through
the
system.
This
(7)
of
such
triple
p
FIG.
2.
stability
diagram
the (a)
double
whichand
shuttles
one
electron
through
the system.
process
is illustrT
more
detail
in Schematic
Fig.
5. more
for
small,
(b)system
intermediate,
large
interdot
wedot
obtain
is g1(2)
depicted
in
detail
in Fig. on
5.of the
changes
the
voltage
V
of such (c)
triple
points
arecoupling.
distinguished, (!) and ("
1(2), (b)
without
affecting
the
charge
20
van der Wiel et al.: Electron transport through double quantum dots
Linear Response Stability Diagram FIG. 29. Network of capacitors and voltage nodes used to calculate the electrostatic energy of a single quantum dot.
FIG. 30. Network of capacitors and voltage nodes used to calculate the electrostatic energy of a double quantum dot.
2. Single quantum dot
j
0
1
L#
1
L
L
g#
g
g$
1
g
R
R
R#
1
R$
1
1
1
jj
1
L
g
1
0
R
cc
1
N
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1
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k!0,k"j
%
1$
#
#
0$
1
L
L
g
g
R
R&
2
1
jk
kj
0
jk
"1
3. Double quantum dot
1(2)
c
cc
cv
c
v
vc
vv
v
c
c
v
1
L#
2
R#
c
c
g1 #
2
R$
g2 #
L
L
g1
g1
2
R
R
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g2
2
R
g2
v$
1
2
1
g1 $
m#
1
2$
2
g2 $
m#
2
1$
1
m
m
2
1
2
m
2
1
v
cv
L$
1
2
"1
cc #
1
2
m
m
1
m
1
L
L
g1
g1
2
R
R
g2
g2
L
R
Rev. Mod. Phys., Vol. 75, No. 1, January 2003
1(2)
1(2)
Experiments on Carbon nanotube quantum dot Churchill et al, Phys. Rev. Lett. 102, 1066802 (2009)
Two level system: Quantum Bit The Bloch Sphere Mixed states are sta;s;cal mixtures of pure states and can be inside the Bloch sphere. Ref: Lecture by H. Bluhm
Universal control of single Qubit �
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Spin decoherence ;me Spin dephasing ;me Role of Nuclear spins Hyperfine interac;on Overhauser field operator H HF = S1.h1 + S2 .h2
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After initializing into (0,2)S, detuning ε is swept adiabatically
with respect to tunnel coupling through the S-T+ resonance
(quickly relative to S-T+ mixing) , followed by a slow ramp
(τA~1 µs) to large detuning, loading the system in the ground
Spin swap and coherent oscilla;ons state of the nuclear fields
↑↓ . An exchange pulse of
duration τE rotates the system about the z axis in the Bloch
sphere from
↑↓ to ↓↑ . Reversing the slow adiabatic
passage allows the projection onto (0,2)S to distinguish states
↑↓ and ↓↑ after time τE. Typically, τS = τSƍ = 50 ns. (B)
PS as a function of detuning and τE. The z-axis rotation angle
φ = J(ε)τE/ƫ results in oscillations in PS as a function of both ε
and= τ2n
E.!Inset: Model of PS using J(ε) extracted from S-T+
resonance condition assuming g* = -0.44 and ideal
measurement contrast (from 0 to 1). (C) Rabi oscillations
measured in PS at four values of detuning indicated by the
dashed lines in (B). Fits to exponentially damped cosine
function with amplitude, phase, and decay time as free
parameters (solid curves). Curves are offset by 0.3 for clarity.
(D) Faster Rabi oscillations are obtained by increasing tunnel
coupling and by increasing detuning to positive values,
resulting in π-pulse time of ~350 ps.
Hahn Echo: Undoing dephasing due to random nuclear field Spreads out ! Pulse Focus in Pumping the nuclear field polariza;on !"#$% &'()*$+,-$%./$01&23)&$0(%,
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T0
The S-­‐T+ transi;on transfers net angular momentum into the nuclear bath hence polarizing it ! pn
Hence the final step of controlling the ! B Z achieved J
S
Conclusion-­‐1 •  Desired ini;aliza;on of quibit •  Universal gate opera;ons •  Spin-­‐to-­‐Charge read outs Introduc;on: magne;c tunnel junc;ons: Spin Valve Effect Spin-­‐dipole-­‐tronics → polariza;on Spin-­‐dipole-­‐tronics → polariza;on Out of n
L , n
R Plane !! Proximity induce exchange field Single channel ballistic conductor
!
#"
#"
Non-interacting
scattering picture
!
ferromagnet
h
ferromagnet
Resonance condition : 2! + 2# " = 2'n with ! = kh
Linearized quadratic band dispersion : k = k F +
eff
Effective exchange field: Eex
=
E & EF
!v F
!vF
(#% & #$ )
2h
A. Cottet et al.,Europhysics Letters 74, 320 (2006)
Proximity induce exchange field Single channel ballistic conductor
!
#"
#"
Non-interacting
scattering picture
!
ferromagnet
h
ferromagnet
Resonance condition : 2! + 2# " = 2'n with ! = kh
Linearized quadratic band dispersion : k = k F +
eff
Effective exchange field: Eex
=
E & EF
!v F
!vF
(#% & #$ )
2h
A. Cottet et al.,Europhysics Letters 74, 320 (2006)
µL
U
µR
The Model: H = H Dot + H Lead + HTunnel
µL
U
µR
The Model: H = H Dot + H Lead + HTunnel
H Dot = ε Nˆ + U Nˆ ↑ Nˆ ↓
HTunnel = ∑r k σ τ trσ τ dσ+ Cr kτ
H Lead = ∑r kτ ε r kτ Cr+kτ Cr kτ
µL
U
µR
The Model: H = H Dot + H Lead + HTunnel
U >> T , !R / L
#r = 2 " t 2! r
here, H Dot
∑ k δ (ε r k τ − ω ) ≈ ν rτ ;ν r ↑ > ν r ↓
= ε Nˆ + U Nˆ ↑ Nˆ ↓
t r σ τ = σ Exp[−i
HTunnel = ∑r k σ τ trσ τ dσ+ Cr kτ
H Lead = ∑r kτ ε r kτ Cr+kτ Cr kτ
D.O.S 1  
χ r .σ ] σ
2
Nˆ = ∑ σ dσ+ dσ
Konig, Martinek, PRL 90, 166602 (2003)
Braun, Konig and Martinek, PRB, 195345 (2004) Baumgartel, Hell, Das and Wegewijs, work in progress Microscopic deriva'on: real 'me transport theory 1)  Kine;c equa;on for the reduced density operator ( p̂ ) for the dot is obtained ˆ (t ) = Trlead p
ˆ full (t )
p
•
t
p(t ) = −i L p(t ) + ∫ dt ʹ′ W (t − t ʹ′) p(t )
−∞
Laplace transform •
p = 0 = −i L p + W p
state limit Stedy t → ∞ : p(t ) → p
Zero frequency stedy Spin degenerate state Kernal State: L p = 0
2) Expand the density matrix of the dot in appropriate basis ⎛ p00
⎜
pˆ ⇒ ⎜ 0
⎜ 0
⎝
0
p↑↑
p↑↓
0 ⎞ ⎛ p s
⎟ ⎜
↑
p↓ ⎟ = ⎜ 0
p↓↓ ⎟⎠ ⎜⎝ 0
0
(1 / 2) p D + S ZD
S xD + i S yD
0
⎞
⎟
D
D
Sx − i S y
⎟
(1 / 2) p D − S ZD ⎟⎠
3) Perturba;on theory: diagramma;c evalua;on of “W” in first order in hence genera;ng the desired stedy state kine;c equa;on. Γr
 
H Z = −γ B. S
 
S = i [ H , S ] = γ S × B
Z
Braun, Konig and Martinek, PRB, 195345 (2004) µL
U
µR
accumulated Spin Baumgartel, Hell, Das and Wegewijs, work in progress Spin induced transport resonance in SET regime Transport regeim: Spin in over damped 2 ΓR = ΓL
µL
θ = .9 π
BL
U
I
µR
BL
BR
BR
accumulated Spin Vg
SET regeim spin is cri;cally damped: Leads to large precession Baumgartel, Hell, Das and Wegewijs, work in progress Baumgartel, Hell, Das and Wegewijs, PRL (2011) The density matrix ;#%-(%3%>):,/-2
Quadrupole transport and Entalgement flow U8(7M;:)V"#$%&
Bell Basis L[\"O"1*34,0*%'8%*(23(?.*1*(2]"
%8:'S"$7*:")*%$;:)"e:;;"#)'):"O""
"YR7'8*7$(;'*%a:8f"#)'):
45-()+,360,5'.*%
1'1*(2%"9:;#$
%
!"#$%&"'&%#()*($+
D:*&:;"E"?';?7;'):8"$:*)7*M')%J:;+"%&"!W>"
'02%3(6%377,1,.32-'(%'8%9:;%
Hell, Das and Wegewijs, work in progress L[\"%&c:?)%(&"O"*(23(?.*1*(2%-(E*72-'(F
Current in the bias window, Concluding Remarks