Transcript Slide 1

AC-driven double quantum dots as
spin pumps and spin filters
R. Sánchez1, E. Cota2 , R. Aguado1 and G. Platero1
1Instituto
2Centro
de Ciencia de Materiales de Madrid – CSIC, Cantoblanco, Madrid 28049, Spain
de Ciencias de la Materia Condensada-UNAM, Ensenada, México
“Nanoscale Dynamics and Quantum Coherence”
Catania october 2005
Motivation
One of the most important requirements for any spin-based
electronics devices is the ability to generate spin polarized
currents.
Control of spin dynamics of electrons
spintronics
Proposals for generating spin-polarized currents: spin
injection by ferromagnetic metals or DMS.
Semiconducting quantum dots can be used as spin filters
or spin pumps: We propose a system for realizing both spin
filtering and spin pumping with unpolarized leads, by using
an ac-driven double QD connected in series in the presence
of a uniform magnetic field.
Current Rectification by Pauli Exclusion in a Weakly
Coupled Double Quantum Dot System
K. Ono , D.G. Austing, Y. Tokura and S. Tarucha, Science, vol 297 (02)
Spin Blockade
K. Ono , D.G. Austing, Y. Tokura and
S. Tarucha, Science, 297 (02)
J.Iñarrea et al.
two main peaks I/V in a double QD. The spin blockade region
is the plateau between the peaks.
K. Ono , D.G.
Austing, Y. Tokura
and S. Tarucha,
Science, 297 (02)
Allowed and forbidden transitions in artificial hydrogen
and helium atoms: Electrical pump and probe experiments.
T. Fujisawa et al.,
Nature 2002
T=100mK
The relaxation time
does not involve
spin-flip (10 ns)
Orbital relaxation
Average number of tunneling
electrons per pulse:
< nt > = Ip(th + tl)/ e
1/ Gd = 100ns
1/ Gs = 3ns
< nt > » Gd t 1s [1- exp(- th / t
1/ Gs £ t 1s -
2p
1s - 2 p )]
2 p < 1/ Gd
The relaxation
time involves
spin-flip
200 s
Our System:
Hamiltonian
Hˆ  Hˆ l  Hˆ DQD  Hˆ T
Hˆ l 

  L , R k 
ˆ
ˆL H
ˆR H
ˆ
H

H
DQD
QD
QD
LR
 k cˆk†  cˆk 
L
Hˆ QD
   L dˆL† dˆL  U L nˆ L nˆ Lone level in left dot

R
Hˆ QD
   R dˆR† dˆR  U R nˆ R  nˆ R one level in right dot

Hˆ L  R  t LR  (dˆL† dˆR  h.c.)

Tunnelling between dots
Hˆ T   (VL cˆk†L dˆL  h.c.)   VR (cˆk†R dˆRi  h.c.)
k L
ik R
eL ( R ) (t ) = eL ( R ) ±

VAC
cos wt
2
1
 z z ;  z  g B B z Zeeman splitting
2
Blum K., 1981,
Density Matrix:
Master equation for the reduced density matrix:
r = T rLy
Theory and
Applications,
(Plenum)
From Liouville equation
r&(t ) s ¢s = - iws ¢s r (t ) s ¢s
i
f
h
+ å Wsm r mm m¹ s
- g s¢s r s¢s
å
s¢

ˆ (t )
ˆ , ˆ (t )
i
 H
t
éHˆ , rˆ (t )ù f
êë T
ú
û
Wks r ss
( s  s)

reversible dynamics:
s
coherent effects
irreversible dynamics
relaxation
k¹ s
decoherence
(s ¹ s¢)
r ss : Occupation probability of the level s
r s ' s Coherence and phase of superpositions of dot levels s,s’
g s ¢s its real part is responsible for the time decay of the off-diagonal DM elements
(coherences)
1
Re {g }= ( å W + å W
)
2
:
s's
ks
k¹ s
ks '
k¹ s '
Wmn
transition rates due to tunnelling through the contacts
Wmn = Gls { n d Ls m
Gls
2
fl (wmn - ml )dN ¢, N + 1
Transition amplitude
Electrons coming onto left dot from the left lead
Electrons leaving from left dot onto the left lead
+ m d Ls n
2
(1- fl (wnm - ml ))dN ¢, N - 1}
We consider that the AC does not affect the transition probability through the contact
barriers
We include spin-flip processes:
W­ ¯
Electronic dot states are affected by intrinsic degrees of freedom: hyperfine coupling,
S-O interaction: spin relaxation and decoherence.
T1
: Spin relaxation time: includes processes where the electron spin is flipped
The spin relaxation rates
At low T:
T2
W­ ¯ / W¯­ = eD z / KT
1/T1 = W­ ¯ + W¯­
W­ ¯ ? W¯­
: Spin decoherence time (decay of the off-diagonal elements of the DM) it
destroys the information about the relative phase in a superposition of ­ and ¯
it accounts as well for the intrinsic spin decoherence which is present even with no
coupling to the leads
1
T2*
Closed system
Open system
GL + GR
1
1
1
g=
=
+ *+
gate operations for quantum computation
T2
2T1 T2
2
Coherent manipulations of electron spins:
must be performed faster than T2
The current through the drain contact barrier:
I L® R (t ) =
å
W
' r s , s (t )
s ,s s ,s
'
Where s states are such that the right dot is occupied and
s’ are states with one electron less
Results:
U1   L
N 2
Open system
UR=1.3, UL=1
a)
Appearance of interdot triplet
blocks pumping: spin blockade
unpolarized contacts
 6  ,  ˆ , 
r 4 = 0, ­ rˆ 0, ­
0, ­ ® ¯, ­ ® 0, ­ ¯ ® 0, ­
NO!
0, ­ ® ­ , ­ ® 0, ­ ­
Spin down polarized contacts
* Too high in
energy!
b) •Pumping of  spins is realized for
fully spin down polarized injection
from left lead :
0, ­ ® ¯, ­ ® 0, ­ ¯ ® 0, ­
t (1t )
Pauli principle is used for
filtering the electronic spin in
a QD
Spintronics in quantum dots
An AC gate voltage acts as spin pump in
a DQD with unpolarized leads
E
L
E
=U1
=U2
U2 -z
R
E = z
E =0
NO!!
(= z)
 E
(=U2) - E
 E
(=U2) - E =0
Spin pump follows the next sequence

U2 -z
 U2 R
< R
Spin-polarized pumping in a double quantum dot,
E. Cota, R. Aguado, C. E. Creffield and G. Platero,
Nanotechnology 14, 152 (2003)
Hamiltonian
ˆ
ˆL H
ˆR H
ˆ
H

H
DQD
QD
QD
LR
two levels in the right dot:
R
 ˆ† ˆ
Hˆ QD
   Ri
d Ri d Ri  U R ( nˆRi nˆRi   nˆR1 nˆR 2 ' )  JS1  S2
 ,i
i
 '
Including intradot triplet
state in the right dot:
S= 0
____
Dz b
S0 =
- ­ ¯-
De b
­ ¯ ­ ¯­
- ­-
- ¯-
- ­-
- ¯-
T+ = ­ ­ *
S=1
T- = ¯¯*
- ¯­ - ­ ¯T0 =
­ ¯* + ¯­ *
2
ET- = D e + 2D z + EST + U
ET0 = D e + D z + EST + U
2
r r
EST = JS1S2
ET+ = D e + EST + U
As a consequence of Hund’s rule
ES0 = D z + U
the intradot exchange J is
ferromagnetic: J<0
spin filter
­ ¯ ­ ¯­
S0 =
b)
Spin
down
2
AC
GR
AC
GL
GL
(¯­ , ­ )Û (­ , ¯­ )Þ (­ , ­ )Þ (¯­ , ­ )
current (¯­ , ­ )Û
GR
(­ , ¯­ )Þ (¯­ , ¯­ )Þ (¯­ , ­ )
c)
T+ = ­ ­ *
Spin
up
current
AC
GR
AC
GL
GL
(¯­ , ­ )Û (¯, ­ ­ )Þ (¯, ­ )Þ (¯­ , ­ )
GR
(¯­ , ­ )Û (¯, ­ ­ )Þ (¯­ , ­ ­ )Þ (¯­ , ­ )
current peaks also at
w
N
AC
= wAC / N
Absorption of N photons
An external B breaks the degeneracy for one and two-electron states, but
there is a degeneracy in the three electron sector for D ZL = D ZR
w = w­
AC
(¯­ , ­ )Û (¯, ­ ­ * )
spin up current
æ (­ ¯* + ¯­ * )ö÷
çç
÷
÷
(¯­ , ­ )Û çç­ ,
÷
÷
2
çè
ø
AC
The degeneracy is broken as
spin down current
D ZL ¹ D ZR
for D ZL » 0.6D ZR
P­ » 100 0 0
The width of the peaks changes in a non trivial way
as a function of w
WR = 2tLRJN(VAC / w)
J=-.2
G= 0.001
tLR=5 G
VAC = w
D ZL = D ZR
S0
D ZL , R = 0.0026
1-photon
D e = 0.4
UL=1., UR=1.3
AC
GR
(¯­ , ­ )Û (­ , ¯­ )Þ
GL
(­ , ­ )Þ (¯­ , ­ )
3-photons
S1
2-photons
T
The width of
the peaks is
determined by G
if G ? WR and
it is determined
by WR for:
G = WR
1-photon triplet
D ZL = D ZR
AC
(¯­ , ­ )Û (¯, ­ ­ * )
I­
(­ ¯, ­ )
æ (­ ¯* + ¯­ * )ö÷
çç
÷
÷
(¯­ , ­ )Û çç­ ,
÷
÷
2
çè
ø
AC
I¯
æ (­ ¯* + ¯­ * )÷
ö
çç
÷
® (­ ¯, ­ )
÷
(­ , ¯) ® (­ ¯, ¯)Û çç¯,
÷
÷
2
çè
ø
AC
I¯ + I­
w1 AC
D
L
Z
¹ D
R
Z
w1 AC ¹ w2 AC
(¯­ , ­ ) Û (¯, ­ ­ * )
*
* ö
w2 AC æ
­
¯
+
¯
­
(
)÷÷
ç
(¯­ , ­ ) Û ççç­ ,
çè
*
* ö
AC æ
­
¯
+
¯
­
(
)÷÷
çç
÷
(¯­ , ­ )Û çç­ ,
÷
÷
2
çè
ø
(­ ¯, ­ )
÷
÷
÷
ø
2
I¯
æ (­ ¯* + ¯­ * )÷
ö
çç
÷
÷
(­ , ¯) ® (­ ¯, ¯) Û çç¯,
÷
÷
2
çè
ø
AC
(­ ¯, ­ ¯) ® (­ ¯, ­ )
I¯
Excited states
D e = 0.35
VAC = 0.7
AC
VAC = 0.14
GR
GL
(¯­ , ­ )Û (¯, ­ ­ )Þ (¯, ­ )Þ (¯­ , ­ )
*
Spin-up current sensitive to
relaxation processes if
W ­ ¯³ GL
w
W­¯
GL
(¯, ­ )Þ (­ , ­ )Þ (¯­ , ­ )
w
Pumped current near resonance:
=0.6 for different relaxation rates.
Inset: FWHM of I as a function of the relaxation rate for strong ( black dots)
and weak (red squares) field intensity.
E. Cota et al., PRL, 107202 (2005)
Rabi frequency
æVAC ö
÷
WR = 2tLRJ n çç
÷
çè w ÷
ø
G = WR
High field
1/ 2
æg ö
FW = 2WR çç ÷
÷
÷
çè Gø
G» WR
1/ 2
æ gG ÷
ö
çç1 +
÷
çè W 2 ÷
÷
R ø
W­ ¯
Low field
FW = 2g = 2 / T2
Black dots: low VAC Red squares: high VAC
Measuring the width of the current peak as a function of
information on the spin decoherence time:
GL , GR
one gets direct
GL + GR
1
1
1
g=
=
+ *+
T2
2T1 T2
2
For
VAC / w = 1.
WR = 0.044
D ZL ¹ D ZR
G = WR
1/ 2
æg ö
FW = 2WR çç ÷
÷
÷
çè Gø
For
VAC / w = .2
WR = 0.001
for G ; WR
FW is linear with
g
: FW=2 g. Measuring FW at G ; WR
we obtain the decoherence time for the closed system:
g=
1
1
1
=
+ *+G
T2
2T1 T2
1/ 2
æ gG ÷
ö
çç1 +
÷
çè W 2 ÷
÷
R ø
1
1
1
=
+ *
T2
2T1 T2
Including PAT through the contact barriers
Ù
-
U (t ) = e
Unitary Transformation
i
h
t
òt0
Ù
()
H ac t ' dt '
¥
m H 'T (t ) n =
å
J n (a / 2)einwt m H T n
n= ­ ¥
¥
Wmn = 2p
å
n= ­ ¥
J n (a / 2)å
2
2
( g mn fl (wmn + nw - ml )dN
l
2
VAC
a=
w
m , Nn + 1
+ g mn (1 - fl (wmn + nw - ml ))dNm , Nn - 1 )
Pumped Current as a function of frequency
UR=1.3, UL=1.
2-photons
S0
1-photon
Current through
S0
double occupied
singlet S0 in the
right dot
Spin down current
PAT
ˆ
ˆ
­ ¯, ­ ‡ Interdot
ˆˆ ˆˆˆ† ­ , ¯­ ® ­ ¯, ­ ¯ ® ­ ¯, ­
Spin up current through PAT at the leads
PAT
ˆ
ˆ
­ ¯, ­ ¯ ¾ ¾ ¾ ¾
® ­ ¯, ¯ ‡ Interdot
ˆˆ ˆˆˆ† ¯, ­ ¯ ® ­ ¯, ­ ¯
PAT
Rightlead
G = WR
I max
G
;
2
VAC
a=
w
I= 0
J1 (a ) = 0
Comparison of I versus VAC including (non including)
PAT through the contacts.
D ZL = D ZR
wR ; U R
wL ; U L
wR
wL
wL
PAT
PAT
GR
(¯­ , ­ )Left® (¯, ­ )intÛerdot (¯­ , 0)Þ (¯­ , ­ ) I ­ < 0
contact
PAT
GR
GR
GL
(¯­ , 0)intÛerdot (­ , ¯)Þ (­ , ¯­ )Þ (­ , ­ )Þ (¯­ , ­ ) I ¯ > 0
wL ; U L
Tuning the AC parameters pure spin current
(not charge current) could be achieved
CONCLUSIONS
We propose a new scheme for
realizing both spin filtering and spin
pumping using ac-driven double
quantum dots coupled to unpolarized
leads.
•
•Spin polarization of the current can be
manipulated (including fully reversing)
by tuning the frequency of the ac field.
• The width in frequency of spin-up
pumped current gives information on
the relaxation and decoherence times.
•PAT through the contacts limits the
spin filter effect only at high VAC .
Future work
Cotunnel including PAT in the contacts
Analysis of the relaxation and
decoherence times T1 and T2 at finite T.
EARLY STAGE RESEARCHER MC RTN POSITION
Applications are encouraged for a two-year position as an “early stage
researcher” in the Madrid node of this Marie Curie Research and Training
Network (RTNNANO). The successful candidate will initiate his/her contract
shortly before completion of Ph. D., afterwards continuing in Madrid as a postdoc.
Starting date: between January 1 and (ideally) April 1, 2006.
Research interests in the Madrid node include electron entanglement and
decoherence, spintronics, electron and heat AC transport, and NS
transport.
Interested applicants should contact one of the following:
Francisco Guinea (ICMM-CSIC)
Gloria Platero (ICMM-CSIC)
Fernando Sols (Universidad Complutense de Madrid) (contractor)
Closing date: November 30, 2005.
T1=2.58 ms
at B=0.02T