Bhaskaran Muralidharan Dept. of Electrical Engineering,

Indian institute of technology Bombay HRI Allahabad29/02/2016

Currents through Quantum dots

Single Spins: An exciting frontier

Read-Out of spinsElzerman et.al., (2004)

Initailization of spins

Ono et.al., Science, (2002)Spin decoherenceKoppens et.al., Science, (2005)

Coherent ManipulationKoppens et.al., Nature, (2006)

http://qt.tn.tudelft.nl/research/spinqubits/

Preview

3

Dual Resonance Un-identical baths

Fin Structure

EXPERIMENT

THEORY THEORY

Lµ

Rµ

a

ab

b

γ γLµ

Rµ

a

ab

b

γ γ

aN bN aN bN

ba NN = ba NN ≠

S Buddhiraju and B Muralidharan, JPCM, 26, 485302, (2014)

K. Ono et al. PRL 04

Outline

Spin Correlation effects in

Quantum-dot transport

Spin Blockade Transport: Multiple NDR

K. Ono et al. Science 02

Spin Blockade Transport: Role of Non-equilibrium Scattering Processes Hysteretic behavior

K. Ono et al. PRL 04

x

IntroducingFock SpaceTransport

Regimes of Transport

H + U

€

Σ1

€

Σ2

€

µ1

€

µ2

€

Σs

U-> Self Consistent field

RµLµ

SCFU U n= < >

SCF Regime Works for

€

Γ ≥U

2^N many electron levels

00

11

0110

Fock space approach CWJ Beenakker (1991)

CB Regime Works for

€

Γ <<U

NN × NN 22 ×

Self Consistent Field

€

Σ1

€

Σ2

€

µ1

€

µ2

€

Σs

NEGF-SCF

H + U Single particle view point:

RµLµ εLγ

�Rγ

�

><− nUε

U

Let us try to introduce the Fock space View Point:

A small pocket of transport problems?Coulomb Blockade effects

Spin Correlation effects

x

Spin Correlation coupled to Hot-scatterers

Park et al, nature 02

Ono et al, Science 02

Bi-stability

Fock space picture

ε

Lµ RµRγ

�Lγ

�

ε

0

Rγ

�Lγ

�

RµLµU,ε

N=1

N=0

0N=0

ε ε N=1

U+ε2 N=2

How Coulomb Blockade transport works

ε ε

0

Rµ

εLµ

Rµε

Lµ

U+ε

Rµε

Lµ

U+ε

NEGF-SCF

CB

Correlations matter even for a minimal model!

0N =

1N =

2N =U+ε2

Muralidharan et al., PRB06

Generalized Viewpoint

RµLµ

NN ×

NN 22 ×nN =

nN =

1+= nN

Lµ

Rµ1trε

2trε

2trε1trε

Given a bias point a set of Fock states are probabilistically distributed! !These may be viewed as transition Energies in the one-particle picture !

}{ iP

Fock space master equations

{ , }N i

{ 1, }N j−

{ 1, }N j+

,, 1, , 1, , 1, 0N i

N i N j N i N j N i N jj j

dPR P R P

dt → ± ± → ±+ − =∑ ∑

,{ }N iPFock space probability distribution

,, 1, , 1, , 1,

N iN i N j N i N j N i N j

j j

dPR P R P S

dt → ± ± → ±+ − =∑ ∑

{ , }N i { , }N j

With scattering

Part II

Spin Correlation effectsin

Quantum-dot transport

Spin Blockade Transport:Multiple NDR

K. Ono et al. Science 02

Spin Blockade Transport:Role of Non-equilibrium

Scattering ProcessesHysteretic behavior

K. Ono et al. PRL 04

x

Coulomb Blockade v/s Spin Blockade

Rµε

LµU+ε

Coulomb Blockade itself does not differentiate the spin degree

Lµε↑

ε↓

Rµ

Spin Degree of freedom results in zero current

LµRµ

ε↑

ε↓

X

Finite Current Flows

LµRµ

ε↑

ε↓

X

Uε↓+

X

Spin Degree of freedom + Coulomb Blockade Spin Blockade!

Spin Blockade regime in Double Quantum Dots:

Lµ

2εRµ

1ε

No Ferromagnetic Contacts! What is the Blockade mechanism?

Current Blockade!

But why does current flow at all?

What is the mechanism for NDR

NDR: Conventional Viewpoint

I

V

Lµ Rµ LµRµ

Lµ

Rµx

a)

b)

L. Esaki, RTD Phenomenon 1972

No band edges in our case! What makes Spin blockade NDR novel?

NEED FOCK SPACE VIEWPOINT

NDR due to “dark states”

* *C A AC CR M fγ→ =

2CA RDM C H A=

C

RµLµ

RµLµ

A0N n=

0 1N n= +

Transition Rates reflect on the symmetry properties CA

ACR τ

1~→

B

BAε BAε

CAεI

V

a)

b)

NDR from the dark state model

Muralidharan and Datta, PRB07

LAB

RBA

RCA τττ +>

LAB

RBA

RCA τττ +≤

RAB

LBA

LCA τττ +≤

Spin Blockade regime in Double Quantum Dots:

Lµ

2εRµ

1ε

No Ferromagnetic Contacts! What is the Blockade mechanism?

222 U+ε

Lµ

2ε

Rµ

1ε

X

Explains Current Blockade!

But why does current flow at all?

Mechanism for NDR

“Dark” State model: Double Quantum Dots:

Lµ

2ε

Rµ

1ε

Lµ

2ε

Rµ

1ε

Lµ

2ε

Rµ

1ε

BN=1

TS N=2

RµLµ

SBεLµ

RµSBε

TBε

Under Special Conditions Triplet State Can be Dark!

Results

TheoryExperiment

Muralidharan et.al., JCEL 08Muralidharan and Datta, PRB 07

Ono et. al., science 02

Pauli Blockade: A Broader Perspective

Lµ

2ε

Rµ

1ε

Off state

Lµ

2ε

Rµ

1ε

X

Permits manipulation of single Electron spin detected by a current Measurement !Host Nuclei can also assist!

Part III

Spin Correlation effects in

Quantum-dot transport

Spin Blockade Transport: Multiple NDR

K. Ono et al. Science 02

Spin Blockade Transport: Role of Non-equilibrium Scattering Processes Hysteretic behavior

K. Ono et al. PRL 04

x

Preview

24

Dual Resonance Un-identical baths

Fin Structure

EXPERIMENT

THEORY THEORY

Lµ

Rµ

a

ab

b

γ γLµ

Rµ

a

ab

b

γ γ

aN bN aN bN

ba NN = ba NN ≠

S Buddhiraju and B Muralidharan, JPCM, 26, 485302, (2014)

K. Ono et al. PRL 04

Spin-Blockade Toy Model

25

❑ Single QD with single nuclear bath

❑ Spin-down polarized right contact

❑ Blockade lifted by Spin-Flip transitions

N

Lµ

Rµ

γ γ+ε

−ε

0

−+−+ HFH

Hyperfine mediation

Spin-flip at the cost of nuclear-flop

Apply B field externally

appB

26

+−

0

− +

0

N

Analysis: Fermi Golden rule

❑ Z-component – Mean Field Approximation

FI = average nuclear z-polarization

!

!❑ X-Y component – Fermi’s Golden

Rule to give spin-flip rate

27

Electron dynamics:

Nuclear spin dynamics: 

 

 

 

No Overhauser field

❑ Energy of T ❑ Decreases under applied

magnetic field

❑ No effect on energy of S

28

S T

B

With Overhauser field

29

Overhauser field: !It can either oppose the resonance or aid it via negative or positive feedback !Feedback: Origin of Hysteresis !!

S T

B appB ovB

Including Overhauser field

30

❑ Negative feedback from gµB to JeffFI during forward sweep❑pseudo-linear build-up of FI

❑Hysteresis: resonance breaking

❑ Positive feedback during reverse sweep – rapid rise of FI

S T

B appB ovB

Recap

31

Dual Resonance Un-identical baths

Fin Structure

EXPERIMENT

THEORY THEORY

S Buddhiraju and B Muralidharan, JPCM, 26, 485302, (2014)

K. Ono et al. PRL 04

Two Dots, Two Nuclear Baths

32

•  

Lµ

Rµ

t

a

ab

b

γ γaa /ε bb /εaaU abU bbU

aN bN

Double Resonance due to Two-Electron states

❑ Two-electron states at B = 0 ❑ Three S = 1 states (T) ❑ Three S = 0 states (S)

❑ Triplets are blocking states ❑ TWO RESONANCES:

❑ T+1 – S1 ❑ T-1 – S2

❑ T+ and T- move in opposite directions under B and FI.

33

appB0T1T+

1T−

1S

2S

1/ 2B+1/ 2B−

1/ 2A−1/ 2A+

N=1 N=2

34

Double Resonance:

Electronic Structure

0S

1S

bbaababa 1/01/01/01/0 δξβα +++

( )S1,1 ( ) ( )SS 2,0/0,2

1/01/0 βα ≡ Identical Nuclear baths

Unlike Nuclear Baths1/01/0 βα ≠

No Difference Overhauser field

Difference Overhauser field

Lµ

Rµ

t

a

ab

b

γ γaa /ε bb /εaaU abU bbU

aN bN

35

Electronic Structure: Continued

0S

1S

1/01/0 βα ≡ Identical Nuclear baths

Unlike Nuclear Baths1/01/0 βα ≠

No Difference Overhauser field

Difference Overhauser field

baT

abT

babaT

bbaababaS

=

=

+=

+++=

−

+

1

1

0

1/01/01/01/0

2/)(

δξβα

Two-Dot Two-Bath Hamiltonian

❑ Z-component – Mean Field Approximation

F = average nuclear z-polarization

!

!❑ X-Y component – Fermi’s Golden

Rule to give spin-flip rate

36

Electron dynamics:

Nuclear spin dynamics: 

 

 

 

Fermi’s Golden rule: One-bath variable vs

two-bath variables

37

 

 

Lµ

Rµ

a

ab

b

γ γ

aN bN

Lµ

Rµ

a

ab

b

γ γ

N

Identical Nuclear Baths & Near-Simultaneous Resonances

38

❖ Novelty of two nuclear baths: Matrix elements between singlet and triplet elements non-zero simply due to incoherent addition between the two baths. Difference Overhauser field NOT required !!

❖ Two dragged resonances ❖ Each contributes a triangular

current traces ❖ S u p e r p o s i t i o n : f l a t t o p p e d

hysteretic behavior

Superposition

39

a)

2/1b

0

1

1+T1−T

1=N

2=N

2

Un-identical Nuclear Baths: Difference Overhauser field

40

Fin structure at the two ends of the hysteretic sweep as noted in the experiments

S Buddhiraju and B Muralidharan, JPCM, 26, 485302, (2014)

bbaababaTS 1/01/01/01/0/ δξβα +++=

Results

41

Ideal Dual Resonance Un-identical baths

Fin Structure

EXPERIMENT

THEORY THEORY

S Buddhiraju and B Muralidharan, JPCM, 26, 485302, (2014)

Summary❑ Key Points: ❖ One Dot toy example

❖ Hyperfine interaction Hamiltonian ❖ Dragged resonance due to Overhauser Field ❖ Triangular current trace

❖ Double Dot, Two-Bath:

42S Buddhiraju and B Muralidharan, JPCM, 26, 485302, (2014)

yz

x

RL BBB���

+=

S�

Density matrix formalism: Spin dynamics

( )

( )

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡×+

⋅−−=

⎟⎠

⎞⎜⎝

⎛−

−+

−−

Γ=

++−

Γ=

⎥⎥⎦

⎤

⎢⎢⎣

⎡×+

⋅−−=

⎟⎠

⎞⎜⎝

⎛ +=⎟

⎠

⎞⎜⎝

⎛ −=⎟

⎠

⎞⎜⎝

⎛ −=

∫

∑=

αα

ααααααα

αααααα

αα

αα

αα

α

αααααα

τ

εεπ

εετ

τ

ρρρρρρ

BSmSmpS

mpJq

J

EEf

UEEf

dEm

pB

Uff

BSmSmpS

mpJqdt

Sd

SiSS

r

qS

r

RL r

q

xyz

����

��

�

�

����

�

���

,

2

'

,

, ,

2

211221122211

ˆˆˆ2

)(1)(ˆ

,)()(1

,ˆˆˆ

2

22,

22,

22

ε

U+ε2U,ε

TkB<<Γ�

2

3

1

0

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡×+

⋅−−= α

α

ααααααα τ

BSmSmpSmpJq

Jr

qS��

����

,

2 ˆˆˆ

2

Injection Relaxation Precession

( ) ,ˆˆˆ

2, ,

2

∑= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡×+

⋅−−=

RL r

q BSmSmpSmpJqdt

Sdα

αα

αααααα τ

����

��

QD Spin dynamics v/s STT

LL T,µRR T,µε

U+εLL mp �, RR mp �,

∫ ⎟⎠

⎞⎜⎝

⎛−

−+

−−

Γ=

εεπαααα

αα EEf

UEEfdEmpB )(1)(ˆ '

�

�

Precession Damping

Injection Relaxation Precession

LL mp �, RR mp �,

yz

x

RL BBB���

+=

S�

Thermoelectrically induced spin precession!Pure spin current due to spin precession!Y Tserkovnyak et.al., PRL (2002)

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡×+

⋅−−= α

α

ααααααα τ

BSmSmpSmpJq

Jr

qS��

����

,

2 ˆˆˆ

2

Injection Relaxation Precession

Spin precession-spin current

B Muralidharan and M Grifoni, PRB (2013)

Spin batteries/Maxwell’s demon

Heat Engines with demons

ChannelSource Drain

111 NE µ− 222 NE µ−

Out of Equilibrium System “Demon”

0E

11 ,NE 22 ,NE

0

00

02

222

1

111

021

21

≤Δ−−

+−

=++

=+

STNE

TNEEEE

NN

µµ0

00

0

0

2

222

1

111

021

21

≤+−

+−

=++

=+

TE

TNE

TNEEEE

NN

µµ

ChannelSource Drain

111 NE µ− 222 NE µ−

Reservoir/Bath at T0

0E

11 ,NE 22 ,NE

S=klnW

Examples of nano-device “demons”

ChannelSource Drain

111 NE µ− 222 NE µ−

Out of Equilibrium System

0E

11 ,NE 22 ,NE

nano-device “demons”+ info battery

00

≤Δ−Δ=Δ

≥Δ

STEFStot

0

00

02

222

1

111

021

21

≤Δ−−

+−

=++

=+

STNE

TNEEEE

NN

µµ

v/s

ENERGY INFORMATION

nano-device “demons”+ info battery

21 SSJH��⋅=

D U

du

u+D d+U

nano-device “demons”+ info battery

State of the Demons

Current Stops to flow eventually!2ln

2ln0

NkSNkS

S

f

i

=Δ

=

=

2µ1µ

2ln2ln

2ln0

NkTWNkSNkS

S

f

i

≤Δ

=Δ

=

=

Where does the energy come from?!

Connection with Maxwell’s Demon

Reservoirsat T1 and T2

W

Qin QoutHT CT

Does it no violate any known laws orCommon sense?!If not, what is the catch???!Demon exorcism by Szilard (1929)!

Connection with Maxwell’s demon

00

≤Δ−Δ=Δ

≥Δ

STEFStot

00

≤Δ−Δ=Δ

≥Δ

STEFStot

Energy Information

Connection with Maxwell’s demon/Landauer principle

2µ1µ

2ln2ln

2ln

2lnln0

NkTENkTWNkS

NkWkSS

Erase

f

i

≥

≤Δ

=Δ

==

=

µΔµΔ−=Δ ∫

final

initialudNW

u+D d+U

Connection with Maxwell’s demon/Landauer principleState of the Demons

Discharging Randomizing the bit

Erasurecharging

2ln2ln

2ln

NkTENkTWNkS

Erase ≥

≤Δ

=Δ

CHARGE Spin

Energy

Nano-spin-energy group

AcknowledgmentsSiddharth Buddhiraju (student IITB) Prof. Supriyo Datta (Purdue University) Prof. Milena Grifoni (University of Regensburg, Germany)

THANK YOU FOR YOUR

ATTENTION!