New hints from theory for pumping spin currents in quantum circuits Michele Cini Dipartimento di...

New hints from theory for pumping spin currents in

quantum circuitsMichele Cini

Dipartimento di Fisica, Universita’ di Roma Tor Vergata and

Laboratori Nazionali di Frascati, INFN

Advanced many-body and statistical methods in mesoscopic systems II

Brasov, Romania, September 1, 2014

†wires L R h n, n+1,

,

H =H +H =t c c +h.c.= standard tight-binding

with no bias.n

2

wires ring ring-wiresH =H +H +H

Model :Michele Cini and Enrico Perfetto, PRB 84 245201 (2011); Michele Cini and Stefano Bellucci, J. Phys.: Condens. Matter 26, 145301 (2014)Michele Cini and Stefano Bellucci, Eur. Phys. B 14 , 87, 106 (2014)


3

00 0

Flux through the ring= generates complex hopping:

.2 2

t t e , = , .

If flux depends on time,

Faraday law E.

ringih h

ring ring ring

ring ring

Adre hc

fluxonN N N c e

d e d dc dr

dt cN dt dt

bond

dV

dt

Both

( ) in plane f

( ) a

ield (no flux)

( ) interact with spi snd n .B t

B t

B t

‰

‰

SO spin-orbit parameter

where, with the identification

Ring Hamiltonian

( ),

1

(

of ring site with si

) ( exp exp[ ]( )

te 1,

ring ringi ring

ring

ring ring iring

SO

H H i

N

H i ct

t i iN

†1, ,

† †, , , ,

5

. .) ( )

( )( ),

5.79375*

( )

10

ˆ ˆ( ) .

i B z

B i i i i

B

z i i

c h c i

B t c c c c

eV

Tesla

t

i n n

B

‰

Advanced many-body and statistical methods in mesoscopic systems II 4

Method for numerical simulations :

5

Transport theory: Partition-free approach-M. Cini, PRB 22,5887 (1980); Phys. Rev. B89,239902 (2014).

*, 1,

,

2Im ( ) ( ) , where n site,

1q= equilibrium eigenstate of H(t=0), f Fermi function

1

retarded functions i ( ) ( ) ( )

q

B

r rhn q n q n q

q

q

K T

rn q

etJ f G t G t

e

G t n U t q t

Quantum evolution is discretized (time slicing technique)

and many-electron determinantal wave functions updated

by computing new eigenstates and overlaps with old ones.


Definition of Laterally connected ring

Symmetrically connected rings: no magnetic moment

current in wires chirality in ring magnetic moment

The ring is tangent to the circuit

6

Physical model with spin and spin-orbit interaction

7

We also need the adjacency graph for spin-orbitals

spin-up electron states

spin-down electron states

Star =Dirac monopole mimicks the spin-orbit interaction

The vertical bonds represent the effects of in plane field B (t)


Spinless case

Advanced many-body and statistical methods in mesoscopic systems II8

9

Flux ( )t

Tight-binding model with time-dependent magnetic flux

Simplest case: spinless model with B perpendicular to ring

M.Cini and E. Perfetto, Phys. Rev. B 84, 245201 (2011)

( ) pumping! vary the flux and current flows in circuit!

Generally the ring remains excited after the flux is inserted.

t

10

Flux ( )t

Simplest case: spinless model with B perpendicular to ring

We may avoid leaving the ring excited by letting it swallow integer fluxons. Then H is the same at beginning and at the end.

Finding: the only effect is pumping!

Bchirality direction of pumping

11

pumping by an hexagonal ring – insertion of 6 fluxons

Pumped charge staircase

Time dependent flux can be used to pump charge. What kind of pumping is that?


12

Introduced by Thouless (1983):

for a 1d spatially periodic system and time-periodic adiabatic system with H(x+a)=H(x) and H(t+T)=H(t).

waves : ( , , ) ( , , )iqxn nBloch q x t e u q x t

2( )Hamiltonian for periodic function reads: ( , , ) ( , )

2n

p qu H q x t V x t

m

Quantum Pumping: several kinds are known.

0adiabatic current: charge

2 2

Tn nqt n qt

n BZ BZ

dq dqj Q dt

where [ ] Berry curvaturen n n n nqt

u u u ui

q t t q

q

t

Flux of curvature =Berry phase

13

q

tFlux of curvature =Berry phase

1) Berry phase needs at least 2 parameters

Strong implications:

( )

( )

( ,0) ( , ) ( )

(0, ) (2 , ) ( )

q

t

i q

n n

i tn n

u q u q T e same physical state

u t u t e same physical state

1[ 1 0 1 0 ]

2n t t q qQ

Single-valued y Qn=integer (Chern number)

2) Charge at each cycle is quantized

14

Other case:Quantum Pumping in linear systems

P.W Brower (1998) has shown that in linear systems one gets two-parameter pumping

Cohen (PRB 2003) established properties of pumping in linear response theory

If the response is linear, one needs at least two parameters R1 and R2

H=H(R1,R2)

1 1 2 2

Pumped charge is quantized in adiabatic cycle:

ˆ( ) , where [ ( ), ] (Kubo)

Chern number

Berry i i

cycle

iQ Jdt G dR G dR B dS G J t R

Q

15

Charge is not quantized- not an adiabatic result (if pulse time grows charge decreases).

We got 1-parameter pumping (only flux varies)

Classically, magnetic moments of rings would be linear with applied bias. Because of quantum effects, the magnetic response of quantum rings is cubic (Cini, Perfetto and Stefanucci, Phys. Rev. 2010).

Present case: Quantum Pumping from rings

Avron,Raveh and Zur (Rev. Mod. Phys.) analyzed the adiabatic response of circuits with rings in adiabatic approximation and found no pumping.

The one-parameter pumping is in line with that.

16

Nonadiabatic pumping: inserting a flux quantum in the ring in a time of the order of 10 h/ hopping matrix element we shift about an electron.

characteristic hopping time of the system= h/hopping matrix element

17

Equivalent bond concept

The effect of magnetized

If flux depends on time

ring on the outside

,

Faraday

circuit is comparable to a

phase drop of order of a

law E.

few times α in

bondring ring

d e d d dc dr V

dt cN dt dt dt

the equivalent renormalized bond

. . a potential drop of order .bondVi e

Introducing spin and magnetic interactions


18

19

Computer experiment : Ring in (x,y) plane, Rotating B( ) in (y,z) planet

1= (in units)

12.5ht

250 atom ring Half filling (EF=0)

=0 (No spin-orbit interaction)SO

Rotating B experiment

Pumped charge current on first bond of (left and right)

leads:

it is even i.e. same at right and left of the ring.

5

J=1 with t 1 means

4*10 Ampere

h eV


20

it is odd i.e. opposite at left and right of the ring

Pumped spin current on first bond

of right lead

Besides charge, some spin is pumped too

Pumped spin current on first bond

of left lead


5

J=1 with t 1 means

4*10 Ampere

h eV

21

in a fixed field B

Pumps alternating charge and spin currents in the rotating frame.

.

Rotating ring experiment


charge spin

22

2 2, ,

† †, 1, , 1,

Current in fixed frame in direction of field B

in terms of the current in the ring frame

j (n)=Cos ( ) ( ) ( )Cos( ) j2 2 2 2

(c c c )

m m m sf

sf m m m m

j Sin j Sin

t

j

Rotating ring experiment

5J=1 with t 1 means 4*10 Ampereh eV

taking the spin polarization ax

Fixed frame

spin current J -J : it is d.c.

and 10% of the charge current. The spin-orbit interaction i

is along

s not inv l ed

B

v

o

23

24

Pumped spin current with axis

fixed in rotating frame, on first bond of left lead

z

Pumped spin current B

fixed in intertial frame, on first bond of left lead

0.05SO

Pumped spin current B

fixed in intertial frame, on first bond of left lead

0.1SO

Effects of the spin-orbit interaction


How to make a magnetic current

Spin-up electrons move to left and spin down electrons to right.

A pure spin current does not move charge, but magnetization. It is even (same in both wires)

Magnetic field B (t)

in plane of ring with even number of sites

at half filling

25

Spin up

Spin down

Same spin current pumped in both wires (it is even).

710 (0.1 pA)

maxHexagonal ring , B

1

50

SO

T

hopping

5hopping

ttime in units of

h

(If t 1 , 1 3.9*10 Ampere).eV J

26

for spin = - for spin J J

B (Tesla)

hopping

t( )h

t

No charge current

810 0.01 pA

B (Tesla)

maxHexagonal ring , B 50T

hopping

t( )h

t for spin = - for spin J J

Same current pumped in both wires

hopping

t( )h

t

27

No charge current

The stationary pure spin current produces no magnetic field; the electric field has a special pattern (seen in a plase orthogonal to the wire):

wireelectric field

Requres: bipartite lattice+half filling

Parity P x -x

broken by spin-orbit interaction

Spin Reflection :

broken by spin-orbit interaction

holds at any time,

but charge current is g

Symmetry:

enerally a

P

ll

owed

Symmetry:

Advanced many-body and statistical methods in mesoscopic systems II29

The analytical theory of magnetic current generation

Model is bipartite if ring has even number of sites

the adjacency graph for spin-orbitals is also bipartite


Vertical bonds due to in plane B

Ground state properties of our model at half filling

Theorem: each site is exactly half filled at any time

This arises from the fact that the system is bipartite.

Bipartite system sign change of red orbitals changes the sign of H .

But sign change of red orbitals is a gauge.

The spectrum is symmetric on each sand ite E E

At half filling in the ground state the lower half of the LDOS is occupied on each site

1At half filling each site has occupancy .

2

Ground state properties of our model at half filling

Theorem: each site is exactly half filled.

No time-dependent flux in ring,

It follows that the charge is totally pinned on each site.

32

But how does the spin current arise?

Recall the equivalent bond concept of the spinless model with dynamical flux

B(t) pumps spin current because spin-up electrons can do a trip to the down-spin sector where they gain opposite phases. This works like a time-dependent and spin dependent phase.

B( ) in planetB( ) in planet

The mechanism for spin current generation

33

2

Short spike of B(t) with intensity B which lasts time T

sin( )( )h Bspin SO

t BTJ

spin-up ring states =effective bond

spin-down ring states =effective bond

pure spin current, the same on both wires

6pA

ring with 100 sides , sudden switching B 100 at absolute 0T

4pA

ring with 100 sides , sudden switching B 100 at Room TemperatureT

The effect is robust!

What happens at finite temperatures?



pure spin current, the same on both wires

34Advanced many-body and statistical methods in mesoscopic systems II

What happens if we depart from half filling?

F ring with 100 sides and U=0, sudden switching B 100 at room T and E =0.02thoppingT

Charge current is small

8pA

0.2pA

0.8pA

8pAhopping

t=8h

t

hopping

t=8h

t

Spin current Charge current

35

Thought experiment: (M. Cini, submitted for publication)

1) Store magnetization in reservoirs

2) Isolate magnetized reservoirs

3) Connect magnetized reservoirs with wire: a spin current is generated


37

spin current spin current

2 0 4 0 6 0 8 0 1 0 0t

2468

1 01 21 4

B T e s la


38

2 0 4 0 6 0 8 0 1 0 0t

4 .0 1

4 .0 2

4 .0 3

4 .0 4n U p

Spin Up population along zon left cube

2 0 4 0 6 0 8 0 1 0 0t

3 .9 7

3 .9 8

3 .9 9

4 .0 0n D o w n

Spin Down population along zon left cube

2 0 4 0 6 0 8 0 1 0 0t

3 .9 7

3 .9 8

3 .9 9

4 .0 0n U p

Spin Up population along zon right cube

2 0 4 0 6 0 8 0 1 0 0t

4 .0 1

4 .0 2

4 .0 3

4 .0 4n D o w n

Spin Down population along zon right cube


39

2 0 4 0 6 0 8 0 1 0 0 0 .0 0 0 5

0 .0 0 0 5

0 .0 0 1 0Spin Up along zcurrent between Cubes

2 0 4 0 6 0 8 0 1 0 0

0 .0 0 1 0

0 .0 0 0 5

0 .0 0 0 5

Spin Down along zcurrent between Cubes

Delay proportional to length of connection

Oscillatorty polarization


40

30 atom ring, Cubes replaced by 4-atom rings connected by 200 atom leads t1=30 t2=35 Currents observed at centre of storage-ring connection External current is purely spin

40 80t

20

40

B T e s la

20 40 60 80 100t

1. 10 7

1. 10 7

J

40 80t

J external

The frequency of the external oscillations is reduced by dividing by 3 the external wire band width (here internal wires are 50 atoms long, external wires 100 atoms long)


41

30 atom ring, Cubes replaced by 4-atom rings connected by 200 atom leads t1=30 t2=35 Currents observed at centre of storage-ring connection Note delay- External current is purely spin

40 80t

20

40

B T e s la

20 40 60 80 100t

1. 10 7

1. 10 7

J

40 80t

J external

The frequency of the external oscillations is reduced by dividing by 3 the external wire band width (here internal wires are 50 atoms long, external wires 100 atoms long)


Temperature dependence is mild!

42

Conclusions

A time-dependent magnetic field in the plane of the bipartite ring at half filling pumps a pure spin current (= magnetic current) in the external circuit, driven by the relativistic spin-orbit interaction.

Ballistic rings asymmetrically connected to wires and pierced by a time-dependent magnetic field can be used to pump charge. Semiclassical approximations are qualitatively wrong. The phenomenon is purely quantum and nonlinear (violates Brower theorem).

Rotating magnetic fields pump spin-polarized currents- No spin-orbit interaction is involved.

Rotating rings in fixed magnetic fields pump spin- polarized currents with or without the effects of spin-orbit coupling

This current keeps its polarization totally at room temperature and partially if the carrier concentration deviates from half filling. It can be stored as magnetization and later released in a controllable way.


Thank you for your attention!

New hints from theory for pumping spin currents in quantum circuits Michele Cini Dipartimento di...

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