Coupled electromechanical effects in the electronic properties of nanostructures

Sanjay Prabhakar and Roderick Melnik The MS2Interdiciplinary Research Institute

Wilfrid Laurier University, Waterloo, Ontario, Canada

Prof. Luis Bonilla Gregorio Millan Institute, Spain

www.ms2discovery.wlu.cawww.m2netlab.wlu.ca

• Coupled effects in nanostructures of different materials

• g-factor and spin relaxation in III-V semiconductor QDs

• Berry phase in Quantum Dots

• Conclusions

Outline

Theoretical model:

lmikiklms CU εε21

=

spninlmilmi

nniklmiklmik

PVeDVeC+∂−=

∂+=εε

εσˆ

We solve Navier and Maxwell equations

0

0

=∂

=∂

ii

ijj

D

σ

The total electromechanical energy density for nanostructures (nanowires,QDs,supperlattices)

Prabhakar, Melnik, Bonilla JAP (2013)

AlN/GaN QDs

Electron

Hole

Motivation: Experimental observation of ripples seen in graphene sheet

•Thermal fluctuation should induce smaller ripple waves which does not match with the experimental observation Nature Nanotechnology (2009)

•Ripples can be induced as a consequence of adsorbed OH molecules in random sites.

Thompson-Flagg et al., EPL (2009)

Ripples produced by edge effects alone in graphene sheet, 10nmx10nm.

Ripples produced by 20% coverage of OH in graphene sheet, 10nmx10nm.

•Theoretical model has been developed by Bonilla and Carpio PRB (2012)

Theoretical model:

lmikiklms CU εε21

=

tF kextikk /=∂ σ

This provides two coupled Navierequations as:

( ) ( ) tFuCCuCC xextyxyx /6612266211 =++∂+∂( ) ( ) tFuCCuCC yextxyyx /6612211266 =++∂+∂

( )kekext qxqF cosτ=

We solve these two coupled equations to investigate the ripple waves in graphene.

The total elastic energy density for thetwo dimensional graphene sheet

Future work

Prabhakar, Melnik, Bonilla PRB (2014)

Prabhakar, Melnik, Bonilla PRB (2016)

Influence of ripple waves in the Band diagram of graphene: (7)

( )( )

( )( )⎟

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−−−−

=

+

−

−

+

y,xUAPAPy,xU

y,xUAPAPy,xU

H

0000

0000

eApPand,PPP yx +=±=±

•Our goal is to treat strain tensor components as a pseudomorphic vector potential.

•Then, we investigate the influence of electromechanical effects on the band structure of single and bilayer graphene.

( )yyxxF PPvH σσ +=Hamiltonian for single and bilayer graphene can be written as

( ) aA xxyyxy /0,,2 βεεε −=

(8)Band diagram of graphene at Dirac point

0 200 400 6006789

14.515.015.516.0

Temperature [oK]

Ener

gy (m

eV)

•Level crossing points for zigzag states can be observed.

•Such level crossings are absent in the states formed at the center of the graphene sheet due to the presence of three-fold symmetry.

Graphene Quantum Dots:Prabhakar, Melnik, Bonilla

PRB (2014)

Influence of ripple waves in the band diagram of graphene nanoribbons

( ) ( )( )( )

( )L

BqxB

aA

BgeApeApvH

es

xxyyxy

zsByyyxxxF

τ

βεεε

σμσσ

∝=∇=

−=

+−+−=

00

0

,cosBAx

/0,,221

-0.5 0.0 0.5-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

(c)

0.0 0.3 0.6 0.9 1.20.0

0.8

1.6

2.4

-0.5 0.0 0.51E-3

0.01

0.1

1

10

100Sublattice A L=3nm L=3.8nm L=5nm

ky[nm

-1]

|Δε|

(eV

)

-0.5 0.0 0.50.311

0.312

0.313

0.314

ky[nm-1]

Ener

gy (e

V) Sublattice A

Sublattice B

p-spin down

p-spin up

Δε

(eV)

1/L [nm-1](b)

(a) ky[nm-1]

Ener

gy (e

V)

Note that

( ) ( )L−+−−≈−= 24/2/1ˆcos 4200 qqBxqBBs

Prabhakar, Melnik, BonillaPRB (2016)

•Pseudo-spin splitting energy is in the range of meV and can not be neglected for smaller ribbon width.

•Sublattices A and B have different energy spectra.

Pseudo-spin life time caused by in-plane phonon modes Prabhakar, Melnik, Bonilla

PRB (2016)

( ) ..||2

.

,chbek

Airu q

rikkj

tl kr

kph +Ξ= ∑

=α

α α

α

ωρh

The interaction of electron and in-plane phonon modes is written as

( )( )∫∑ +−>

Schematics of spin single electron transistors (SET): QDs Bandyopadhyay et al.,

PRB (2000)

Prabhakar, RaynoldsPRB (2009)

DRz Η+Η+Η+Η=Η 0( ) Β+++Ρ=Η Β zbyax σμω 02220

2

0 g21m

21

m2

r

Hamiltonian of quantum dots in III-V semiconductors

•The lack of structural inversion asymmetry leads to Rashba spin-orbit coupling

( )xyyxRR Εe Ρ−Ρ=Η σσγh

( )yyxxDD meE Ρ+Ρ−⎟⎠⎞

⎜⎝⎛=Η σσγ

3/2

2

2hh

•Bulk inversion asymmetry leads to Dresselhaus spin-orbit coupling

( )Β

−=

Β

+−

μεε 2/1,0,02/1,0,0g

0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 1 7 5-0 .4-0 .20 .00 .20 .40 .60 .81 .0

g /

g 0

Q D s ra d iu s (n m )

E 1 e 4 v c m E 1 e 5 v c m E 2 e 5 v c m E 3 e 5 v c m E 4 e 5 v c m E 5 v c m E 6 v c m E 7 e 5 v c m E 8 e 5 v c m E 9 e 5 v c m E 1 e 6 v c m

1TB =•Wave function overlap: electric fields can “move” the wave function to sample different materials (e.g. GaAs has g = -0.44; AlGaAs has g = +0.4)

•Better understanding of g-factor is the key parameter for the design of QD devices•Electrical control of “g” (physical mechanisms)

( )Β

−=

Β

+−

μεε 2/1,0,02/1,0,0g

g-factor changes its sign

Theory: Prabhakar and RaynoldsPRB 79, 195307 (2009)

Experiment: Jiang and YablonovitchPRB 64, 041307 (2001)

Spin states in InAs QDs: Experiment vs Theory

Experiment: Takahashi et. al PRL 104, 246801 (2010)

1 2 3 46

8

10

12

14 Perturbation results Numerical results

Magnetic Field, B (T)

Ener

gy (m

eV)

2.7 2.8 2.9 3.0 3.12.3

2.4

2.5

2.6

2.7

Δε = 65 μeVε

2−ε

1

ε3−ε

1

En

ergy

(meV

)

Magnetic Field, B (T)

Prabhakar, Melnik, RaynoldsPRB 84, 155208 (2011)

Phonon mediated spin transition rates

( )( )∫∑ +−>

Phonon mediated spin transition rates

GaAs QDs

Dresselhaus case Rashba case

D. Loss group PRB 71 (2005)•Cusp-like structure can be seen for the pure Rashbacase in the phonon mediated spin-flip rate

• However, spin-flip rate is a monotonous function of the magentic fields for the pure Dresselhaus case

Why? Need some Math

Why only Rashba spin-orbit coupling gives cusp-like structures?Prabhakar, Melnik, Bonilla

PRB 87, 235202 (2013)

•Bulk g-factor is –ve. Only Rashba coupling has accidental degeneracy which provides the cusp-like structure in the spin-flip rate.

GaAs QDs

Dresselhaus case Rashba case

Also see D. Loss group PRB 71, 205324 (2005)

•Accidental degeneracy point is also called the spin hot spot.

•Since the spin-flip rate at or nearby the level crossing point is enhanced by several order of magnitudes, it provides the most favorable condition for the design of spin based logic gates

Never find the accidental degeneracy point

( )( )∫∑ +−>

Phonon mediated spin transition rates: anisotropic effects

( )( )∫∑ +−>

Geometric phase factors accompanying adiabatic changes (Berry Phase)

According to Schrödinger Equation, the state of the system evolves

( )tΨ

( )( ) ( ) ( )tΨt

itΨtRΗ∂∂

= hˆ

( ) ( ) ( ) ( )RnRΕRnRΗ n=ˆAt any instant,

M. Berry Proc. R. Soc. Lond. A 392,45-57(1984)

( ) ( )( ) ( )( ) ( )( )tRntiγtREdtitΨ nt '

n' expexp

0 ⎭⎬⎫

⎩⎨⎧−= ∫h Prabhakar, Melnik, Bonilla

PRB ( 2014)

Dynamical Phase Factor Berry phase

( ) ( ) ( ) ( ) ( ) ( )( ) ( )( )∑∫∫ ≠ Ε−Ε

Η∇×Η∇•−=

nm nm

RR

Cn RR

RnRˆRmRmRˆRndSIm 2γ

220

0

41

2 cc

B

;

Bg;

ωωωω

μρ

+=Ω±Ω=

=ΔΔ±Ω=

±

−± h

Manipulation of spin through Berry phasein III-V semiconductor QDs

( )2222

2

20

2100 2

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−±Ω

=

−+−−

−+±

ρα

ραξ

ξωπγDR

/,,r

h

hm

Prabhakar, Melnik, Bonilla PRB (2014)

(13)

•We apply non-degenerate perturbation theory • We find the Berry phase in QDs as

•Interplay between Rashba and Dresselhaus spin-orbit couplings in the Berry phase has been explored•Sign change in the g-factor has been observed•Level crossing in the Berry phase can be obtained•Berry phase is higly sensitive to the magnetic fields, QDs radii and the electric fields along z-direction

0.2 0.4 0.6 0.8 1.05.25

5.28

5.31

5.34

5.37

5.40(ii)

1.02 1.05 1.082.1122.1202.128

αR/αD

Δ

ε (m

eV)

0.3 0.6 0.90.4

0.8

1.2

αR/αD

g/g 0

(i)

GaAs QDs

Ber

ry p

hase

αR/αD

Extension of Berry Phase for degenerate case:disentangling operator method

For non degenerate state,

For a degenerate state, geometric phase factor is replaced by a non-Abelianunitary operator acting on the initial states within the subspace of a degeneracy

( ) ( ) ( )0ˆdttEiexp babT

a UT Ψ⎭⎬⎫

⎩⎨⎧ −=Ψ ∫h

Non-Abelian Unitary transformation=abÛ F. Wilczek and A. Zee; PRL 52, 2111, (1984)

( ) ( )( ) ( )( ) ( )( )tRntiγexptREdtiexptΨ nt

0

'n

'

⎭⎬⎫

⎩⎨⎧ −= ∫h

Dynamical Phase Factor Berry phase

•We seek to apply the Feynman disentangling operators to find the exact evolution operators for the Hamiltonian associated to QDs

Quantum dot orbiting in a closed path in the plane of 2DEG

tsinmRP 0x ωω−= tcosmRP 0y ωω=

( ) ( )xyxy PPiPPH αββα −−=± m

( )⎥⎦

⎤⎢⎣

⎡+−= ∫

πφσφσφ

2

00 cossinexp yxso

ad lRdiTU

For the pure Dresselhaus case;

Prabhakar, Raynolds, Inomata, MelnikPRB 82, 195306 (2010)

Consider both Rashba and Dresselhausspin-orbit couplings

•We find the evolution operator and investigate the interplay between the Rashba and the Dresselhausspin-orbit couplings• We apply the Feynman disentangling operators scheme to find the exact evolution operator.

Results: Evolution of spin dynamics during the adiabatic movement of the QDs in the plane of 2DEG

Rashba case Dresselhaus case Mixed case

E=5x105 V/cm (black)E=106 V/cm (red)R0 =500nm

•Spin-flip transition probability is enhanced with the gate controlled electric fields.

•Periodicity of the propagating waves is reduced with increasing electric fields which provides a shortcut to flip the spin rapidly.

•Periodicities of the propagating waves are different for the pure Rashba and pure Dresselhaus cases. As a result, we find the spin echo due to a superposition of Rashba and Dresselhaus spin waves.

Spin-echo

Prabhakar, Raynolds, Inomata, Melnik. PRB 82, 195306 (2010)

Coish et. al (PRL 2012); Spin-echo found in heavy holes Interacting with nuclear spins

0 1 2 3 4 5 6-1.2

-0.8

-0.4

0.0

0.4

0.8 ω = 1/ps ω = 2/ps ω = 3/ps

dγ /

dθ

Rotation angle, θ [rad]

0.2 0.4 0.6 0.8 1.05.25

5.28

5.31

5.34

5.37

5.40(ii)

1.02 1.05 1.082.1122.1202.128

αR/αD

Δ

ε (m

eV)

0 .3 0 .6 0 .90 .4

0 .8

1 .2

α R /α D

g/g 0

(i)

GaAs QDs

Ber

ry p

hase

αR/αD

We propose a method to flip the spin completely by an adiabatic transport of quantum dots.

Prabhakar, Melnik, InomataAPL (2014).

Sign change in the g-factor is reflected in the manipulation of spin via Scalar Berry phase.

0 2 4 6 8 10 1210-6

10-11041091014 Pure Dresselhaus

Magnetic Field, B (T)

Tran

sitio

n ra

te

a=b=4 a=1, b=16

Exact evolution operator is found via Feynman disentangling operators scheme. Spin echo dynamics is observed.

Spin hot spot due to anisotropy effect in Dresselhaus spin-orbit coupling

Prabhakar, Melnik, BonillaPRB (2014), EPJB (2015)

Prabhakar, Raynolds, Inomata, Melnik, PRB (2010)

Prabhakar, Melnik and Bonilla; PRB (2013)

Summary:

Prabhakar, Melnik, BonillaPRB (2016)

Sanjay Prabhakar and Roderick Melnik�The MS2Interdiciplinary Research Institute �Wilfrid Laurier University, Waterloo, Ontario, Canada�Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Phonon mediated spin transition ratesSlide Number 16Phonon mediated spin transition rates: anisotropic effectsSlide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23