Semiconductor quantum dots with spin-orbit …...Semiconductor quantum dots with spin-orbit...

Semiconductor quantum dots with spin-orbit interaction

Mikio Eto Faculty of Science and Technology,

Keio University, Japan

Spintech6 (Matsue, Jul.31- Aug.5, 2011)

Outline

1. Introduction1.1. Spin-orbit interaction1.2. Spin Hall effect

2. Spin Hall effect in 2DEG with artificial potential

4. Search for Majorana fermions4.1. Topological quantum computer4.2. Majorana fermion

3. Semiconductor quantum dot3.1. Coulomb oscillation3.2. Spin Hall effect in quantum dot

Outline





Atom in vacuum

+Ze−e +Ze

−e

• Positive charge rotation makes a magnetic field

( )sl

s

⋅−=⋅−=

⎟⎠⎞

⎜⎝⎛ =−=

hh

h

32

20

effsSO

BBs

24

2 2

rmZeBH

me

πμ

μμ

μ

μ Different from Diracequation by factor 2(Thomas factor)

• Magnetic dipole moment of electron spin

( ) ( ) vrllvrB ×==−×−

= mrm

Zer

Ze hh ,

44 30

30

eff πμ

πμ

1.1. Spin-orbit (SO) interaction

Biot-Savart law

( )

( )

( )rZeUU

cm

rrZe

cme

rmZeH

2

022

20

22

32

20

SO

41

4

4

1 2

24

−=∇×⋅−=

=×⋅=

×=⋅−=

πε

πε

πμ

pσ

rEEps

prlsl

h

h

hhh

( )matrices Pauli 21 σs =

• Relativistic effect: 2mc2=1MeV in the denominator isenergy gap between particle and antiparticle

+Ze−e

( )2

SO 2 , ⎟

⎠⎞

⎜⎝⎛−=∇×⋅=

mcUH h

hλλ pσ

Spin-orbit interaction in semiconductors (I)

• Valence band in compound semiconductors: consists mainly of p orbitals (l=1) SO interaction

( )[ ]

21,

23

211 :

21 222

=±=+=

−−+=⋅

jslj

slslsl

2/1,23 :LH

,2/3,23 :HH

±==

±==

z

z

j/j

j/j

2/1,21 ±== zj/j

• Conduction band: consists mainly of s orbital (l=0)No SO interaction?

( )UH ∇×⋅= pσh

λSO

P: matrix element between conduction and valence bandsE0: band gapΔ0: SO splitting in valence band

• Roughly speaking, band gap corresponds to particle-antiparticle energy gap in Dirac equation.

• k-p perturbation theory for conduction band• SO interaction is enhanced, particularly in narrow-gap

semiconductors (InAs, InGaAs)

( ) ⎥⎦⎤

⎢⎣

⎡Δ+

−= 200

20

2 113 EE

Pλ

Spin-orbit interaction in semiconductors (II)

Large α:Nitta et al., PRL (1997);Grundler, PRL (2000);Sato et al., JAP (2001).

(1) Rashba SO interaction

( )

( ) ( )yxxy pp

UH

σσαα

λ

−=×⋅=

∇×⋅=

hh

h

zpσ

pσ

ˆ

RSO

2DEGx

y

E in z direction

eEzU =

( )λα eE=

• U: external potential• Electric field perpendicular to 2D electron gas in

InGaAs/GaAs heterostructure

• Inversion symmetry breaking in III-V compound semiconductors

• U: crystal field

( ) ( )yxyxyxyyxx ppppppH σσγσσβ 22DSO −++−=

h

(2) Dresselhaus SO interaction

• Same order as Rashba SO (α~β) in GaAs

• Time reversal symmetry,Kramers’ degenerate

• One-body problemSO interaction

SO interaction vs. magnetic field (Zeeman effect)

Magneticfield

SO interaction

time reversal symmetry:E+(k)=E−(−k)

(Kramers degeneracy)

yxx pm

pH σαh

+=2

2

1.2. Spin Hall effect

• “Spin injection” without ferromagnet, magnetic field• Intrinsic and extrinsic spin Hall effect (SHE)

(a) Hall effect: Lorentz force by magnetic field(b) Extrinsic SHE: SO interaction + impurity scattering(c) Intrinsic SHE: topological structure of valence band

S.Murakami, N. Nagaosa and S.-C. Zhang, Science (2003)

• V: Centrally symmetric potential in 3D (e.g. screened Coulomb potential by charged impurity)

( ) ( )[ ] ( ) ( )

( )drdV

rrV

rVrVrVrVV

2

~

1

1

λ

λ

−=

⋅+=∇×⋅+= slpσh

• Semi-classical theory

- “skew scattering”- “side jump” effect

Extrinsic SHE

• Optical experiment (Kerr rotation) byKato et al., Science (2006)

- Ascribable to extrinsic SHE- Quantitatively explained by semiclassical theory

H.-A. Engel, B. I. Halperin, and E. I. Rashba,PRL 95, 166605 (2005).

In this talk• Quantum mechanical formulation of SHE for 2DEG in

semiconductor heterostructures with “single impurity”- Artificial potential by single antidot, STM tip- InAs quantum dot with SO interaction

quantumdot

Outline




3. Semiconductor quantum dot3.1. Coulomb oscillation and Kondo effect3.2. Spin Hall effect in quantum dot

mkEEV

m 2 ,~

*2

222 hh==⎥

⎦

⎤⎢⎣

⎡+Δ− ψψ

( )drdV

rrV 2

1 λ−=

• 2D Schrödinger equation (effective mass equation) with axially symmetric potential V(r)

z

2DEGx

y

• lz and sz are conserved in 2D.

: same sign as V(r) when |V(r)|monotonically decreases with r.

2.1. Formulation of SHE for 2DEG

( ) ( )[ ] ( ) ( ) zzslrVrVrVrVV 1~ +=∇×⋅+= pσ

h

λ

• Scattering enhanced for lz >0• suppressed for lz <0

sz=1/2:

(opposite effects)sz=−1/2:

extrinsic spin Hall effect

( ) ( )( ) ( ) ( )

( ) ( ) ( )⎪⎩

⎪⎨

⎧

−=−

+=+=+=

2/1 21

2/1 21

~

1

1

1

zz

zz

zz

slrVrV

slrVrVslrVrVV

cf. Partial wave expansion:Eto and Yokoyama, J. Phys. Soc. Jpn. 78, 073710 (2009).

( )

( ) ⎟⎠⎞

⎜⎝⎛ −−

== ∑∞

−∞=

42cos2~

cos

πππ

θθ

mkrkr

krJ

ekrJiee

m

m

imm

mikrikx

phase shift: 2/1 ,for ±==±zzm smlδ

( ) ( ) ⎟⎠⎞

⎜⎝⎛ +−−→ ±± ±

mi

mmmkre

krrRkrJ m δππ

πδ

42cos2~

• Incident wave

(Bessel function; asymptotic form at )• Incident wave + scattered wave

(*) 3D: textbooks by Landau-Lifshitz, Mott-Massey

Partial wave expansion*: ( )L3, ,2 ,1 ,0 ±±±== mlz

∞→r

Appendix

mmm −

± = δδ- (time reversal symmetry)

000 δδδ ≡= −+- SO interaction does not work on S wave (m=0):

( )

( ) ( )

21for

21

*2

from determined is

12

2

2

22

±=

=⎥⎦

⎤⎢⎣

⎡±+⎟⎟

⎠

⎞⎜⎜⎝

⎛−+− ±±

±

z

mm

m

s

ERRrVmrVrm

drd

rdrd

m

rR

h

( ) zz slrV1−+ ≠ mm δδ

Appendix

( )

( ) ( )

( )∑

∑∞

=

∞

=

±

−+

−+

−=

⎭⎬⎫

⎩⎨⎧

−++−=

±=

1

22

1

222

sin21

cos2121

0

m

ii

m

iii

meek

B

meeeki

A

BAf

mm

mm

θπ

θπ

θ

δδ

δδδ

• Scattering amplitude (cross section: )

Spin polarization: −Pz for −θ direction

( ) ( )( ) ( )

( )2222

22*Re2

BAAB

ffff

Pz+

=+

−=

−+

−+

θθθθ

( ) ( ) 2θθσ ±± = f

• Spin polarization of scattered wave in θ direction

−+ ≠ mm δδ

Appendix

( ) ( ) ( ) ( )⎥⎦⎤

⎢⎣⎡ −±−−= ra

amraVraVrV δλθθ 00

2.2. SHE by a tunable potential well

• Fermi wavelength 2π/Radius of well a: ka=

• Strength of SO interaction: λ/a2=0.01 (at ka=1)

k: 10~100nm1, 2, 3

Resonant scattering of S wave (m=0) andP wave (m=0) at some values of |V0|(unitary limit)

centrifugalpotential

Unitary limit: 2πδ =±

m

• Scattering probability of partial waves (m=0,1)

• Spin polatization in −y direction (θ=−π/2)

Around resonance, spin polarization is enhanced.

2/12/1

−=+=

z

z

ss

~30%

SHE exists, butis very small.

ka=2, V0 > 0• Repulsive potential

Conductance and Pz Pz for each channel in incident wave

25%

73%

Channel 1 can be injectedselectively using QPC

−+

−+

+−

=GGGGPz and eConductanc

Application to 3-terminal spin filter

2.3. Conclusions

• Formulation of extrinsic spin Hall effect (SHE) in 2D using partial wave expansion

• Enhanced SHE by resonant scattering by tuning attractive potential

• Three-terminal device including an antidot for spin injection, showing polarization ~30% for two channels and ~70% for single channel

ReferencesM. Eto and T. Yokoyama, J. Phys. Soc. Jpn. 78, 073710(2009); T. Yokoyama and M. Eto, PRB 80, 125311 (2009).

Outline





• Quantum dots: zero-dimensional systems of nano-meter scale

• Transport through “quantum levels” in quantum dots• Quantum levels are controlled by gate voltage.

peak structure of current

Coulomb oscillation(Coulomb blockade between peaks)

3.1. Coulomb oscillation in quantum dot

What are “quantum levels”?

1. In absence of electron-electron interaction, “quantum levels” are single-electron energy levels.

2. In presence of electron-electron interaction (charging energy), increase in energy to put an electron on the dot (electro-chemical potential):

1−−= NNN EEμ

Constant interaction model with spin-degenerate levels

( )

( )UNEE

UNNE

NNNN

N

iiN

1

,2

1

1

1

−+=−=

−+=

−

=∑

εμ

ε

Discrete energy levels are occupied consecutively.Peak positions reflect the energy levels.

UU U+ΔεUU

U

32

24

23

12

11

+=+=+=

=

εμεμεμεμ

- Coulomb peak: resonant tunneling at kBT<Γ- Coulomb blockade regime: Kondo resonance at T<TK

Condition to observe Coulomb oscillation, blockade

(level spacing), (Charging energy) >> kBT, Γ

• Quantum fluctuation: “level broadening” Γ(due to finite lifetime by tunnel coupling to the leads)

( )

( )( )22

22

2

;,

21

2

,21

RL

RL

nknTkRL

VV

VV

dHk

+==Γ

+=

−= ∑=

πντ

νπ

εεδαπτ α

h

h

h

• InAs quantum dot: strong SO interaction- Y. Igarashi et al., PRB 76, 081303(R) (2007).- S. Takahashi et al., PRL 104, 246801(2010).

- Energy level splitting bySO interaction: 0.23meV

- Kondo effect

- C. Fasth et al., PRL 98, 266801 (2007).- A. Pfund et al., PRB 76, 161308(R) (2007).

3.2. Spin Hall effect in quantum dot

“Spin Hall effect” at quantum dot

• Strong SO interaction is present only in quantum dot• Multi-terminal system

- unpolarized current is injected from lead S- spin-polarized current to leads D1, D2,…

“Spin filter” effect

quantumdot

cf. Previous work for “open quantum dot” without tunnelbarriers: Krich and Halperin, PRB 78, 035338 (2008).

Model• Two energy levels in a quantum dot (minimal model)• Single channel in leads• No magnetic field: wavefunctions are real

1221 ,

2εεεεε −=Δ

+=d

2leads ofNumber

≥N

tunable by VG

( )UH ∇×⋅= pσh

λSO

212

02211

SOSO

SOSO

σh ⋅=

==

iH

HH

Quantization axis // hSO

⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔ±ΔΔ−

+=±=SO

SO1 dot, 2

1i

iH dσ

mε

( ) ( )122SO Ui ∇×= ph hλ

SOSO h=Δ

• SO interaction in the quantum dot

For details, seeEto and Yokoyama,

Poster WP-62

Three-terminal system

Γ≡Γ=Γ 1DS

1,321

2D21D22D11D1

2S1S

=−=

=

,,,,

,,

eeeeee

Γ=ΓΓ=Γ

Γ=ΓΓ=Γ

2 (d) (c)

5.0 (b)2.0 (a)

2D

2D

2D

2D

Γ=Δ 2.0SO

Γ=ΔΓ=−=Δ 2.012 εε

ΔSO=0.23 meVΓ ~ 1 meV

Γ≡Γ=Γ 1DS

1,321

2D21D22D11D1

2S1S

=−=

=

,,,,

,,

eeeeee

Γ=ΓΓ=Γ

Γ=ΓΓ=Γ

2 (d) (c)

5.0 (b)2.0 (a)

2D

2D

2D

2D

Γ=Δ 2.0SO

Γ=ΔΓ=−=Δ 2.012 εε

(1) Large spin polarization around current peakEnhancement of SHE by resonant tunneling

(2) Level spacing Δ ~ broadening ΓTwo levels should contribute to transport

(3) Control of SHE by tuning ΓD2 (tunnel coupling to D2)


3.3. Enhanced SHE by Kondo resonanceKondo effect in quantum dot• Spin S=1/2 in quantum dot + Fermi sea in leads• Spin-singlet state (S=0)

Many-body ground state(Binding energy:

Kondo temperature TK)

Resonant level at EFwith width of TK

• Addition and extraction energies ( )2

R2

L

B ,,

VV

TkEE

+=Γ

Γ>>−+

πν

Γ>>−+ ,, BTkEE

⎩⎨⎧

−=−=−+=−=

−

+

01

02

εμμμμεμμ

EUE

Coulomb blockade with single electron

• Sequential tunnel process is forbidden.• Higher-order tunnel process “cotunneling”

• Spin-flip by cotunneling

• Anti-ferromagnetic coupling between localized spinand conduction electrons

[ ] ( ) ⎟⎠⎞

⎜⎝⎛ +=⋅=+= −+↓

+↑+∑ EE

VJJccSJH kkkk

kk11 ,22 2

''

' sSL

Ground state with antiferromagnetic coupling

• Two interacting spins:

• One spin and Fermi sea:

( )21212

1Grd ↑↓−↓↑=

Conduction electrons coherently couple with a localizedspin (spin is completely screened).

Spin-singlet state

Kondo singlet state(Many-body state)

• T>>TK: Spin S=1/2 is localized in the quantum dot (small G by Coulomb blockade)

• T<<TK: Kondo singlet state is formed. Resonant tunneling through the singlet state.

heG

22=

Conductance

Two levels in quantum dot

12 εε −=Δ

(1) TK < Δ: level 2 is irrelevant

Spin S=1/2 in level 1 is screened out(conventional “spin SU(2)” Kondo effect)

(2) TK > Δ:Both pseudo-spin (levels 1 and 2) and spin S=1/2 arescreened out (SU(4) Kondo effect)

Crossover of (1) and (2): slave-boson mean-field theorycf. J. S. Lim et al., PRB 74, 205119 (2006).

Two-terminal system

−+ = ,1,1 GG

∞=U

Kondo regime : − εd << Γ

charge fluctuationregime: −εd ~ Γ


Γ≡Γ=Γ 1DS

1,321

2D21D22D11D1

2S1S

=−=

=

,,,,

,,

eeeeee

Γ=ΓΓ=Γ

Γ=ΓΓ=Γ

2 (d) (c)

5.0 (b)2.0 (a)

2D

2D

2D

2D

Γ=Δ 2.0SO

TK decreases with decreasing εd:- SU(4) Kondo: TK > Δ- SU(2) Kondo: TK < Δ

Γ=−=Δ 2.012 εε∞=U

3.4. Conclusions

• Formulation of spin Hall effect at quantum dot- needs two levels in the dot, more than two leads

• Enhancement of SHE by resonant tunneling and Kondo resonance (SU(4) Kondo effect)

[Observation by “inverse SHE”]• Ferromagnet (magnetization p) + InAs quantum dot:

Hamaya et al., APL (2007)

−+−

++

= ,1,11 2cos1

2cos1 GpGpG θθ θ: angle between

magnetization and hSO

[Reference]Eto and Yokoyama, J. Phys. Soc. Jpn. 79, 123711 (2010).

Outline


2. Spin Hall effect with artificial potential



• Conventional quantum computer(1) qubit: single electron spin

4.1. Topological quantum computer

(2) Easy to manipule using ESR, etc.(3) Decoherence problem: continuous phase error

↓+↑=ψ 10 CC

↓+↑⇒↓+↑ θieCCCC 1010

overcome by “quantum error correction”

• Topological quantum computer(1) Many-body states of interacting electrons

(2) Topologically protected:robust against local perturbation

(3) Manipulation by exchanging quasi-particlesNon-Abelian braiding statistics

states) ground e(degenerat ∑ Ψ=j

jjCψ

1 2 3

1 2 3 4 …

( ) ( )KK ,3,2,1,3,1,2 Ψ=Ψ

( ) ( )KK ,3,2,1,3,1,2 Ψ−=Ψ

( ) ( )KK ,3,2,1,3,1,2 Ψ=Ψ αie

Boson

Fermion

“Anyon”

Suggested for high Tc cuprates∞=U 2D,

1 2 3 4 …

( ) ( )∑ Ψ=Ψj

jiji C KK ,3,2,1,3,1,2 Non-Abelian

1 2 3 1 2 3

2 3 1 3 1 2

≠2112 TTTT ≠

Candidates• Fractional Quantum Hall effect with ν=5/2

• Half-quantum vortex in p-wave superconductorStrontium ruthenate Sr2Ru2O4: Science 331, 186 (2011).

( ) ( )↓↓−↑↑±Δ=Δ2

10 yx ikk

1,1 ±== zLL 1=S

“chiral states”clockwise vs. anticlockwise

Spin triplet“equal spin pairing”

- Usual spin-singlet superconductor

( ) ( )e

hn2

flux 2 =Φ⇒Δ=+Δ θπθ

- p-wave (chiral) superconductore

hn22

flux =Φ

Bogoliubov-de Gennes eq. yields quasi-particleoperators

( ) ( )EEvu−=

Ψ+Ψ=+

++

γγ

γ

( ) ( )00 ===∴ + EE γγ

1=zL

Vortexcore

1−=zL

4.2. Majorana fermions

• Creation and annihilation operators are self conjugate

- possibly neutrino- Half-quantum vortex at E=0

γγ =+

• 2n half-quantum vortices: ( )niii 2,,2,1 L==+ γγ

( )( )⎩

⎨⎧

−=+=

−+

−

2/2/

212

212

iii

iii

ididγγγγ

n “usual” fermions0,1== +

iii ddn

2n degenerate states

* Majorana fermion is a “half” of usual fermion

• 2n degenerate states• Non-Abelian braiding statistices

1

1

=−=

>−=

jiTTTTTT

jiTTTT

jijiji

ijji

* Majorana fermions enable topological quantum computation

D. A. Ivanov, PRL 86, 268 (2001).

4.3. Majorana fermions in semiconductor

• Rashba spin-orbit interaction

2DEG

• S-wave superconductivity (proximity effect) + magnetic field

1D

Similar to chiral superconductivity

cf. Wray’s talk: topological insulator + superconductivity

(1) Semiconductor thin film

(2) InAs nanowire

J.D.Sau et al., PRL 104,040502 (2010)

J. Alicea et al., Nat. Phys. 7,412 (2011)

A pair of Majoranas

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Semiconductor quantum dots with spin-orbit …...Semiconductor quantum dots with spin-orbit...

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