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. Introduction 1.1. Spin-orbit interaction 1.2. Spin Hall effect 2. Spin Hall effect in 2DEG with artificial potential 3. Semiconductor quantum dot 3.1. Coulomb oscillation 3.2. Spin Hall effect in quantum dot 4. Search for Majorana fermions 4.1. Topological quantum computer 4.2. Majorana fermion

Outline 1. Introduction 1.1. Spin-orbit interaction 1.2. Spin Hall effect 2. Spin Hall effect in 2DEG with artificial potential 3. Semiconductor quantum dot 3.1. Coulomb oscillation 3.2. Spin Hall effect in quantum dot 4. Search for Majorana fermions 4.1. Topological quantum computer 4.2. Majorana fermion

1.1. Spin-orbit (SO) interaction Atom in vacuum −e

+Ze −e

+Ze

• Positive charge rotation makes a magnetic field

( μ0 − r )× (− v ) μ 0 Ze hl Beff = Ze = , 3 3 4π r 4π m r

Biot-Savart law hl = mr × v

• Magnetic dipole moment of electron spin ⎛ ⎝

μs = −2 μ B s ⎜ μ B = H SO

eh ⎞ ⎟ 2m ⎠

μ 0 Ze 2 h (hl ⋅ s ) = − μs ⋅ Beff = − 2 3 4π 2m r

Different from Dirac equation by factor 2 (Thomas factor)

H SO

μ0 Ze 2 h (hl ⋅ s ) =− 2 3 4π 2m r

hl = r × p

eh = s⋅(p× E) 2 2 2m c

Ze r E= 4πε 0 r 2 r

h =− σ ⋅ ( p × ∇U ) 2 2 4m c

1

− Ze 2 U= 4πε 0 r 1

+Ze

1 s = σ (Pauli matrices ) 2 H SO

λ

⎛ h ⎞ = σ ⋅ ( p × ∇U ), λ = −⎜ ⎟ h ⎝ 2mc ⎠

2

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

−e

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

[

]

1 (l + s )2 − l 2 − s 2 2 1 3 1 j = l + s : j = 1± = , 2 2 2

l⋅s =

HH : j = 3 / 2, j z = ±3 / 2 , LH : j = 3 / 2, j z = ±1 / 2 j = 1/ 2, j z = ±1 / 2

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

Spin-orbit interaction in semiconductors (II) • k-p perturbation theory for conduction band • SO interaction is enhanced, particularly in narrow-gap semiconductors (InAs, InGaAs)

H SO =

λ h

σ ⋅ ( p × ∇U )

P2 λ= 3

⎡ 1 ⎤ 1 ⎢ 2− 2⎥ ( ) E E + Δ 0 0 ⎣ 0 ⎦

P: matrix element between conduction and valence bands E0: band gap Δ0: SO splitting in valence band

• Roughly speaking, band gap corresponds to particleantiparticle energy gap in Dirac equation.

(1) Rashba SO interaction • U: external potential • Electric field perpendicular to 2D electron gas in InGaAs/GaAs heterostructure H RSO = =

λ h

α

h

σ ⋅ ( p × ∇U ) σ ⋅ ( p × zˆ ) =

U = eEz

α h

E in z direction

x 2DEG

y

(p σ y

x

− p xσ y ) (α = eEλ ) Large α: Nitta et al., PRL (1997); Grundler, PRL (2000); Sato et al., JAP (2001).

(2) Dresselhaus SO interaction • Inversion symmetry breaking in III-V compound semiconductors • U: crystal field β H DSO = (− p xσ x + p yσ y ) + γ ( p x p 2yσ x − p y p x2σ y ) h • Same order as Rashba SO (α~β) in GaAs

SO interaction

• Time reversal symmetry, Kramers’ degenerate • One-body problem

SO interaction vs. magnetic field (Zeeman effect)

Magnetic field

p x2 α H= + p xσ y SO interaction 2m h

time reversal symmetry: E+(k)=E−(−k) (Kramers degeneracy)

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)

Extrinsic SHE • V: Centrally symmetric potential in 3D (e.g. screened Coulomb potential by charged impurity) λ ~ V = V (r ) + σ ⋅ [ p × ∇V (r )] = V (r ) + V1 (r )l ⋅ s h 2 dV V1 (r ) = −λ r dr

• Semi-classical theory - “skew scattering” - “side jump” effect

• Optical experiment (Kerr rotation) by Kato 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

quantum dot

Outline 1. Introduction 1.1. Spin-orbit interaction 1.2. Spin Hall effect 2. Spin Hall effect in 2DEG with artificial potential 3. Semiconductor quantum dot 3.1. Coulomb oscillation and Kondo effect 3.2. Spin Hall effect in quantum dot 4. Search for Majorana fermions 4.1. Topological quantum computer 4.2. Majorana fermion

2.1. Formulation of SHE for 2DEG • 2D Schrödinger equation (effective mass equation) with axially symmetric potential V(r) ⎡ h h k ~⎤ Δ + V ⎥ψ = Eψ , E = ⎢− 2m ⎣ 2m * ⎦ 2

2

z

2

y

x

λ ~ V = V (r ) + σ ⋅ [ p × ∇V (r )] = V (r ) + V1 (r )lz sz h 2 dV : same sign as V(r) when |V(r)| V1 (r ) = −λ r dr

2DEG

monotonically decreases with r.

• lz and sz are conserved in 2D.

1 ⎧ ⎪V (r ) + 2 V1 (r )l z ~ V = V (r ) + V1 (r )l z s z = ⎨ 1 ⎪V (r ) − V1 (r )l z 2 ⎩

sz=1/2: sz=−1/2:

(s z = +1 / 2) (s z = −1 / 2)

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

SHE exists, but is very small.

Application to 3-terminal spin filter Conductance and PGz + −G − Conductance and Pz = G + +G −

Pz for each channel in incident wave 73%

25%

Channel 1 can be injected selectively using QPC

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

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

Outline 1. Introduction 1.1. Spin-orbit interaction 1.2. Spin Hall effect 2. Spin Hall effect in 2DEG with artificial potential 3. Semiconductor quantum dot 3.1. Coulomb oscillation 3.2. Spin Hall effect in quantum dot 4. Search for Majorana fermions 4.1. Topological quantum computer 4.2. Majorana fermion

3.1. Coulomb oscillation in quantum dot

• 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)

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):

μ N = E N − E N −1

Constant interaction model with spin-degenerate levels

N ( N − 1) EN = ∑ ε i + U, 2 i =1 μ N = E N − E N −1 = ε N + (N − 1)U N

U

U+Δε

U

μ1 = ε 1 μ2 = ε1 + U μ3 = ε 2 + 2U μ 4 = ε 2 + 3U

Discretepeak: energy levels are occupied - Coulomb resonant tunneling at kconsecutively. BT kBT, Γ

• Quantum fluctuation: “level broadening” Γ (due to finite lifetime by tunnel coupling to the leads) 2 2π = ∑ α , k H T d n δ (ε k − ε n ) τ α = L, R; k h

1

(

)

2π 2 2 = ν VL + VR h 1h 2 Γ= = πν VL + VR 2τ

(

2

)

3.2. Spin Hall effect in quantum dot • 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 by SO interaction: 0.23meV - Kondo effect

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

“Spin Hall effect” at quantum dot 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 cf. Previous work for “open quantum dot” without tunnel barriers: 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

Number of leads N ≥2

εd =

ε1 + ε 2

, Δ = ε 2 − ε1

2 tunable by VG

• SO interaction in the quantum dot H SO =

λ h

σ ⋅ ( p × ∇U )

Quantization axis // hSO

1⎛ −Δ H dot, σ = ±1 = ε d + ⎜⎜ 2 ⎝ ± iΔ SO

Δ SO = hSO

1 H SO 1 = 2 H SO 2 = 0 2 H SO 1 = i hSO ⋅ σ 2 ihSO 2 = (λ h ) 2 ( p × ∇U ) 1

m iΔ SO ⎞ ⎟⎟ Δ ⎠ For details, see Eto and Yokoyama, Poster WP-62

Three-terminal system

ΓS = ΓD1 ≡ Γ eS,1 eS,2 = 1 2 eD1,1 eD1,2 = −3 , eD2 ,1 eD2 ,2 = 1

(a) ΓD 2 = 0.2Γ (b) ΓD 2 = 0.5Γ (c) ΓD 2 = Γ (d) ΓD 2 = 2Γ ΔSO = 0.2Γ

ΔSO=0.23 meV Γ ~ 1 meV

Δ = ε 2 − ε1 = 0.2Γ

Δ=Γ

Three-terminal system

Δ = ε 2 − ε1 = 0.2Γ

Δ=Γ

ΓS = ΓD1 ≡ Γ eS,1 eS,2 = 1 2 eD1,1 eD1,2 = −3 , eD2 ,1 eD2 ,2 = 1

(1) Large spin polarization around current peak (a) ΓD 2 = 0.2Γ Enhancement of SHE by resonant tunneling (b) ΓD 2 = 0.5Γ

(2) (c)Level ΓD 2 = Γspacing Δ ~ broadening Γ Two= 2levels should contribute to transport (d) Γ Γ D2

(3) ΔSOControl = 0.2Γ of SHE by tuning ΓD2 (tunnel coupling to D2)

3.3. Enhanced SHE by Kondo resonance Kondo 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 EF with width of TK

Coulomb blockade with single electron

E + , E − >> k BT , Γ • Addition and extraction energies

E + , E − >> k BT , Γ

(

Γ = πν VL + VR 2

2

)

⎧E + = μ2 − μ = ε 0 + U − μ ⎨ − ⎩ E = μ − μ1 = μ − ε 0

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

• Spin-flip by cotunneling

• Anti-ferromagnetic coupling between localized spin and conduction electrons 1 ⎞ + 2⎛ 1 H = 2 J ∑ S + ck '↑ ck ↓ + L = 2 JS ⋅ (s )k 'k , J = V ⎜ + + − ⎟ E ⎠ ⎝E kk '

[

]

Ground state with antiferromagnetic coupling • Two interacting spins: Grd =

(

1 ↑ ↓ −↓ ↑ 1 2 1 2

2

)

Spin-singlet state • One spin and Fermi sea:

Kondo singlet state (Many-body state) Conduction electrons coherently couple with a localized spin (spin is completely screened).

• T>>TK: Spin S=1/2 is localized in the quantum dot (small G by Coulomb blockade) • T1

TiT jTi = T jTiT j i − j = 1 D. A. Ivanov, PRL 86, 268 (2001).

* Majorana fermions enable topological quantum computation

4.3. Majorana fermions in semiconductor • Rashba spin-orbit interaction 2DEG

1D

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