Learning to Love Disorder: Spin-charge Conversion and Other - - PowerPoint PPT Presentation

Learning to Love Disorder:

Spin-charge Conversion and Other Interesting Effects in 2D Spin-Orbit Coupled Systems

Miguel A. Cazalilla

National Tsing Hua University & National Center for Theoretical Sciences, Taiwan & Donostia International Physics Center (DIPC) , Spain

“Novel Quantum States in CMT NQS2017” 京都大学基礎物理学研究所（２０１７年１１月２日)

Acknowledgements

ChunLi Huang Taiwan Yidong Chong NTU (Singapore) Giovanni Vignale U of Missouri (US) Hung-Yu Yang NTHU → Boston College Xianpeng Zhang NTHU → DIPC (Spain) Junhui Zheng (&Ms. Zheng) Taiwan → Frankfurt Hector Ochoa (UCLA) Aires Ferreira Univ of York (UK) Mirco Milletari Singapore

On the importance of disorder

近藤さん

Disorder can be useful…

Quantum Hall Plateaux John Bardeen

Outline: Part I

Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

Outline: Part II

Topology = Dimensional reduction? Impurity in a 1D Channel w and w/o interactions: Kane & Fisher Magnetic Impurity near the non-interacting edge of a 2D QSHI Magnetic Impurity near the interacting edge of a 2D QSHI

Outline: Part I

Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

SHE and Symmetry (2D)

2D rotations Jz = Lz + Sz

[H, Jz] = 0 [U(1) group]

Charge current (2D vector)

J = (Jx, Jy)

J α =

• J α

x , J α y

• Rank-2 Tensor?

(2D vector)

J α → J z ⊕ (J x, J y)

Under reflection x ➝ -x and y ➝ y and Sz ➝ -Sz

Jx → −Jx Jy → Jy vs. J z

y → Jz x

Jy → −J z

y Different sign determined by TRS (by Onsager’s reciprocity)

J z

y = θSHE Jx

Jx = −θSHE J z

y

Symmetry arguments are fine, but what are the mechanisms?

AH Castro Neto & F Guinea Phys Rev Lett (2009)

Motivation: Functionalized Graphene

Chemisorption: H, F, … Physisorption:

Cu, Ag, Au, Th, In,…

C Weeks et al Phys Rev X (2012)

Substrates:

Z Wang et al Phys. Rev. (2016) B Wang et al 2D Mater (2016)

σ Ω

Ω

σ

SiO2 Graphene TMD Si

(a) (b)

σ Ω Ω σ Ω

TMD = WSe2, MoS2

R

(V) σ(mS)

(a) (b) (c) h-BN SiO₂/p-Si V V WS₂

Extrinsic Mechanisms for SHE

elastic channel up down

δ

− δ

σsH ∼ const

elastic channel up down

σsH ∼ 1 Nimp

Quantum Side Jump Skew Scattering (nimp ⪡ 1)

Nagaosa, Sinova, Onoda, MacDonald, Ong, Rev. Mod. Phys. (2010)

nimp < 1%

Extrinsic SHE: Mott’s Scattering

Tkp = Akp 1 + Bkp · s

2 x 2 Scattering matrix (to all orders…)

k

p

θ

Unpolarized!

ρin(p) = 1 21 ρout(k) = Tkpρin(p)T †

kp = 1

2TkpT †

kp

hSziout = Tr [szρout(k)] = Re ⇥ A∗

kpBz kp

⇤ / sin θ

Polarization! Electron-atom scattering

Bkp = S(θ) (k × p) h cos θ = ˆ k · ˆ p i

(3D Rotational Sym.) SOC

θ

Forces “emerge” from collisions (akin to the Bernoulli principle)

Forces from collisions

“Orbital” pseudo magnetic field from collisions

F s ∝ nimp ˆ ysz Z dθ sin θ Re h Akp

• Bz

kp

∗i (J = Jxˆ x) Lorentz-like force Bz

kp = S(θ) (k × p) · ˆ

z ∝ sin θ

Graphene: Resonant Enhancement

Graphene

γ = spin Hall angle (T = 0) Graphene is very prone to resonant scattering

A Ferreira, T Rappoport, MAC, AH Castro Neto Phys Rev Lett (2014)

Linear Density of States

Yang et al. Science (2013)

⇢(✏) ∼ |✏|

SHE in CVD Graphene

J Balakrishnan et al Nat. Comm. (2014)

Spin Hall Angle Electric Conductivity Model: Kane-Mele SOC impurities plus scalar resonant scatterers

Outline: Part I

Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

provided INVERSION SYMMETRY IS BROKEN must be related to

2D Magnetoelectric Effect & Symmetry

(pseudo-vector under 3D rotation)

M = (Mx, My, Mz) = Mz ⊕ Mk =(Mx, My)

J

M

Under reflection x ➝ -x and y ➝ y and Sz ➝ -Sz

My = αCISP Jx Jx = ˜ αCISP My

Charge current (2D vector)

J = (Jx, Jy)

CISP from Edelstein Effect

HR = BR(k) · s BR(k) = αR (ˆ z × k)

kx ky

E

My 6= 0

Indirect DMC: Eldestein Effect

K Shen, G Vignale & R Raimondi PRL (2014) R Raimondi, P Schwab, C Gorinni, and G Vignale Ann Phys (Berlin) 2012

J

Two-step process

J z

SHE

(a) (b)

My

Rashba SOC

HR = BR(k) · s BR(k) = αR (ˆ z × k)

δSy

:

¼ −2mαJz

y − δSy

τEY

Outline: Part I

Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

k

p

Tkp = Akp 1 + Bkp · s

Revisiting Mott: Anisotropic Spin Precession

Understanding disorder from one impurity Unpolarized ρin(p) = 1

21 ρout(k) = Tkpρin(p)T †

kp = 1

2TkpT †

kp

hSi = Tr [sρout(p)] = Re ⇥ A∗

kpBkp

⇤ + i

• B∗

kp ⇥ Bkp

• Mott’s scattering

ASP

No ASP scattering! Bkp = S(θ) (k × p) ∝ B∗

kp ⇒ ASP = 0

Isotropic TR-invariant

3D Rotational Invariance broken to 2D rotation

Bkp = Bk

kp + ˆ

z Bz

kp

ASP in 2D Materials

MM Glazov, E Ya Sherman, VK Dugaev Physica E (2010) JP Binder et al Nature Physics (2016)

“Zeeman-like” (Rashba disorder fluctuations) Orbital-like 2D Mott’s scattering

Bz

kp = S(θ) (k × p) · ˆ

z

ˆ z

k

p

Tkp = Akp 1 + Bkp · s

C Huang, Y Chong, and MAC, Phys Rev B (2016)

Bz

kp ∝ sin θ

• ut of plane

Anisotropic Precession Scattering (ASP)

θ

Bk

kp

Electron current → spin polarization perp. to E (i.e. My) ASP rate = i

• Bkp ⇥ B⇤

kp

• / iBk

kp

• Bz

kp

⇤ ⇥ ˆ z / hSyiout Scattering angle in-plane

C Huang, Y Chong, and MAC, Phys Rev B (2016)

Kohn & Luttinger: Strong disorder

C Huang, Y Chong, and MAC, Phys Rev B (2016)

Charge Spin

C-L Huang Yidong Chong

@t⇢k + eE · rkn0

k

~ = I1[⇢k, mk], ⇣ ⌘

~ @tmk + ⇣ ~H nimp ~ Re Bkk ⌘ ⇥ mk = I2[⇢k, mk].

I1, I2 ∝ nimp

Collision integral

Impurity (Rashba) self-energy

k k

δnk = nk − n0

k

δnk = ρk1 + mk · s

Collision Integral

Scattering rate ∝ Probability = (Amplitude) × (Amplitude)∗

=

×

T +

kp

k p

T −

kp

k

p

T +

kp

k

p

2

nimp nimp nimp

Drude relaxation rate Conductivity

c1

Skew scattering rate SHE / ISHE

c2

Anisotropic spin precession (ASP) scattering rate CISP

c3

I1 = nimp 2⇡~ Z d2p  c1(⇢p − ⇢k) + c2 · (mp − mk) − c3 · (mp + mk)

• (✏p − ✏k)

I2 = nimp 2⇡~ Z d2p  c1(mp − mk) + c2(⇢p − ⇢k) + c3 (⇢p − ⇢k) + K

• (✏p − ✏k)

Charge Spin

C Huang, Y Chong, and MAC, Phys Rev B (2016)

Keldysh: Smooth weak disorder

C Huang, M Milletari, and MAC, arxiv:17061316 (2017) (accepted in PRB)

Spin current

J

Charge current

J

M

Magnetization

(Non-equilibrium spin polarization)

θsH

a)

αR αasp

  Jx J z

y

My   =   θsH τDαasp −θsH τDαR τEYαasp −τEYαR     Jx J z

y

My   + σD   Ex   Linear Response & Direct Magnetoelectric Coupling (DMC)

J z

y

Ex

My

Onsager Rij = ✏i✏jRji

(TR parity ✏i = ±1)

O E O

C Huang, Y Chong, and MAC, Phys Rev B (2016)

Spin current

J

Charge current

J

M

Magnetization

θsH

a)

αR αasp

10 20 30 40 50

• 0.4
• 0.2

0.0 0.2 0.4 10 20 30 40 50

σcisp = σD (θsHαRτs + αaspτs) .

• Eq. (1) to firs

My = σcispEx, w

Extrinsic CISP? Yes! Two mechanisms

Edelstein effect: SHE ×Rashba

ASP Scattering

Controlled by gating C Huang, Y Chong, and MAC, Phys Rev B (2016)

Outline: Part I

Extrinsic Spin Hall Effect and Mott Scattering Indirect Magneto-electric Coupling: Edelstein Effect Direct Magneto-electric Coupling: Anisotropic Spin Precession What about experiments? Non local probes, Hanle, and all that

Electric field

Jc Js

Nonlocal Voltage

C1

C4 C3

C2

Direct Inverse

E

SHE

Nonlocal Resistance

(or Mott’s double scattering)

m

Direct Magnetoelectric Coupling (ASP)

C Huang, Y Chong, and MAC, arXiv:1702.04955 (2017) J Hirsch PRL (1999) E M Hankiewiz et al PRB (2004) D Abanin et al PRB (2007)

Hanle effect is qualitatively different for different spin- charge conversion mechanism. It may become asymmetric.

Anomalous Hanle spin precession

C Huang, Y Chong, and MAC, Phys. Rev. Lett. (2017)

• 10
• 5

5 10

• 3
• 2
• 1

1 2 3 wLts Rnlêrxx H 10-3L

SHE>>ASP

• 10
• 5

5 10

• 3
• 2
• 1

1 2 3 wLts Rnlêrxx H 10-3L

ASP>>SHE

• 10
• 5

5 10

• 3
• 2
• 1

1 2 3 wLts Rnlêrxx H 10-3L

SHE~ASP ∝ Bk Bk

τJ

Bk

τJ

LETTERS

PUBLISHED ONLINE: 17 MARCH 2013 | DOI: 10.1038/NPHYS2576

Colossal enhancement of spin–orbit coupling in weakly hydrogenated graphene

Jayakumar Balakrishnan1,2†, Gavin Kok Wai Koon1,2,3†, Manu Jaiswal1,2‡, A. H. Castro Neto1,2,4 and Barbaros Özyilmaz1,2,3,4*

Hanle precession Non-local resistance

“Spooky” Non-local signals…

AA Kaverzin and BJ van Wees PRB (2015)

Weakly Hydrogenated Graphene No Hanle precession!?

Strain-Induced Pseudo–Magnetic Fields Greater Than 300 Tesla in Graphene Nanobubbles

• N. Levy,1,2*† S. A. Burke,1*‡ K. L. Meaker,1 M. Panlasigui,1 A. Zettl,1,2 F. Guinea,3
• A. H. Castro Neto,4 M. F. Crommie1,2§

C B

Experiment

2nm 30 JULY 2010 VOL 329 SCIENCE

F

A B C

4Å 0Å

K K0

LETTERS

Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering

• F. Guinea1*, M. I. Katsnelson2 and A. K. Geim3*

Opposite Pseudo-magnetic field

• n opposite valleys

by TRS

X

Strained, semi-classically

y x (b)

A Chavez et al Phys Rev B (2010) Real Magnetic field Non uniform strain

˙ r = uk, ˙ k = (eE + ⌧z ˙ r ⇥ Bs)

Semiclassical equations of motion

uk = rk✏k ✏k = ±vF |k| Bs = r × As(r)

@tnk + ˙ r · rrnk + ˙ k · rk ⇥ n0

k + nk

⇤ = I[nk]

Semiclassical Transport

δnk = ρk1 + Pk · τ

Describes quantum coherence between valleys

Strain induced Valley Hall Effect

Non uniform (shear) strain (~ 1%)

X-P Zhang, CL Huang, and MAC 2D Materials (2017)

ωc ⌧ max{τ −1

D , kBT}

Valley currents do not exhibit the Hanle effect!

Interplay of VHE and SHE

X-P Zhang, CL Huang, and MAC in preparation

Hanle oscillations are suppressed!

2 4 6 8 10

• 0.5

0.0 0.5 1.0 = 0.2T = 0.0T = 0.4T = 1.0T

Non uniform strain Hydrogen+ nonuniform strain

Outline: Part II

Topology = Dimensional reduction? Impurity in a 1D Channel w and w/o interactions: Kane & Fisher Magnetic Impurity near the non-interacting edge of a 2D QSHI Magnetic Impurity near the interacting edge of a 2D QSHI

Low energy description: Topology = Dimensional Reduction?

QSHI

Vimp

Quantum Spin Hall Insulators

Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells

• B. Andrei Bernevig,1,2 Taylor L. Hughes,1 Shou-Cheng Zhang1*

T

SCIENCE VOL 314 15 DECEMBER 2006

R14,23 / Ω

(Vg – Vthr) / V

M König et al Science (2007) S Tang et al Nat. Phys (2017)

LETTERS

Quantum spin Hall state in monolayer 1T’-WTe2

• Figure 1 | Two-terminal characteristics of WTe devices. a, Structure and lattice constants of
• =0.20 μm,

Z Fei et al Nat. Phys (2017)

Absence of conductance quantization

In samples with long edge channels two-terminal conductance is NOT quantized! DEVIATIONS FROM 2e2/h The mechanism for backscattering is not clear, but conductance exhibits quantum critical SCALING with T and V

Review: G Dolcetto, M Sassetti, T L Schmidt arxiv:1511.0614 (2015)

Are 1D models of the edge channels always complete?

Outline: Part II

Topology = Dimensional reduction? Impurity in a 1D Channel w and w/o interactions: Kane & Fisher Magnetic Impurity near the non-interacting edge of a 2D QSHI Magnetic Impurity near the interacting edge of a 2D QSHI

Structureless Impurity in 1D

Weak impurity: Potential scattering

Non-interacting electrons

Strong scatterer limit |✏0| t

t0 = t2 ✏0

H = −t X

n

h c†

ncn+1 + c† n+1cn

i +✏0c†

0c0

Strong impurity: Weak tunneling link

H = −t X

n

h c†

ncn+1 + c† n+1cn

i −t0 h c†

1c−1 + c† +1c−1

i

T(✏) = G0(✏) + G0(✏)V T(✏) G0(✏) = (✏+ − H0)−1

Lippmann-Schwinger Equation Conductance from Landauer-Buttiker

T (✏) ∼ T(✏) G(✏) = |T (✏)|2 = |T0|2 = const.

(For ε near the center of the band)

The Kane-Fisher Problem: Impurity in 1D

C Kane and MPA Fisher Phys Rev Lett 1992

Weak impurity: Potential scattering Strong impurity: Weak tunneling link

Add interactions

ω ∼ kBT ρ(T) ∼ T

1 K −1

H = X

n

h −t ⇣ c†

ncn+1 + c† n+1cn

⌘ +V c†

nc† n+1cn+1cn

i + ✏0c†

0c0

Tunneling density of states

STM tip

ρ(ω) ∼ |ω|

1 K −1

dt0 d` = ✓ 1 − 1 K ◆ t0

G(T) ∼ |t0(T)|2 ⇢ → 0 K < 1 (V > 0) → t K > 0 (V < 0)

F Guinea and M Ueda Z Phys B (1991)

Another way to understand it …

Hartree approximation to interactions

Hint = V X

n

c†

nc† n+1cn+1cn ! HMF A int

= V X

n

[hnninn+1 + hnninn+1]

Impurity induced Fiedel Oscillation

hnni ⇠ sin(2kF xn) xn

D Yue, L Glazman, K A Mateev Phys Rev B (1994)

Solve external + Hartree potential

Transmission coefficient develops log singularity (becomes a power-law when resumed)

T (✏) = T0  1 + c|R0|2V log

• ✏F − ✏

D

• + · · ·
• ∼ |✏ − ✏F |c|R0|2V

Phase diagram

V > 0 V < 0 V = 0

C Kane and MPA Fisher Phys Rev Lett 1992

✏0 = 0 (t0 = t)

t0 = 0 (✏0 = +∞)

Impurity strength Interaction

Experiment and Numerics

MAC and JB Marston Phys Rev Lett (2001)

0.0 10 5.0 10

• 3

1.0 10

• 2

1.5 10

• 2

2.0 10

• 2

4 8 12 16 20 24 Exact solution for V= 0 Td-DMRG for V = 0 Td-DMRG for V/w = +0.5 Td-DMRG for V/w = - 0.5

current across the junction J(t)

time t

.

Numerics: Time-dependent DMRG

• 101

G (µS) T (K)

Experiment Z Yao et al Nature (1999) Scaling

Power-law conductace

Outline: Part II

Topology = Dimensional reduction? Impurity in a 1D Channel w and w/o interactions: Kane & Fisher Magnetic Impurity near the non-interacting edge of a 2D QSHI Magnetic Impurity near the interacting edge of a 2D QSHI

Backsatterer in a non interacting TI

H0 = HKM = −t X

hi,ji

c†

icj + iλSO

X

hhi,jii

νijc†

iszcj

Semi-infinite Kane-Mele model (with zigzag edge) No interactions: Solve Scattering problem (Lippmann-Schwinger equation)

|Ψs(kx)i = |Φs(kx)i + VintG0(✏)|Ψs(kx)i

Vimp = λimp sx

+ TRS breaking impurity (backscatter)

↑

Hs

0(kx, ˆ

)Φs(kx, y) = ✏Φs(kx, y),

Look for solutions of the form

Φs(kx, y) ∼ e−κy

Spectrum of the pristine system

Transmission coefficient

Impurity on 1st atomic row No coupling to bulk states (zero weight on the 1st atomic row)

Indendent of the energy (like 1D system)

Energy dependent scattering Beyond 1st row

What is going on?

Vimp

Impurity in the bulk of the crystal

Vimp = λimp sx

G0(✏) ∼ ✏ |✏| < ∆ 2 (Im G0(✏) = 0)

Bulk Green’s function (particle-hole symmetric)

1 − G0(✏)Vimp = 0 ⇒ 1 − g0imp ✏ sx = 0

Bound state?

T(✏) = Vimp + VimpG0(✏)T(✏)

T(✏) = Vimp 1 − G0(✏)Vimp

Lippmann-Schwinger Equation

✏Bound = ✏0 ∼ ± 1 imp

Bound state?

1 − G0(✏)Vimp = 0 ✏ ∈ Reals

Fitting an effective low energy model

0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.0 0.2 0.4 0.6 0.8 1.0 Ε Λimp40 Vc VB Ε0 20 30 40 50 60 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Λimp

Transmission coefficient Model parameters (N = 2)

Edge states

Bound states resonate with edge Use discrete symmetries of the microscopic model

(TRS + π-rotation, TRS+p-h transformation)

Heff = HB + H+ [u, t+ (0)] + H− [d, t− (0)] , HB = ivF Z dx †sz@x + VBa0 †(0)sx (0), H±[f, ] = ±✏0

• f †f − 1

2

• + Vca1/2

⇥ f † + h.c. ⇤

Two-level Fano model

Outline: Part II

Topology = Dimensional reduction? Impurity in a 1D Channel w and w/o interactions: Kane & Fisher Magnetic Impurity near the non-interacting edge of a 2D QSHI Magnetic Impurity near the interacting edge of a 2D QSHI

Adding interactions

Non-resonant case = Kane-Fisher

✏F 6= ±✏0

V 0

B ' VB

 V 2

c

✏0 ✏F + V 2

c

✏0 + ✏F

• Integrate out both bound states

Edge Edge

Resonant case

✏F = ±✏0

Back to Kane and Fisher

✏0 ∝ V 0

B

Integrate out one bound state

tc

✏0 ∝ V 0

B

dyB d ln ξ = (1 − K) yB + y2

t .

dyt d ln ξ = 1 − K/4 − (1 − δF )2K1/4 yt + yt(yB + vB), dδF d ln ξ = 4(1 − δF )y2

t ,

dvB d ln ξ = (1 − K)vB.

Tunneling flows to strong coupling also for moderately attractive interactions!!

tc

✏0 ∝ V 0

B

Side-coupled resonant level

M Goldstein and R Berkovits Phys Rev Lett (2011)

Adding interactions: RG Flow

Compare to Kane-Fisher

dyB d log ξ = (1 − K)yB

Broadening of transmission resonance at low T

↓ T

• 1.0
• 0.5

• Resonant enhancement of Skew Scattering in

Graphene

• Direct coupling between non equilibrium spin

polarization and charge current induced by impurities: Anisotropic spin precession (ASP)

• ASP can lead to negative non local resistance and

asymetry in the Hanle precession

• Strong coupling limit of a magnetic impurity near the

edge of a QSHI induces resonant states and non-trivial behavior of the transmission at resonance

Summary & Conclusions

おおきに～