Wang Yao

The University of Hong Kong

Valley and spin physics in2D transition metal dichalcogenides

Prof. Xiaodong Cui(HKU)

Prof. Xiaodong Xu(U of Washington)

Acknowledgement 

Funding

CollaboratorsGroup @ HKU

Zhirui GongHongyi Yu Guibin Liu Pu Gong

Prof. Di Xiao(Carnegie Mellon U)

Outline

Valley physics from inversion symmetry breaking

Spin‐valley coupling in monolayer TMDCs

Interplay of spin, valley & layer in bilayer TMDCs

Exciton Dirac spectra in monolayer TMDCs

Valley index of Bloch electron

Valley index of Bloch electronDegenerate energy extrema of Bloch bands in momentum space

Long lifetime of valley polarization expectedIntervalley scattering suppressed by large k-space separation

In atomically thin 2D crystals: graphene, BN, MoS2 etc.

Valley polarization

Valley index of Bloch electron

Lesson from spintronics

ValleytronicsValley for encoding information

How to distinguish the valleys?

Control of the valley dynamics?

Measurable quantities associated with valley index?

= 0 = 1

Beenakker et al., Nat Phys. 07”Shayegan et al., PRL 06”

Valley vs spin for information processing

Magnetic moment

Hall effect

Spin ValleyIndex of Bloch 

electronAssociatedphysical phenomena

Optical selection rule

WY, Xiao & Niu, PRB 08”

Xiao, WY & Niu, PRL 07”

Valley physics from inversion symmetry breaking

Valley can be manipulated in ways similar to spin

Key quantities: Berry curvature & orbital magnetic moment

Hall effect

Valley contrasting properties by ISB

Time-reversal symmetry

Space-inversion symmetry

k k

k k

m m k k

m m k k

Valley contrasting properties

– Necessary condition: inversion symmetry breaking (ISB)– Opposite & m for a time reversal pair of valleys

Both symmetries 0 k 0m k

2

2

Example: graphene with staggered sublattice potential

H = at(�kx σx + ky σy )+∆2σz − λ�

σz − 1

2sz

Massive Dirac fermion:

Berry curvature 1 ( 1)z at valley K (-K)

Valley contrasting Berry curvature

Xiao, WY & Niu, PRL 99, 236809 (2007)

Valley Hall effect

Gapped energy dispersion

gapped Dirac cones

Valley optical selection rule

Magnetic moment 1 ( 1)z at valley K (-K)

magnetic moment of valley pseudospin

WY, Xiao & Niu, PRB 77, 235406 (2008)

Valley selection rule of interband transition

K ‐ KGapped energy dispersion

gapped Dirac cones

Layered structure suitable for extracting monolayer by mechanical exfoliation

Bulk or even‐layers Monolayer

with inversion symmetry

without inversion  symmetry

MX2

2D transition metal dichalcogenides

z

x

Top view

x

z Even‐odd oscillation of SHGZeng, et al. Sci Rep 13”

Indirect bandgap Direct bandgap

Splendiani et al., NL 10”Mak et al., PRL 10”

Monolayer group VIB TMDCs

H = at(�kx σx + ky σy )+∆2σz − λ�

σz − 1

2sz

|φci = |dz2 i

|φ⌧v i =1p2(|dx 2 − y2 i + i�|dxy i )Basis:

Hamiltonian:

a ∆ t 2λM oS2 3.193 1.66 1.10 0.15

W S2 3.197 1.79 1.37 0.43

M oSe2 3.313 1.47 0.94 0.18

W Se2 3.310 1.60 1.19 0.46

cos c v

Valley optical selection rule

K ‐ K

k 4cos22cos2

Degree of circular polarization:

Massive Dirac fermions at the band edge

Valley Hall effectBerry curvature: k 3t2

2(2 3k 2a2t2 )3/2

Valley index: 1 1 at K (-K) valley

eV

Xiao, Liu, Feng, Xu & WY, PRL 108, 196802 (2012)

(neglecting SOC)

Optical generation of valley polarization

Cui group @HKU: Zeng, Dai, WY, Xiao & Cui, Nature Nano. 12”

Parallel works: Heinz group @Columbia (Nat. Nano. 12”); PKU‐CAS group (Nat. Comm. 12”) 

Optical pump of valley polarization

K ‐ K

Electrically tunable polarized PL in biased bilayer MoS2

Polarized PL under circular polarized excitation in monolayer MoS2

Optical detection of valley polarization

Pump with ‐ light => e‐h pairs in valley K

Valley polarization of e (h) => Faraday rotation

Valley polarization of e‐h pair => polarized photoluminescence

Xu group @ UW: Wu, Ross, Liu et al., Nature Physics 13”

Valley optical selection rule

Controllable inversion symmetry breaking by perpendicular electric field

Absence of Hanle effect: magnetic field do not couple K & –K

Strong Coulomb binding: valley excitons with optical selection rules

σ+ Incident

Energy (eV)

1.60 1.65 1.75

Gat

e (V

)

Photon Energy (eV)

Xo

X+

X-

X -’

60

0

-60

40

20

-40

-20

1.70

σ- Incident

Black: σ+Red: σ-

-5V

Xo

450

300

150

0

-5V

+10VX- Xo

360

240

120

+10V

-60VX+150

75

0 1.60 1.65 1.70 1.75 1.60 1.65 1.70 1.75

-60V

Energy (eV)

Valley polarization of excitons & trions

‐K K‐K K

Detection

eehh

eehhhh

eehhee

σ-σ+

Prof. Xiaodong Xu

Jones, Yu et al., Nature Nano. 13”

monolayer WSe2

Optical generation of valley coherence

Valley optical selection rule

Linear polarized light excite two valleys in linear superposition

Possibility to address valley coherence in macro systems

(Jones, Yu et al., Nature Nanotech 13”)

= +

Linear polarized PL: polarization angle coincide with excitation

Optical injected valley coherence can survive carrier relaxations

DetectionBlack: HRed: V

H Incident

X- Xo

+10V

Xo

-5V

X+-60V

1.60 1.65 1.70 1.75Energy (eV)

σ+ IncidentDetectionBlack: σ +

Red: σ -

-5V

Xo

450

300

150

0

-60VX+

150

75

0 1.60 1.65 1.70 1.75Energy (eV)

+10VX- Xo

360

240

120

PL In

tens

ity (c

ount

s/se

cond

)Excitonic valley coherence in ML WSe2

Valley polarization Valley coherence

Only X0 has linearly polarized PL

PL angle (degre

e)

In cid ent an gle (d egree)

Polarization

120

60

0

180

120600 180

0.4

0.2

0.0

C

Jones, Yu et al., Nature Nano. 13”

‐K K ‐K K

Valley coherence of X‐ broken by exchange w extra electron

Optical orientation of valley pseudospin

|K> + ei2 |-K>

|-K>

|K> 

Valley pseudospin of electron‐hole pair (exciton)

(e‐h pair in valley K)

(e‐h pair in valley ‐K)

Outline

Valley physics from inversion symmetry breaking

Spin‐valley coupling in monolayer TMDCs

Interplay of spin, valley & layer in bilayer TMDCs

Exciton Dirac spectra in monolayer TMDCs

Out of plane spin

In plane spin

2D crystal with mirror symmetry

E(, k) E(, k)

Spin orbit coupling has to be out‐of‐plane, i.e.   f (k)sz

Spin-valley coupling in monolayer

Time reversal symmetry

f (k) f (k)

mirror sym    +    time reversal sym    +    broken inversion sym

Hsoc zsz

Inversion symmetry

f (k) f (k)

Spin-valley coupling in monolayer

2D crystal with mirror symmetry  =>  SOC f (k)sz

z 1 z 1

Spin-valley coupled massive Dirac fermions

• Spin‐valley coupling of hole (~ 0.15 eV in MoX2, ~ 0.4 eV in WX2) 

Basis:

Hamiltonian:ra

H = at(�kx σx + ky σy)+∆2σz − λ�

σz − 1

2sz

|φci = |dz2 i

|φ⌧v i =1p2(|dx2 − y2 i + i�|dxy i) (m 2)

(m 0) On-site SOC:

L S LzSz 12

(LS LS )

• Spin‐valley coupling of electron (O(1) － O(10) meV)

K - K

• Spin and valley flip suppressed

• Valley Hall accompanied by spin Hall

Sign difference between MoX2 & WX2

K ‐ K

K ‐ K

WX2

MoX2

Guibin Liu et al., PRB 88, 085433 (2013)

mainly from coupling to remote m=±1 d band 

mainly from mix in of porbitals 

Spin dependent optical selection rule

Valley optical selection rule

K ‐ K

K ‐ K

AB

Valley & spin optical

selection rule

Selective excitation of valley & spin controlled by light polarization & freq

Xiao, Liu, Feng, Xu & WY, PRL 108, 196802 (2012)

WY, Xiao & Niu, PRB 77, 235406 (2008)

Outline

Valley physics from inversion symmetry breaking

Spin‐valley coupling in monolayer TMDCs

Interplay of spin, valley & layer in bilayer TMDCs

Exciton Dirac spectra in monolayer TMDCs

AB stacked TMDC bilayer & multilayers

• Neighboring layers are 180o rotation of each other

• 180o rotation switch the valleys but leave spin unchanged

AB stackingK -K

Hsocu zsz

Hsocl zsz

Hsoc z zsz

z 1

z 1

z 1 z 1

• Valley and layer dependent spin splitting:

Gong et al., Nat. Comm. 4, 2053 (2013). 

AB stacking

K -K

Hopping at K:

Top L Bottom LHopping amplitude~ 0.1 eV

~ 0.15 eV for MoX2~ 0.4 eV for WX2

Top Layer

Bottom Layer

Suppression of interlayer hopping

• Interlayer hopping conserves spin and in‐plane momentum

Energy cost:

w SOC

w/o SOC

WS2 thin films

Suppression of interlayer hopping

Zeng, Liu, et al. Scientific Reports 3, 168 (2013)

w SOC

PL from WS2 :

PL from WSe2 :

IA B

Suppression of interlayer hopping

Zeng, Liu, et al. Scientific Reports 3, 168 (2013)

Prof. Xiaodong Cui

Spin & valley dependent layer polarization

Spin and valley dependent layer polarization:

Band edge carrier near K points:! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! cos 2! , !!!!!cos 2! ≡!

! ! ! !!!

K

-K

Two‐sets of bands localized in opposite layers

~ 85% in MoX2~ 95% in WX2

~ 100%

Gong et al., Nat. Comm. 4, 2053 (2013). 

Conduction band at ±K:     hopping vanishes in leading order=> even larger ratio of  over t, 

Spin Hall & Spin circular dichroism

K -K

Bilayer optical selection rule:

Spin circular dichroism in bilayer

-K

K

sin2

cos2

Gong et al., Nat. Comm. 4, 2053 (2013). 

E

E

Spin Hall in bilayer

ME effects from spin-layer locking

K

-K

tB0

1

-10

zs

Valley ‐K

Valley K

• Electrically tunable spin Larmor precession

0

-0.4

0.4

0 40 80

Ez

B0 0 40 80tB0

Valley dependent precession frequencies

Spin‐layer locking

Gong et al., Nat. Comm. 4, 2053 (2013). 

• Oscillation of layer (electric) polarization in magnetic field

K K

K

K

K

-K

Spin doublet couples to both electric & magnetic fields, in different ways

K

-K

Spin‐layer locking

ME effects from spin-layer locking

Valley conditioned spin rotations

BzEz

Bx

t

Valley dependent spin splitting by E & B fields in z direction

Valley dependent spin resonance by oscillating Bx

K

-K

K -K

K

K

K

K

Faraday geometry

Gong et al., Nat. Comm. 4, 2053 (2013). 

Electrically & magnetically driven ESR

K K

K K

K K

K K

Bx

Ez

t +_

Bz

t

K -K

Electrically driven ESR and magnetically driven ESR

Valley dependent interference of electric & magnetic fields

Voigt geometry

K

-K

Spin‐layer locking

E

v

c

⇑

Lower LayerUpper Layer

ɷ1ɷ2 c v‐ɷ1 ɷ2‐ =

σ+

σ-

Nor

mal

ized

PL

150V

60V

90V

120V

Energy (eV)1.61.651.71.75

ɷ1ɷ2

Evidence of spin-layer locking in bilayer PL

~ 100%

~ 95%

Prof. Xiaodong Xu

Jones, Yu, et al., Nat. Phy. 10, 130, 14”

PL from trion in BL WSe2

Electrically induced Zeeman splitting

K

150V

Energy (eV)1.6 1.65 1.7 1.75

Excitation: V

Black: VRed: H

ɷ1ɷ2150V

Energy (eV)1.6 1.65 1.7 1.75

Energy (eV)1.6 1.7 1.75

Jones, Yu, et al., Nat. Phy. 10, 130, 14”

VV

XIntralayerXInterlayer

X- Xo

+10V

1.60 1.65 1.70 1.75Energy (eV)

Monolayer WSe2

Interlayer & intralayer trionBilayer WSe2

Intralayer X‐: valley coherence suppressed, similar to monolayer

Interlayer X‐: valley coherence preserved, no exchange with excess electron

bottom layer has lower energy for excess electron

Outline

Valley physics from inversion symmetry breaking

Spin‐valley coupling in monolayer TMDCs

Interplay of spin, valley & layer in bilayer TMDCs

Exciton Dirac spectra in monolayer TMDCs

Tightly bound valley excitons in monolayer

1.60 1.65 1.75

Gat

e (V

)

Photon Energy (eV)

Xo

X+

X-

X -’

60

0

-60

40

20

-40

-20

1.70

Ultra strong coulomb binding

X0 binding energy: 0.5 – 1 eVBohr radius: ~ 1 nm

Large effective mass & reduced screening in 2D

Valley configurations

K -K

K -K

=

=

σ+

Valley‐orbit coupling

Trion binding 30 meVK -K

V(k)

strong e‐h exchange

Valley-orbit coupling of exciton

Effective valley‐orbit coupling

~ 10-2K

light cone

longitudinal branch

transverse branch

~ 2 meV

ωu

ωd

ω0~ 10-3K

probability for e-h to overlap~ aB

2

Coulomb in 2D

linear in k

strong coupling: VOC splitting >> radiative decay

chirality of 2

vanish at k = 0

rotation symmetry

Hongyi Yu et al. arXiv 1401.0667

Effect of tensile strain

-0.01 0 0.01

-0.01

0.01

0

kx / K

k y /Klight cone

2K J0

J

strain breaks rotational symmetry

I = 2 VOCin-plane

Zeeman field

one I = 2 cone

two I = 1 cones

light cone

2J0

20JcK

Linearly dispersed Dirac saddle point

Yu et al. arXiv 1401.0667

Gapped Dirac cone of trion

K -K K -K K -K K -K

exchangeexchange

Negatively charged trions

• Indexed by polarization of emitting photon + spin of extra electron (s)

• Exchange coupling with the extra electron

• An effective out-of-plane Zeeman field conditioned on the extra spin

valley pseudospin of recombining e-h pair ()

K -K K -K K -K K -K

exchangeexchange

E

10

Trion brightness

qX-1 0.991.01 -1 -1.01-0.99

0

-5

5

Ener

gy (m

eV)

Berry curvature (10

4Å2)

0

-2

2

≈≈

/ K

Gapped Dirac cone of trion

Trion valley Hall

Summary

Valley dependent Hall current, magnetic moment, optical selection rule from inversion symmetry breaking

A pair of time reversal symmetric valleys may play similar roles like spin in electronic applications

Strong spin‐valley coupling in monolayer TDMCs: valley control enables spin control

Coupling of layer pseudospin to valley & spin in bilayers: magnetoelectric effects, valley conditioned spin control

e‐h exchange of the tightly bound excitons: strong valley‐orbit coupling, strain tunable Dirac spectra