Valley and spin physics in 2D transition metal dichalcogenides
Valley and spin physics in 2D transition metal dichalcogenides
Wang Yao The University of Hong Kong
Acknowledgement Collaborators
Group @ HKU
Hongyi Yu Guibin Liu
Pu Gong
Zhirui Gong Prof. Xiaodong Xu (U of Washington)
Prof. Xiaodong Cui (HKU)
Funding 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 electron Degenerate energy extrema of Bloch bands in momentum space In atomically thin 2D crystals: graphene, BN, MoS2 etc.
Long lifetime of valley polarization expected Intervalley scattering suppressed by large k-space separation
Valley polarization
Valley index of Bloch electron Valleytronics Valley for encoding information
Beenakker et al., Nat Phys. 07” Shayegan et al., PRL 06”
How to distinguish the valleys? Control of the valley dynamics?
Lesson from spintronics Measurable quantities associated with valley index?
= 0
= 1
Valley vs spin for information processing Index of Bloch electron
Associated physical phenomena
Magnetic moment Hall effect Optical selection rule
Spin
Valley
Xiao, WY & Niu, PRL 07”
WY, Xiao & Niu, PRB 08”
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
k k
m k m k
Space-inversion symmetry
k k
m k m k
Both symmetries
k 0
m k 0
Valley contrasting properties –
Opposite &
–
Necessary condition: inversion symmetry breaking (ISB)
m for a time reversal pair of valleys
Example: graphene with staggered sublattice potential 2
2
Massive Dirac fermion:
∆ σ ˆz − 1 Hˆ = at( kx σˆx + ky σˆy ) + σˆz − λ sˆz 2 2
Valley contrasting Berry curvature Berry curvature
z 1 (1) at valley K (-K)
Valley Hall effect
Gapped energy dispersion
gapped Dirac cones Xiao, WY & Niu, PRL 99, 236809 (2007)
Valley optical selection rule Magnetic moment
z 1 (1) at valley K (-K)
magnetic moment of valley pseudospin
Gapped energy dispersion
Valley selection rule of interband transition K
‐K
gapped Dirac cones WY, Xiao & Niu, PRB 77, 235406 (2008)
2D transition metal dichalcogenides Top view
MX2
x z
Layered structure suitable for extracting monolayer by mechanical exfoliation
Bulk or even‐layers
z x
Monolayer
Indirect bandgap
Direct bandgap
with inversion symmetry
without inversion symmetry
Even‐odd oscillation of SHG Zeng, et al. Sci Rep 13”
Splendiani et al., NL 10” Mak et al., PRL 10”
Monolayer group VIB TMDCs
Massive Dirac fermions at the band edge Hamiltonian: Hˆ
= at( kx σ ˆx + k y σ ˆy ) +
1 2
∆ σ ˆz − 1 sˆz SOC) σ ˆz − λ (neglecting 2 2
Basis: |φ⌧ v i = p (|d x 2 − y 2 i + i |dx y i )
a
|φc i = |dz 2 i
Valley index: 1 1 at K (-K) valley
M oS 2 W S2 M oSe2 W Se2
3.193 3.197 3.313 3.310
∆ 1.66 1.79 1.47 1.60
Valley Hall effect Berry curvature:
t 1.10 1.37 0.94 1.19
2λ 0.15 0.43 0.18 0.46
eV
3t 2 k 2( 2 3k 2 a 2 t 2 )3/2
K
‐K
Valley optical selection rule
Degree of circular polarization:
k
4 cos 2 2 cos2
cos
c v
Xiao, Liu, Feng, Xu & WY, PRL 108, 196802 (2012)
Optical generation of valley polarization Optical pump of valley polarization
Valley optical selection rule K
Pump with ‐ light => e‐h pairs in valley K
Optical detection of valley polarization Valley polarization of e (h) => Faraday rotation
‐K
Valley polarization of e‐h pair => polarized photoluminescence Absence of Hanle effect: magnetic field do not couple K & –K
Polarized PL under circular polarized excitation in monolayer MoS2 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”)
Electrically tunable polarized PL in biased bilayer MoS2 Controllable inversion symmetry breaking by perpendicular electric field Xu group @ UW: Wu, Ross, Liu et al., Nature Physics 13”
Strong Coulomb binding: valley excitons with optical selection rules
Valley polarization of excitons & trions monolayer WSe2 60 40
Black: σ+ Red: σ-
20
Prof. Xiaodong Xu 360 +10V
+10V X-
240
Xo
Gate (V)
Detection
X -’ e e h X-
0
-5V
450 -5V 300
-20
Xo
-60 1.60
e hh
‐K
-60V
-60V
1.65
X+ 1.70
1.75
Photon Energy (eV)
150 150
h e
-40
120
0
Xo
K
‐K
K
X+
σ-
75 0 1.60 1.65 1.70 1.75 Energy (eV)
1.60 1.65 1.70 1.75 Energy (eV)
σ+ Incident
σ- Incident
σ+
Jones, Yu et al., Nature Nano. 13”
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 Linear polarized PL: polarization angle coincide with excitation Optical injected valley coherence can survive carrier relaxations (Jones, Yu et al., Nature Nanotech 13”)
Excitonic valley coherence in ML WSe2 Valley polarization σ+
K
‐K
K
‐K
Detection Black: σ + Red: σ -
Incident
H Incident
X-
Xo
120 450 -5V 300
Xo
X-
Valley coherence of X‐ broken by exchange w extra electron
Xo
-5V
Only X0 has linearly polarized PL
Xo
150
150
-60V X+
75
180 C 0.4 120
0
0
0.2
60
0
P o larizatio n
0
P L a n g le (d eg ree)
PL Intensity (counts/second)
Detection Black: H Red: V
+10V
360 +10V 240
Valley coherence
X+
0.0 60 120 180 In cid e n t an g le (d e g re e)
1.60 1.65 1.70 1.75 Energy (eV)
-60V
Jones, Yu et al., Nature Nano. 13”
1.60 1.65 1.70 1.75 Energy (eV)
Optical orientation of valley pseudospin Valley pseudospin of electron‐hole pair (exciton) |K> + ei2 |K>
|K> (e‐h pair in valley K)
|-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
Spin-valley coupling in monolayer 2D crystal with mirror symmetry Out of plane spin
In plane spin
E(, k) E(, k)
Spin orbit coupling has to be out‐of‐plane, i.e. f (k)sz
Spin-valley coupling in monolayer 2D crystal with mirror symmetry => SOC f (k)sz Time reversal symmetry
f (k) f (k)
Inversion symmetry
f (k) f (k)
mirror sym + time reversal sym + broken inversion sym
z 1
H soc z sz
z 1
Spin-valley coupled massive Dirac fermions ra
Hamiltonian: Hˆ
= at( kx σ ˆx + k y σ ˆy )+
∆ σ ˆz − 1 ˆz − λ σ sˆz 2 2
1 2
Basis: |φ⌧ v i = p (|dx 2 − y 2 i + i |dx y i ) (m 2) |φc i = |dz 2 i
K
(m 0)
-K
On-site SOC: 1 L S Lz Sz (L S L S ) 2
• Spin‐valley coupling of hole (~ 0.15 eV in MoX2, ~ 0.4 eV in WX2) • Spin and valley flip suppressed • Valley Hall accompanied by spin Hall • Spin‐valley coupling of electron (O(1) － O(10) meV)
Sign difference between MoX2 & WX2 WX2
MoX2
K
‐K
mainly from coupling to remote m=±1 d band
K
‐K
mainly from mix in of p orbitals
Guibin Liu et al., PRB 88, 085433 (2013)
Spin dependent optical selection rule K
Valley optical selection rule
‐K
Valley & spin optical selection rule
WY, Xiao & Niu, PRB 77, 235406 (2008)
K
‐K
B
A
Xiao, Liu, Feng, Xu & WY, PRL 108, 196802 (2012)
Selective excitation of valley & spin controlled by light polarization & freq
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 z 1
AB stacking z 1
K
z 1 -K
z 1
u H soc z sz
l H soc z sz
• Neighboring layers are 180o rotation of each other • 180o rotation switch the valleys but leave spin unchanged • Valley and layer dependent spin splitting: H soc z z sz Gong et al., Nat. Comm. 4, 2053 (2013).
Suppression of interlayer hopping AB stacking
K
-K
Top Layer
Bottom Layer
• Interlayer hopping conserves spin and in‐plane momentum Hopping at K:
Top L
Hopping amplitude ~ 0.1 eV
Bottom L Energy cost:
~ 0.15 eV for MoX2 ~ 0.4 eV for WX2
Suppression of interlayer hopping WS2 thin films
w/o SOC
w SOC
Zeng, Liu, et al. Scientific Reports 3, 168 (2013)
Suppression of interlayer hopping PL from WS2 : Prof. Xiaodong Cui
PL from WSe2 :
A B I
w SOC
Zeng, Liu, et al. Scientific Reports 3, 168 (2013)
Spin & valley dependent layer polarization Band edge carrier near K points: !
!
! ! ! ! ! !!
Gong et al., Nat. Comm. 4, 2053 (2013).
!! ! !
Spin and valley dependent layer polarization: !!
! ! ! ! cos 2! , !!!!!cos 2! ≡
!
!
!
! !!
!
Conduction band at ±K: hopping vanishes in leading order => even larger ratio of over t, Two‐sets of bands localized in opposite layers K
-K
~ 100%
~ 85% in MoX2 ~ 95% in WX2
Spin Hall & Spin circular dichroism K E
-K
Gong et al., Nat. Comm. 4, 2053 (2013).
E
Spin Hall in bilayer
Spin circular dichroism in bilayer
Bilayer optical selection rule:
K
sin 2
-K
cos2
ME effects from spin-layer locking • Oscillation of layer (electric) polarization in magnetic field • Electrically tunable spin Larmor precession Gong et al., Nat. Comm. 4, 2053 (2013).
sz Valley ‐K
0.4
Ez 0 B0
1 0
Valley K
-1
0
40
80
tB0
-0.4 0
40
tB0
80
Valley dependent precession frequencies
K
Spin‐layer locking
-K
ME effects from spin-layer locking K
-K
K
K
K
K
Spin doublet couples to both electric & magnetic fields, in different ways
K
Spin‐layer locking
-K
Valley conditioned spin rotations Ez
Bz
K
t
-K
Bx
Faraday geometry Valley dependent spin splitting by E & B fields in z direction Valley dependent spin resonance by oscillating Bx -K
K K
K
K
Gong et al., Nat. Comm. 4, 2053 (2013).
K
Electrically & magnetically driven ESR Bx
K
-K K K
K K
Ez
t
t
Bz
_
+ K K
K K
Voigt geometry Electrically driven ESR and magnetically driven ESR Valley dependent interference of electric & magnetic fields K
Spin‐layer locking
-K
Evidence of spin-layer locking in bilayer PL PL from trion in BL WSe2 60V
σ+ σ-
E
Normalized PL
90V
Prof. Xiaodong Xu
Electrically induced Zeeman splitting 120V
Upper Layer
K
Lower Layer
c
150Vɷ 2
ɷ1
~ 100%
ɷ1 ɷ2
v
⇑ 1.6 1.65 1.71.75 Energy (eV)
ɷ1 ‐ ɷ2 = c ‐ v
~ 95%
Jones, Yu, et al., Nat. Phy. 10, 130, 14”
Interlayer & intralayer trion Intralayer X V
Bilayer WSe2 ɷ2
150V
Interlayer X
ɷ1 V
Excitation: V
Monolayer WSe2 +10V X-
1.60
bottom layer has lower energy for excess electron
Black: V Red: H
Xo
1.65 1.70 1.75 Energy (eV)
1.6
1.65
1.7 Energy (eV)
1.75
Jones, Yu, et al., Nat. Phy. 10, 130, 14”
Intralayer X‐: valley coherence suppressed, similar to monolayer Interlayer X‐: valley coherence preserved, no exchange with 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 Ultra strong coulomb binding
Valley configurations
Large effective mass & reduced screening in 2D X0 binding
energy: 0.5 – 1 eV
Bohr radius: ~ 1 nm Trion binding 30 meV
strong e‐h exchange
60
Gate (V)
40
=
σ+
-K
K
V(k)
X -’
-K
K
Valley‐orbit coupling
20 X-
0
Xo
-20
K
=
-40 -60 1.60
1.65
X+ 1.70
1.75
Photon Energy (eV)
-K
Valley-orbit coupling of exciton Effective valley‐orbit coupling rotation symmetry
Coulomb in 2D
~a
2 B
chirality of 2
linear in k
probability for e-h to overlap
vanish at k = 0
strong coupling: VOC splitting >> radiative decay light cone
longitudinal branch Hongyi Yu et al. arXiv 1401.0667
transverse branch
~ 10-2K
ωu ~ 10-3K
ω0
~ 2 meV
ωd
Effect of tensile strain strain breaks rotational symmetry in-plane Zeeman field
I = 2 VOC 2 0 J cK
Linearly dispersed Dirac saddle point light cone
Yu et al. arXiv 1401.0667 2J0
light cone
one I = 2 cone
J0 J
0
-0.01
two I = 1 cones 2K
ky / K
0.01
-0.01
0
kx / K
0.01
Gapped Dirac cone of trion Negatively charged trions K
-K exchange
•
K
-K
K
-K
K
-K
exchange
Indexed by polarization of emitting photon + spin of extra electron (s) valley pseudospin of recombining e-h pair ()
•
Exchange coupling with the extra electron
•
An effective out-of-plane Zeeman field conditioned on the extra spin
Gapped Dirac cone of trion Trion valley Hall
-K
K
-K
K
-K
K
exchange
K
exchange
≈
0
0
-2 -5 1
0.99
≈
Energy (meV)
2
qX- / K
-0.99
-1
-1.01
Berry curvature (104Å2)
5
1.01
E
-K
Trion brightness 0
1
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
Wang Yao The University of Hong Kong
Acknowledgement Collaborators
Group @ HKU
Hongyi Yu Guibin Liu
Pu Gong
Zhirui Gong Prof. Xiaodong Xu (U of Washington)
Prof. Xiaodong Cui (HKU)
Funding 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 electron Degenerate energy extrema of Bloch bands in momentum space In atomically thin 2D crystals: graphene, BN, MoS2 etc.
Long lifetime of valley polarization expected Intervalley scattering suppressed by large k-space separation
Valley polarization
Valley index of Bloch electron Valleytronics Valley for encoding information
Beenakker et al., Nat Phys. 07” Shayegan et al., PRL 06”
How to distinguish the valleys? Control of the valley dynamics?
Lesson from spintronics Measurable quantities associated with valley index?
= 0
= 1
Valley vs spin for information processing Index of Bloch electron
Associated physical phenomena
Magnetic moment Hall effect Optical selection rule
Spin
Valley
Xiao, WY & Niu, PRL 07”
WY, Xiao & Niu, PRB 08”
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
k k
m k m k
Space-inversion symmetry
k k
m k m k
Both symmetries
k 0
m k 0
Valley contrasting properties –
Opposite &
–
Necessary condition: inversion symmetry breaking (ISB)
m for a time reversal pair of valleys
Example: graphene with staggered sublattice potential 2
2
Massive Dirac fermion:
∆ σ ˆz − 1 Hˆ = at( kx σˆx + ky σˆy ) + σˆz − λ sˆz 2 2
Valley contrasting Berry curvature Berry curvature
z 1 (1) at valley K (-K)
Valley Hall effect
Gapped energy dispersion
gapped Dirac cones Xiao, WY & Niu, PRL 99, 236809 (2007)
Valley optical selection rule Magnetic moment
z 1 (1) at valley K (-K)
magnetic moment of valley pseudospin
Gapped energy dispersion
Valley selection rule of interband transition K
‐K
gapped Dirac cones WY, Xiao & Niu, PRB 77, 235406 (2008)
2D transition metal dichalcogenides Top view
MX2
x z
Layered structure suitable for extracting monolayer by mechanical exfoliation
Bulk or even‐layers
z x
Monolayer
Indirect bandgap
Direct bandgap
with inversion symmetry
without inversion symmetry
Even‐odd oscillation of SHG Zeng, et al. Sci Rep 13”
Splendiani et al., NL 10” Mak et al., PRL 10”
Monolayer group VIB TMDCs
Massive Dirac fermions at the band edge Hamiltonian: Hˆ
= at( kx σ ˆx + k y σ ˆy ) +
1 2
∆ σ ˆz − 1 sˆz SOC) σ ˆz − λ (neglecting 2 2
Basis: |φ⌧ v i = p (|d x 2 − y 2 i + i |dx y i )
a
|φc i = |dz 2 i
Valley index: 1 1 at K (-K) valley
M oS 2 W S2 M oSe2 W Se2
3.193 3.197 3.313 3.310
∆ 1.66 1.79 1.47 1.60
Valley Hall effect Berry curvature:
t 1.10 1.37 0.94 1.19
2λ 0.15 0.43 0.18 0.46
eV
3t 2 k 2( 2 3k 2 a 2 t 2 )3/2
K
‐K
Valley optical selection rule
Degree of circular polarization:
k
4 cos 2 2 cos2
cos
c v
Xiao, Liu, Feng, Xu & WY, PRL 108, 196802 (2012)
Optical generation of valley polarization Optical pump of valley polarization
Valley optical selection rule K
Pump with ‐ light => e‐h pairs in valley K
Optical detection of valley polarization Valley polarization of e (h) => Faraday rotation
‐K
Valley polarization of e‐h pair => polarized photoluminescence Absence of Hanle effect: magnetic field do not couple K & –K
Polarized PL under circular polarized excitation in monolayer MoS2 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”)
Electrically tunable polarized PL in biased bilayer MoS2 Controllable inversion symmetry breaking by perpendicular electric field Xu group @ UW: Wu, Ross, Liu et al., Nature Physics 13”
Strong Coulomb binding: valley excitons with optical selection rules
Valley polarization of excitons & trions monolayer WSe2 60 40
Black: σ+ Red: σ-
20
Prof. Xiaodong Xu 360 +10V
+10V X-
240
Xo
Gate (V)
Detection
X -’ e e h X-
0
-5V
450 -5V 300
-20
Xo
-60 1.60
e hh
‐K
-60V
-60V
1.65
X+ 1.70
1.75
Photon Energy (eV)
150 150
h e
-40
120
0
Xo
K
‐K
K
X+
σ-
75 0 1.60 1.65 1.70 1.75 Energy (eV)
1.60 1.65 1.70 1.75 Energy (eV)
σ+ Incident
σ- Incident
σ+
Jones, Yu et al., Nature Nano. 13”
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 Linear polarized PL: polarization angle coincide with excitation Optical injected valley coherence can survive carrier relaxations (Jones, Yu et al., Nature Nanotech 13”)
Excitonic valley coherence in ML WSe2 Valley polarization σ+
K
‐K
K
‐K
Detection Black: σ + Red: σ -
Incident
H Incident
X-
Xo
120 450 -5V 300
Xo
X-
Valley coherence of X‐ broken by exchange w extra electron
Xo
-5V
Only X0 has linearly polarized PL
Xo
150
150
-60V X+
75
180 C 0.4 120
0
0
0.2
60
0
P o larizatio n
0
P L a n g le (d eg ree)
PL Intensity (counts/second)
Detection Black: H Red: V
+10V
360 +10V 240
Valley coherence
X+
0.0 60 120 180 In cid e n t an g le (d e g re e)
1.60 1.65 1.70 1.75 Energy (eV)
-60V
Jones, Yu et al., Nature Nano. 13”
1.60 1.65 1.70 1.75 Energy (eV)
Optical orientation of valley pseudospin Valley pseudospin of electron‐hole pair (exciton) |K> + ei2 |K>
|K> (e‐h pair in valley K)
|-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
Spin-valley coupling in monolayer 2D crystal with mirror symmetry Out of plane spin
In plane spin
E(, k) E(, k)
Spin orbit coupling has to be out‐of‐plane, i.e. f (k)sz
Spin-valley coupling in monolayer 2D crystal with mirror symmetry => SOC f (k)sz Time reversal symmetry
f (k) f (k)
Inversion symmetry
f (k) f (k)
mirror sym + time reversal sym + broken inversion sym
z 1
H soc z sz
z 1
Spin-valley coupled massive Dirac fermions ra
Hamiltonian: Hˆ
= at( kx σ ˆx + k y σ ˆy )+
∆ σ ˆz − 1 ˆz − λ σ sˆz 2 2
1 2
Basis: |φ⌧ v i = p (|dx 2 − y 2 i + i |dx y i ) (m 2) |φc i = |dz 2 i
K
(m 0)
-K
On-site SOC: 1 L S Lz Sz (L S L S ) 2
• Spin‐valley coupling of hole (~ 0.15 eV in MoX2, ~ 0.4 eV in WX2) • Spin and valley flip suppressed • Valley Hall accompanied by spin Hall • Spin‐valley coupling of electron (O(1) － O(10) meV)
Sign difference between MoX2 & WX2 WX2
MoX2
K
‐K
mainly from coupling to remote m=±1 d band
K
‐K
mainly from mix in of p orbitals
Guibin Liu et al., PRB 88, 085433 (2013)
Spin dependent optical selection rule K
Valley optical selection rule
‐K
Valley & spin optical selection rule
WY, Xiao & Niu, PRB 77, 235406 (2008)
K
‐K
B
A
Xiao, Liu, Feng, Xu & WY, PRL 108, 196802 (2012)
Selective excitation of valley & spin controlled by light polarization & freq
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 z 1
AB stacking z 1
K
z 1 -K
z 1
u H soc z sz
l H soc z sz
• Neighboring layers are 180o rotation of each other • 180o rotation switch the valleys but leave spin unchanged • Valley and layer dependent spin splitting: H soc z z sz Gong et al., Nat. Comm. 4, 2053 (2013).
Suppression of interlayer hopping AB stacking
K
-K
Top Layer
Bottom Layer
• Interlayer hopping conserves spin and in‐plane momentum Hopping at K:
Top L
Hopping amplitude ~ 0.1 eV
Bottom L Energy cost:
~ 0.15 eV for MoX2 ~ 0.4 eV for WX2
Suppression of interlayer hopping WS2 thin films
w/o SOC
w SOC
Zeng, Liu, et al. Scientific Reports 3, 168 (2013)
Suppression of interlayer hopping PL from WS2 : Prof. Xiaodong Cui
PL from WSe2 :
A B I
w SOC
Zeng, Liu, et al. Scientific Reports 3, 168 (2013)
Spin & valley dependent layer polarization Band edge carrier near K points: !
!
! ! ! ! ! !!
Gong et al., Nat. Comm. 4, 2053 (2013).
!! ! !
Spin and valley dependent layer polarization: !!
! ! ! ! cos 2! , !!!!!cos 2! ≡
!
!
!
! !!
!
Conduction band at ±K: hopping vanishes in leading order => even larger ratio of over t, Two‐sets of bands localized in opposite layers K
-K
~ 100%
~ 85% in MoX2 ~ 95% in WX2
Spin Hall & Spin circular dichroism K E
-K
Gong et al., Nat. Comm. 4, 2053 (2013).
E
Spin Hall in bilayer
Spin circular dichroism in bilayer
Bilayer optical selection rule:
K
sin 2
-K
cos2
ME effects from spin-layer locking • Oscillation of layer (electric) polarization in magnetic field • Electrically tunable spin Larmor precession Gong et al., Nat. Comm. 4, 2053 (2013).
sz Valley ‐K
0.4
Ez 0 B0
1 0
Valley K
-1
0
40
80
tB0
-0.4 0
40
tB0
80
Valley dependent precession frequencies
K
Spin‐layer locking
-K
ME effects from spin-layer locking K
-K
K
K
K
K
Spin doublet couples to both electric & magnetic fields, in different ways
K
Spin‐layer locking
-K
Valley conditioned spin rotations Ez
Bz
K
t
-K
Bx
Faraday geometry Valley dependent spin splitting by E & B fields in z direction Valley dependent spin resonance by oscillating Bx -K
K K
K
K
Gong et al., Nat. Comm. 4, 2053 (2013).
K
Electrically & magnetically driven ESR Bx
K
-K K K
K K
Ez
t
t
Bz
_
+ K K
K K
Voigt geometry Electrically driven ESR and magnetically driven ESR Valley dependent interference of electric & magnetic fields K
Spin‐layer locking
-K
Evidence of spin-layer locking in bilayer PL PL from trion in BL WSe2 60V
σ+ σ-
E
Normalized PL
90V
Prof. Xiaodong Xu
Electrically induced Zeeman splitting 120V
Upper Layer
K
Lower Layer
c
150Vɷ 2
ɷ1
~ 100%
ɷ1 ɷ2
v
⇑ 1.6 1.65 1.71.75 Energy (eV)
ɷ1 ‐ ɷ2 = c ‐ v
~ 95%
Jones, Yu, et al., Nat. Phy. 10, 130, 14”
Interlayer & intralayer trion Intralayer X V
Bilayer WSe2 ɷ2
150V
Interlayer X
ɷ1 V
Excitation: V
Monolayer WSe2 +10V X-
1.60
bottom layer has lower energy for excess electron
Black: V Red: H
Xo
1.65 1.70 1.75 Energy (eV)
1.6
1.65
1.7 Energy (eV)
1.75
Jones, Yu, et al., Nat. Phy. 10, 130, 14”
Intralayer X‐: valley coherence suppressed, similar to monolayer Interlayer X‐: valley coherence preserved, no exchange with 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 Ultra strong coulomb binding
Valley configurations
Large effective mass & reduced screening in 2D X0 binding
energy: 0.5 – 1 eV
Bohr radius: ~ 1 nm Trion binding 30 meV
strong e‐h exchange
60
Gate (V)
40
=
σ+
-K
K
V(k)
X -’
-K
K
Valley‐orbit coupling
20 X-
0
Xo
-20
K
=
-40 -60 1.60
1.65
X+ 1.70
1.75
Photon Energy (eV)
-K
Valley-orbit coupling of exciton Effective valley‐orbit coupling rotation symmetry
Coulomb in 2D
~a
2 B
chirality of 2
linear in k
probability for e-h to overlap
vanish at k = 0
strong coupling: VOC splitting >> radiative decay light cone
longitudinal branch Hongyi Yu et al. arXiv 1401.0667
transverse branch
~ 10-2K
ωu ~ 10-3K
ω0
~ 2 meV
ωd
Effect of tensile strain strain breaks rotational symmetry in-plane Zeeman field
I = 2 VOC 2 0 J cK
Linearly dispersed Dirac saddle point light cone
Yu et al. arXiv 1401.0667 2J0
light cone
one I = 2 cone
J0 J
0
-0.01
two I = 1 cones 2K
ky / K
0.01
-0.01
0
kx / K
0.01
Gapped Dirac cone of trion Negatively charged trions K
-K exchange
•
K
-K
K
-K
K
-K
exchange
Indexed by polarization of emitting photon + spin of extra electron (s) valley pseudospin of recombining e-h pair ()
•
Exchange coupling with the extra electron
•
An effective out-of-plane Zeeman field conditioned on the extra spin
Gapped Dirac cone of trion Trion valley Hall
-K
K
-K
K
-K
K
exchange
K
exchange
≈
0
0
-2 -5 1
0.99
≈
Energy (meV)
2
qX- / K
-0.99
-1
-1.01
Berry curvature (104Å2)
5
1.01
E
-K
Trion brightness 0
1
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
