Topological Valleytronics in Bilayer Graphene Jun Zhu

Department of Physics The Pennsylvania State University

2DCC Webinar, Nov 7, 2017

1 [email protected] Nov 7, 2017 Electronic degrees of freedom Ø Charge electric field, magnet field Ø Spin magnec field, spin-orbit coupling MOSFET

Spin transistor (ongoing research) Magnec tunnel juncon

1338 C. W. J. Beenakker: Colloquium: Andreev reflection and Klein Ø Valley …

K K’ Valley-(controlled) (elec)tronics

FIG. 2. ͑Color online͒ Atomic force microscope image ͑false color͒ of a carbon monolayer covered by two superconducting Al electrodes. From Heersche et al., 2007. 2 [email protected] Nov 7, 2017 spectroscopic measurements by Zhou et al. ͑2006͒ and E k k Bostwick et al. ͑2007͒, the electronic properties of FIG. 3. ͑Color online͒ Band structure ͑ x , y͒ of a carbon monolayer. The hexagonal first Brillouin zone is indicated. The graphene are described by an equation ͑the Dirac equa- conduction band ͑EϾ0͒ and the valence band ͑EϽ0͒ form tion of relativistic quantum mechanics, even though the ͒ conically shaped valleys that touch at the six corners of the microscopic Hamiltonian of carbon atoms is nonrelativ- Brillouin zone ͑called conical points, Dirac points, or K istic. While graphene itself is not superconducting, it ac- points͒. The three corners marked by a white dot are con- quires superconducting properties by proximity to a su- nected by reciprocal-lattice vectors, so they are equivalent. perconductor. We therefore have the unique possibility Likewise, the three corners marked by a black dot are equiva- to bridge the gap between relativity and superconductiv- lent. In undoped grapheme, the Fermi level passes through the ity in a real material. Dirac points. Illustration by C. Jozsa and B. J. van Wees. For example, Fig. 2 shows two superconducting elec- trodes on top of a carbon monolayer. The supercurrent measured through this device by Heersche et al. ͑2007͒ is lace, 1947͒. Near each corner of the hexagonal first Bril- carried by massless electrons and holes, converted into louin zone, the energy E has a conical dependence on each other by the superconducting pair potential. This the two-dimensional wave vector k=͑kx ,ky͒. Denoting conversion process, known as Andreev reflection ͑An- by ␦k=k−K the displacement from the corner at wave dreev, 1964͒, is described by a superconducting variant vector K, one has for ␦kaӶ1 the dispersion relation of the Dirac equation ͑Beenakker, 2006͒. In this Colloquium, we review the unusual physics of Andreev reflection in graphene. For a broader perspec- ͉E͉ = បv͉␦k͉. ͑1͒ tive, we compare and contrast this coupling of electrons 1 6 and holes by a superconducting pair potential with the The velocity vϵ 2 ͱ3␶a/បϷ10 m/s is proportional to coupling of electrons and holes by an electrostatic po- the lattice constant a=0.246 nm and to the nearest- tential. The latter phenomenon is called Klein tunneling neighbor hopping energy ␶Ϸ3 eV on the honeycomb ͑Cheianov and Fal’ko, 2006; Katsnelson, et al., 2006͒ lattice of carbon atoms ͑shown in Fig. 4͒. with reference to relativistic quantum mechanics, where The linear dispersion relation ͑1͒ implies an energy- -k=v of lowץE/បץit represents the tunneling of a particle into the Dirac independent group velocity vgroupϵ sea of antiparticles ͑Klein, 1929͒. Klein tunneling in energy excitations ͑EӶ␶͒. These electron excitations graphene is the tunneling of an electron from the con- ͑filled states in the conduction band͒ or hole excitations duction band into hole states from the valence band ͑empty states in the valence band͒, therefore, have zero ͑which plays the role of the Dirac sea͒. effective mass. DiVincenzo and Mele ͑1984͒ and Se- The two phenomena, Andreev reflection and Klein menoff ͑1984͒ noticed that—even though vӶc—such tunneling, are introduced in Secs. III and IV, respec- massless excitations are governed by a wave equation, tively, and then compared in Sec. V. But first we summa- the Dirac equation, of relativistic quantum mechanics, rize, in Sec. II, the special properties of graphene that govern these two phenomena. More comprehensive re- y ⌿A ⌿Aץx − iץ 0 views of graphene have been written by Castro Neto et − iបv = E . ͑2͒ ͩ ͪͩ⌿ ͪ ͩ⌿ ͪ y 0 B Bץx + iץ al. ͑2006, 2007͒, Geim and Novoselov ͑2007͒, Gusynin et al. ͑2007͒, Katsnelson ͑2007͒, and Katsnelson and No- voselov ͑2007͒. ͓The derivation of this equation for a carbon monolayer goes back to McClure ͑1956͒.͔ II. BASIC PHYSICS OF GRAPHENE The two components ⌿A and ⌿B give the amplitude iK·r iK·r A. Dirac equation ⌿A͑r͒e and ⌿B͑r͒e of the wave function on the A and B sublattices of the honeycomb lattice ͑see Fig. 4͒. The unusual band structure of a single layer of graph- The differential operator couples ⌿A to ⌿B but not to ite, shown in Fig. 3, has been known for 60 years ͑Wal- itself, in view of the fact that nearest-neighbor hopping

Rev. Mod. Phys., Vol. 80, No. 4, October–December 2008 Two-dimensional layered materials

and bilayer graphene

Insulator Semi-conductor Semi-metal Metal Superconductor Topological insulator

h-BN, graphene fluoride, MoS2, WSe2, graphene, NbSe2, Germanene, Silicene, Stanene, hexagonal GaN … 3 [email protected] Geim and Grigorieva, Nature perspecve, 2013 Nov 7, 2017 2um h-BN/bilayer graphene/h-BN

• High sample quality • Sophiscated nanostructures

4 [email protected] Nov 7, 2017 Crystal structure of convenonal semiconductor: Si

Mul-valleys but they are equivalent.

6 [email protected] Nov 7, 2017 1338 C. W. J. Beenakker:Monolayer graphene: two Colloquium: Andreev reflection andinequivalent Klein … valleys

π* K’ K

π FIG. 2. ͑Color online͒ Atomic force microscope image ͑false color͒ of a carbon monolayer covered by two superconducting Al electrodes. From Heersche et al., 2007. K’ K spectroscopic measurements by Zhou et al. ͑2006͒ and E k k Bostwick et al. ͑2007͒, the electronic properties of FIG. 3. ͑Color online͒ Band structure ͑ x , y7 ͒ of a carbon [email protected] monolayer. The hexagonal first Brillouin zoneNov 7, 2017 is indicated. The graphene are described by an equation ͑the Dirac equa- conduction band ͑EϾ0͒ and the valence band ͑EϽ0͒ form tion of relativistic quantum mechanics, even though the ͒ conically shaped valleys that touch at the six corners of the microscopic Hamiltonian of carbon atoms is nonrelativ- Brillouin zone ͑called conical points, Dirac points, or K istic. While graphene itself is not superconducting, it ac- points͒. The three corners marked by a white dot are con- quires superconducting properties by proximity to a su- nected by reciprocal-lattice vectors, so they are equivalent. perconductor. We therefore have the unique possibility Likewise, the three corners marked by a black dot are equiva- to bridge the gap between relativity and superconductiv- lent. In undoped grapheme, the Fermi level passes through the ity in a real material. Dirac points. Illustration by C. Jozsa and B. J. van Wees. For example, Fig. 2 shows two superconducting elec- trodes on top of a carbon monolayer. The supercurrent measured through this device by Heersche et al. ͑2007͒ is lace, 1947͒. Near each corner of the hexagonal first Bril- carried by massless electrons and holes, converted into louin zone, the energy E has a conical dependence on each other by the superconducting pair potential. This the two-dimensional wave vector k=͑kx ,ky͒. Denoting conversion process, known as Andreev reflection ͑An- by ␦k=k−K the displacement from the corner at wave dreev, 1964͒, is described by a superconducting variant vector K, one has for ␦kaӶ1 the dispersion relation of the Dirac equation ͑Beenakker, 2006͒. In this Colloquium, we review the unusual physics of Andreev reflection in graphene. For a broader perspec- ͉E͉ = បv͉␦k͉. ͑1͒ tive, we compare and contrast this coupling of electrons 1 6 and holes by a superconducting pair potential with the The velocity vϵ 2 ͱ3␶a/បϷ10 m/s is proportional to coupling of electrons and holes by an electrostatic po- the lattice constant a=0.246 nm and to the nearest- tential. The latter phenomenon is called Klein tunneling neighbor hopping energy ␶Ϸ3 eV on the honeycomb ͑Cheianov and Fal’ko, 2006; Katsnelson, et al., 2006͒ lattice of carbon atoms ͑shown in Fig. 4͒. with reference to relativistic quantum mechanics, where The linear dispersion relation ͑1͒ implies an energy- -k=v of lowץE/បץit represents the tunneling of a particle into the Dirac independent group velocity vgroupϵ sea of antiparticles ͑Klein, 1929͒. Klein tunneling in energy excitations ͑EӶ␶͒. These electron excitations graphene is the tunneling of an electron from the con- ͑filled states in the conduction band͒ or hole excitations duction band into hole states from the valence band ͑empty states in the valence band͒, therefore, have zero ͑which plays the role of the Dirac sea͒. effective mass. DiVincenzo and Mele ͑1984͒ and Se- The two phenomena, Andreev reflection and Klein menoff ͑1984͒ noticed that—even though vӶc—such tunneling, are introduced in Secs. III and IV, respec- massless excitations are governed by a wave equation, tively, and then compared in Sec. V. But first we summa- the Dirac equation, of relativistic quantum mechanics, rize, in Sec. II, the special properties of graphene that govern these two phenomena. More comprehensive re- y ⌿A ⌿Aץx − iץ 0 views of graphene have been written by Castro Neto et − iបv = E . ͑2͒ ͩ ͪͩ⌿ ͪ ͩ⌿ ͪ y 0 B Bץx + iץ al. ͑2006, 2007͒, Geim and Novoselov ͑2007͒, Gusynin et al. ͑2007͒, Katsnelson ͑2007͒, and Katsnelson and No- voselov ͑2007͒. ͓The derivation of this equation for a carbon monolayer goes back to McClure ͑1956͒.͔ II. BASIC PHYSICS OF GRAPHENE The two components ⌿A and ⌿B give the amplitude iK·r iK·r A. Dirac equation ⌿A͑r͒e and ⌿B͑r͒e of the wave function on the A and B sublattices of the honeycomb lattice ͑see Fig. 4͒. The unusual band structure of a single layer of graph- The differential operator couples ⌿A to ⌿B but not to ite, shown in Fig. 3, has been known for 60 years ͑Wal- itself, in view of the fact that nearest-neighbor hopping

Rev. Mod. Phys., Vol. 80, No. 4, October–December 2008 Sublace inversion symmetry

ˆ ˆ ˆ H = vF (ξ pxσ x + pyσ y ) ξ = ±1 for K and K’ valley E = ±v p K valley ± F K’ valley ! ! ⎛ ⎞ ip⋅r /! ⎛ ⎞ ψA e 1 ψξ = ⎜ ⎟ = ⎜ ⎟ ± ⎜ ⎟ ⎜ eiξθ ⎟ hole ⎝ ψB ⎠ 2 ⎝ ±ξ ⎠

−1 ⎛ py ⎞ θ p = tan ⎜ ⎟ ky ky ⎝ px ⎠ kx kx Zero band gap comes from electron electron A/B inversion symmetry

8 [email protected] Nov 7, 2017 Lace inversion symmetry broken

A band gap opens!

Cao et al, Nat. Comm. 3, 887 (2012)

Also graphene on a Moire lace and gated bilayer graphene 9 [email protected] Xiao et al, Phys. Rev. Le. 108, 196802 (2007) Nov 7, 2017 Bernal (AB)-stacked bilayer graphene

BLG on SiO2

10µm

B1 A1

+ - Δ E B2 A2 - +

Li, … J.Z. PRB 94, 161406(R) (2016) 10 [email protected] Zou, … J.Z. PRB 84, 085408 (2011) Zhang et al, Nature 459, 820 (2009) (F. Wang group) Nov 7, 2017 Electric field induced band gap in bilayer graphene

250 meV

Zhu lab Δ

F. Wang group, IR absorpon(IR absorpon) Δ up to 200 meV

Zou, … J. Z. PRB 82, 081407(R)(2010) Li, … J. Z. aXiv:1708.03644v1 11 [email protected] Zhang et al, Nature 459, 820 (2009) (F. Wang group) Nov 7, 2017 PHYSICAL REVIEW LETTERS week ending PRL 99, 236809 (2007) 7 DECEMBER 2007

where  is the Pauli matrix accounting for the sublattice e@ index, and q is measured from the valley center K m K1;2 zB ; B ; (3) 1;2 †ˆ ˆ 2me 4=3a x^ with a being the lattice constant. In the fol-  † 2 2 2 lowing, we shall focus on the n-doped graphene. General- where me 2
@ = 3a t is the effective mass at the ˆ † † ization to the p-doped graphene is straightforward due to band bottom. This is in close analogy with the Bohr the particle-hole symmetry presented in this system. magneton for the electron spin, where the effective mass Because spin-orbit coupling is extremely weak in gra- becomes the free electron mass. In fact, the analogy goes phene [17], the valley magnetic moment can only be of further because one can also obtain the spin Bohr magne- orbital nature. To study this quantity, we invoke the semi- ton by constructing a wave packet at the bottom of the classical formulation of the wave packet dynamics of positive energy bands of the Dirac theory and calculating Bloch electrons [18]. It has been shown that in addition the self-rotating orbital moment. Therefore, it makes sense to the spin magnetic moment, Bloch electrons carry an to call the orbital moment calculated above as the intrinsic orbital magnetic moment given by m k i e=2@ magnetic moment associated with the valley degree of †ˆÿ † ku H k " k ku , where u k is the periodic freedom, provided one is only concerned with low-energy hpartr ofj ‰ the Bloch †ÿ function, †Šjr Hi k is thej Bloch †i Hamiltonian, electrons near the bottom of the valleys [19,20]. and " k is the band energy [18 ].† It originates from the self- The valley magnetic moment has important implications rotation † of the wave packet. For a two-dimensional system, in valleytronics as it can be inferred from all kinds of the orbital magnetic moment is always in the normal experiments analogous to those on the spin magnetic mo- direction of the plane and may be written as m k z^. Its ment. For example, while spin polarization of electrons can momentum dependence can easily be calculated from † the be created by a magnetic field (Pauli paramagnetism), we tight-binding Bloch states, and is shown in Fig. 1. As we expect a similar valley polarization in graphene due to can see, m k is concentrated in the valleys and has oppo- coupling between a perpendicular magnetic field and the site signs in the† two inequivalent valleys. Analytic expres- valley magnetic moment. Moreover, for typical values of sion can also be obtained from the model Hamiltonian (1)
0:28 eV and t 2:82 eV with a lattice constant a    ˆ in the neighborhood of such valleys: 2:46 A, we find B to be about 30 times of the Bohr magneton. Therefore, the response to a perpendicular mag- 2 2 3ea
t netic field is in fact dominated by the valley magnetic m k z : (2) †ˆ 4@
2 3q2a2t2 moment at low doping in graphene. Interestingly, unlike ‡ † the spin moment which will respond to magnetic fields in It is instructive to consider the low-energy limit (q 0) all directions,  only couples to magnetic fields in the of the orbital magnetic moment ! B In a gapped two-dimensional hexagonal lace, z-direction. Thus, spin and valley magnetic moment can both be determined from the anisotropic Pauli paramag- netism in a tilted magnetic field [21]. K’ Complimentarily, a population difference in the two valleys may be detected as a signal of orbital magnetiza- tion. The orbital magnetization consists of the orbital mo- K ments of carriers plus a correction from the Berry curvature [22]

1 20 d2k

M 2 m k e=@  " k k ; (4) 0.5 10 ˆ 2 2 ‰ †‡ †‰ ÿ †Š †Š

~ ~ e/ h)

2 † What are the valley-contrasng properes? 0 0 Z (eV) where  is the local chemical potential, and the integration (a

How do we control and detect them? − 0.5 − 10 m is over states below the chemical potential. The Berry

~ −1 ~ − 20 curvature k k z^ is defined by k 19 16 13 13 16 19 †ˆ † †ˆ 12 12 12 12 12 12 12 k u k i k u k and its distribution has a similar [email protected] Nov 7, 2017 structurer  h to†j r thatj of†im k . We note that Eq. (4) is for kx (/)a temperatures much lower † than the energy scale of band FIG. 1 (color online). Energy bands (top panel) and orbital structure (roughly given by
), which holds up to room magnetic moment of the conduction bands (bottom panel) of a temperature as the experimentally observed band gap
graphene sheet with broken inversion symmetry. The Berry 0:28 eV [8]. For a two-band model with particle-hole curvature k has a distribution similar to that of m k . The † † symmetry, we have a simple relation between the orbital first Brillouin zone is outlined by the dashed lines, and two magnetic moment and the Berry curvature in the conduc- inequivalent valleys are labeled as K and K . The top panel 1 2 tion band: m k e=@ " k k . Using this relation, shows the conduction and valence bands in the energy range †ˆ † † † Eq. (4) may be further simplified as M 2 e=@ from 1 to 1 eV. The parameters used are t 2:82 eV and
ˆ † 0:28 eVÿ . ˆ ˆ d2k 2 2  k .Whenthetwovalleysareinequilibrium † † R 236809-2 PHYSICAL REVIEW LETTERS week ending NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2012.96PRL 99, 236809 (2007) 7 DECEMBER 2007 where  is the Pauli matrix accounting for the sublattice e@

LETTERS index, and q is measured from the valley center K m K1;2 zB ; B ; (3) 1;2 †ˆ ˆ 2me 4=3a x^ with a being the lattice constant. In the fol-  † 2 2 2 lowing, we shall focus on the n-doped graphene. General- where me 2
@ = 3a t is the effective mass at the ˆ † † ization to the p-doped graphene is straightforward due to band bottom. This is in close analogy with the Bohr the particle-hole symmetry presented in this system. magneton for the electron spin, where the effective mass Because spin-orbit coupling is extremely weak in gra- becomes the free electron mass. In fact, the analogy goes a phene [17], the valleyd magnetic moment can only be of further because one can also obtain the spin Bohr magne- Monolayer orbital nature. To study this quantity, we invoke the semi- tonBilayer by constructing a wave packet at the bottom of the classical formulation of the wave packet dynamics of positive energy bands of the Dirac theory and calculating Bloch electrons [18]. It has been shown that in addition the self-rotating orbital moment. Therefore, it makes sense to the spin magnetic moment, Bloch electrons carry an to call the orbital moment calculated above as the intrinsic orbital magnetic moment given by m k i e=2@ magnetic moment associated with the valley degree of †ˆÿ † ku H k " k ku , where u k is the periodic freedom, provided one is only concerned with low-energy hpartr ofj ‰ the Bloch †ÿ function, †Šjr Hi k is thej Bloch †i Hamiltonian, electrons near the bottom of the valleys [19,20]. and " k is the band energy [18 ].† It originates from the self- The valley magnetic moment has important implications S rotation † of the wave packet. For a two-dimensional system, in valleytronics as it can be inferred from all kinds of the orbital magnetic moment is always in the normal experiments analogous to those on the spin magnetic mo- direction of the plane and may be written as m k z^. Its ment. For example, while spin polarization of electrons can Mo momentum dependence can easily be calculated from † the be created by a magnetic field (Pauli paramagnetism), we tight-binding Bloch states, and is shown in Fig. 1. As we expect a similar valley polarization in graphene due to can see, m k is concentrated in the valleys and has oppo- coupling between a perpendicular magnetic field and the S site signs in the† two inequivalent valleys. Analytic expres- valley magnetic moment. Moreover, for typical values of sion can also be obtained from the model Hamiltonian (1)
0:28 eV and t 2:82 eV with a lattice constant a    ˆ Orbital magnec moment and opcal selecon rules in the neighborhood of such valleys: 2:46 A, we find B to be about 30 times of the Bohr magneton. Therefore, the response to a perpendicular mag- 2 2 3ea
t netic field is in fact dominated by the valley magnetic m k z : (2) †ˆ 4@
2 3q2a2t2 moment at low doping in graphene. Interestingly, unlike e ‡ † the spin moment which will respond to magnetic fields in b It is instructive to consider the low-energy limit (q 0) ! all directions, B only couples to magnetic fields in the of the orbital magnetic moment z-direction. Thus, spin and valley magnetic moment can both be determined from the anisotropic±3/2 Pauli paramag- m = −3/2 +3/2 ±3/2 netism in a tilted magnetic field [21]. j Complimentarily, a population difference in the two m = −1/2 +1/2 ±1/2 valleys may be detected as a signal of±1/2 orbital magnetiza- K j tion.Spatial The orbital magnetization consists of the orbital mo- ments of carriers plus a correction from the Berry curvature T-reversal [22inversion] M 1 20 d2k ±1/2 m = −1/2 +1/2 ±1/2 M 2 m k e=@  " k k ; (4) j 0.5 10 ˆ 2 2 ‰ †‡ †‰ ÿ †Š †Š

~ ~ e/ h)

2 † 0 0 Z K’ (eV) where  is the local chemical potential, and the integration ±1/2 (a ±1/2

−− 0.51/2 − 10 m mj = +1/2 is over states below the chemical potential. The Berry

~ −1 ~ − 20 curvature k k z^ is defined by k 19 16 13 13 16 19 †ˆ † †ˆ 12 12 12 12 12 12 k u k i k u k and its distribution has a similar structurer  h to†j r thatj of†im k . We note that Eq. (4) is for kx (/)a KK’ KKtemperatures much lower † than’ the energy scale of band FIG. 1 (color online). Energy bands (top panel) and orbital structure (roughly given by
), which holds up to room magnetic moment of the conduction bands (bottom panel) of a temperature as the experimentally observed band gap
K and K’ couple to light of opposite circular polarizaons. graphene sheet with broken inversion symmetry. The Berry 0:28 eV [8]. For a two-band model with particle-hole curvature k has a distribution similar to that of m k . The † Decoupled † symmetry, valley we have a simple and relation spin between the orbital Coupled valley-spin excitonic absorption first Brillouin zone is outlined by the dashed lines, and two magnetic moment and the Berry curvature in the conduc- c inequivalent valleys aref labeled as K and K . The top panel 1 2 tion band: m k e=@ " k k . Using this relation, Mak et al, Nat. Nano. 7, 494 (2012) Xiao et al, Phys. Rev. shows theLe. 99, 236809 (2007) conduction and valence bands in13 the energyOptical range spin orientation†ˆ † † † [email protected] Nov 7, 2017 Eq. (4) may be further simplified as M 2 e=@ from 1 to 1 eV. The parameters used are t 2:82 eV and
ˆ † 0:28 eVÿ . ˆ ˆ d2k 2 2  k .Whenthetwovalleysareinequilibrium B B † † R 236809-2 A A

K K’ K K’

Figure 1 | Atomic structure and electronic structure at the K and K′ valleys of monolayer (a–c) and bilayer (d–f) MoS2. a, The honeycomb lattice structure of monolayer MoS2 with two sublattice sites occupied by one molybdenum and two sulphur atoms. Spatial inversion symmetry is explicitly broken. b,The lowest-energy conduction bands and the highest-energy valence bands labelled by the z-component of their total angular momentum. The spin degeneracy at the valence-band edges is lifted by the spin–orbit interactions. The valley and spin degrees of freedom are coupled. c, Optical selection rules for the A and

Bexcitonstatesattwovalleysforcircularlypolarizedlight.d,BilayerMoS2 with Bernal stacking. e,Spindegeneracyofthevalencebandsisrestoredby spatial inversion and time-reversal symmetries. Valley and spin are decoupled. f, Optical absorption in bilayer MoS2.Undercircularlypolarizedexcitation (shown for s2)bothvalleysareequallypopulatedandonlyanetspinorientationisproduced. below, in the absence of valley-specific excitation, the observation a Photon energy (eV) 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 of full photoluminescence helicity is unexpected. The need for 0.4 valley-selective excitation is further confirmed by the weakness of the photoluminescence helicity for the bilayer sample where inver- 0.3 sion symmetry precludes such selectivity. 0.2 Photoluminescence helicity r reflects generally the relationship 0.1 between the excited-state lifetime and the angular momentum relaxation time23. In our case, this corresponds to the relationship Δ R / 0.0 between the exciton lifetime and the hole spin lifetime23 −0.1 B (Supplementary Sections S1,S3). For a quantitative treatment, we A @14 K −0.2 note that the helicity rA of the A exciton emission in monolayer MoS under on-resonance s excitation is determined by the b 1,200 2 2 A steady-state hole valley-spin population, 1,000

800 nA nA r K − K′ A = nA nA 600 K + K′ 400 where nA and nA are, respectively, the populations of excitons in K K′ 200 B the K (hole spin up) and K′ (hole spin down) valleys. By balancing the pumping, recombination and relaxation rates of the exciton Photoluminescence intensity 0 complexes, including both the neutral and charged excitons 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 (Supplementary Section S1), we obtain, for the helicity of the Photon energy (eV) neutral A and charged A2 exciton emission, Figure 2 | Optical absorption and photoluminescence spectra of monolayer 1 1 MoS2. a,DifferentialreflectancespectrumshowingthenarrowAexciton r , r r 1 and the broader B exciton features. Red, yellow and green arrows represent A = 1 2t /t A− = A 1 2t /t ( ) + A AS + A− A−S the three different photon energies used to excite the samples in the photoluminescence measurements. b, Photoluminescence spectrum (not t 1 t 1 where A− A−− denotes the total decay rate of the neutral (charged) polarization resolved) for 2.33 eV (532 nm) excitation. The spectrum consists 1 1 exciton and t− t− is the intervalley relaxation rate of the neutral of B exciton hot luminescence and A exciton luminescence (including the !"AS A−S (charged) exciton. The charged exciton photoluminescence emis- neutral exciton emission and the charged exciton emission, redshifted by sion time is estimated!" to be 5 ps according to time-resolved 40 meV). The lower energy feature is attributed to trapped excitons. # NATURE NANOTECHNOLOGY | VOL 7 | AUGUST 2012 | www.nature.com/naturenanotechnology 495 PHYSICAL REVIEW LETTERS week ending PRL 99, 236809 (2007) 7 DECEMBER 2007

where  is the Pauli matrix accounting for the sublattice e@ index, and q is measured from the valley center K m K1;2 zB ; B ; (3) 1;2 †ˆ ˆ 2me 4=3a x^ with a being the lattice constant. In the fol-  † 2 2 2 lowing, we shall focus on the n-doped graphene. General- where me 2
@ = 3a t is the effective mass at the ˆ † † ization to the p-doped graphene is straightforward due to band bottom. This is in close analogy with the Bohr the particle-hole symmetry presented in this system. magneton for the electron spin, where the effective mass Because spin-orbit coupling is extremely weak in gra- becomes the free electron mass. In fact, the analogy goes phene [17], the valley magnetic moment can only be of further because one can also obtain the spin Bohr magne- orbital nature. To study this quantity, we invoke the semi- ton by constructing a wave packet at the bottom of the classical formulation of the wave packet dynamics of positive energy bands of the Dirac theory and calculating Bloch electrons [18]. It has been shown that in addition the self-rotating orbital moment. Therefore, it makes sense to the spin magnetic moment, Bloch electrons carry an to call the orbital moment calculated above as the intrinsic orbital magnetic moment given by m k i e=2@ magnetic moment associated with the valley degree of †ˆÿ † ku H k " k ku , where u k is the periodic freedom, provided one is only concerned with low-energy hpartr ofj ‰ the Bloch †ÿ function, †Šjr Hi k is thej Bloch †i Hamiltonian, electrons near the bottom of the valleys [19,20]. and " k is the band energy [18 ].† It originates from the self- The valley magnetic moment has important implications rotation † of the wave packet. For a two-dimensional system, in valleytronics as it can be inferred from all kinds of the orbital magnetic moment is always in the normal experiments analogous to those on the spin magnetic mo- direction of the plane and may be written as m k z^. Its ment. For example, while spin polarization of electrons can momentum dependence can easily be calculated from † the be created by a magnetic field (Pauli paramagnetism), we tight-binding Bloch states, and is shown in Fig. 1. As we expect a similar valley polarization in graphene due to can see, m k is concentrated in the valleys and has oppo- coupling between a perpendicular magnetic field and the site signs in the† two inequivalent valleys. Analytic expres- valley magnetic moment. Moreover, for typical values of sion can also be obtained from the model Hamiltonian (1)
0:28 eV and t 2:82 eV with a lattice constant a    ˆ in the neighborhood of such valleys: 2:46 A, we find B to be about 30 times of the Bohr magneton. Therefore, the response to a perpendicular mag- 2 2 3ea
t netic field is in fact dominated by the valley magnetic m k z : (2) †ˆ 4@
2 3q2a2t2 moment at low doping in graphene. Interestingly, unlike ‡ † the spin moment which will respond to magnetic fields in It is instructive to consider the low-energy limit (q 0) Berry curvature Ω(k): magnec field in momentum space ! all directions, B only couples to magnetic fields in the of the orbital magnetic moment z-direction. Thus, spin and valley magnetic moment can both be determined from the anisotropic Pauli paramag- hol hol netism in a tilted magnetic field [21]. e e Non-zero Berry curvature Complimentarily, a population difference in the two 2 2 2! vFΔ valleys may be detected as a signal of orbital magnetiza- Ω()k = τ z 3/2 2 2 2 2 tion. The orbital magnetization consists of the orbital mo- electr electr (Δ + 4k ! vF ) on on ments of carriers plus a correction from the Berry curvature K K’ [22]

1 20 d2k

M 2 m k e=@  " k k ; (4) 0.5 10 ˆ 2 2 ‰ †‡ †‰ ÿ †Š †Š

~ ~ e/ h)

2 † 0 0 Z (eV) where  is the local chemical potential, and the integration (a − 0.5 − 10 m is over states below the chemical potential. The Berry

~ −1 ~ − 20 curvature k k z^ is defined by k 19 16 13 13 16 19 u k i u†ˆk and † its distribution has a similar †ˆ 12 12 12 12 12 12 rk  h †j rkj †i k (/)a structure to that of m k . We note that Eq. (4) is for x 14 † [email protected] Xiao et al, Phys. Rev. Le. 99, 236809 (2007) Nov 7, 2017 temperatures much lower than the energy scale of band FIG. 1 (color online). Energy bands (top panel) and orbital structure (roughly given by
), which holds up to room magnetic moment of the conduction bands (bottom panel) of a temperature as the experimentally observed band gap
graphene sheet with broken inversion symmetry. The Berry 0:28 eV [8]. For a two-band model with particle-hole curvature k has a distribution similar to that of m k . The † † symmetry, we have a simple relation between the orbital first Brillouin zone is outlined by the dashed lines, and two magnetic moment and the Berry curvature in the conduc- inequivalent valleys are labeled as K and K . The top panel 1 2 tion band: m k e=@ " k k . Using this relation, shows the conduction and valence bands in the energy range †ˆ † † † Eq. (4) may be further simplified as M 2 e=@ from 1 to 1 eV. The parameters used are t 2:82 eV and
ˆ † 0:28 eVÿ . ˆ ˆ d2k 2 2  k .Whenthetwovalleysareinequilibrium † † R 236809-2 Hall effect (from real magnec field) Valley Hall effect ( from Berry curvature) k! = −r!× B r! = −k! ×Ω B Ω=0 B=0 Ω≠0

K K K’ K’

How to detect valley polarizaon?

15 [email protected] Xiao et al, Phys. Rev. Le. 99, 236809 (2007) Nov 7, 2017 RESEARCH | REPORTSRESEARCH | REPORTS

in this study is shownin thisin Fig. study 1. A is bias shown voltage in Fig. (Vx) 1. A biastensities voltageP;thedataweretakenwithafocused (Vx) tensities P;thedataweretakenwithafocusedrespectively. All of therespectively. observations All of made the observations below made below is applied along theis short applied channel, along andthe short the lon- channel,laser and beam the lon- (centeredlaser at beam 1.9 eV) (centered located at the1.9 eV)have located been at repeated the have on multiple been repeated devices on [six multiple de- devices [six de- gitudinal current (Ixgitudinal)ismeasured.TheHallvolt- current (Ix)ismeasured.TheHallvolt-center of the device.center Similar of tothe the device. effect Similar of elec- to thevices effect for of monolayers elec- vices and for two monolayers devices for and bilayers two devices for bilayers age (VH,thevoltagedifferencebetweencontactsage (VH,thevoltagedifferencebetweencontactstrical gating (Fig. 1C,trical inset), gating the effect (Fig. 1C, of incident inset), the effect(figs. of S3 incident to S8 and S10)].(figs. S3 to S8 and S10)]. AandB)issimultaneouslymeasured,andabackAandB)issimultaneouslymeasured,andabackphotons is to increasephotons the channel is to increase conductivity the channelIn conductivity Fig. 2A, we showIn the Fig.Vx 2A,-dependence we show the of theVx-dependence of the gate voltage (Vg)isappliedtothesiliconsub-gate voltage (Vg)isappliedtothesiliconsub-sxx,whichindicatesthatphotoconductionisthesxx,whichindicatesthatphotoconductionistheanomalous Hall voltageanomalous (VH)at HallVg =0V[( voltage23 (VH),)atVg =0V[(23), strate in order to continuouslystrate in order vary to the continuously channel varymain the mechanism channel drivingmain mechanism the photoresponse driving the in photoresponsesections 2.4 into 2.6].sections A small 2.4 but to finite 2.6].V AH smallthat but finite VH that doping level. The Vdopingg-dependence level. The of theVg-dependence conduc- our of the device conduc- (30); photocurrentour device ( generation30); photocurrent under generationscales linearly under withscalesVx is linearly observed with underVx is R-L observed under R-L tivity of device M1 (tivitysxx)extractedfromtwo-and of device M1 (sxx)extractedfromtwo-andzero bias is negligiblezero [(23 bias), section is negligible 2.3]. The [(23 change), section 2.3].modulation The change (Fig. 2A,modulation solid red line). (Fig. 2A, This solid is the red line). This is the four-point measurementsfour-point is shown measurements in Fig. 1C. is shownin conductivity in Fig. 1C. within and conductivity without laser with illumina- and withoutsignature laser illumina- of a photoinducedsignature AHE of a photoinduced driven by a net AHE driven by a net Unless otherwise indicated,Unless otherwise all measurements indicated, all measurementstion Dsxx ≡ sxx,lighttion– sDsxxxx,dark≡ assxx a,light function– sxx,dark of asvalley a function polarization. of valley Given polarization. the geometry Given of elec- the geometry of elec- were performed onwere monolayer performed MoS2 onat monolayer 77 K (ex- MoSincident2 at 77 photon K (ex- energiesincidentE (Fig. photon 1D) clearlyenergies showsE (Fig. 1D)trical clearly connections shows trical shown connections in Fig. 1B, a shown positive in Fig. 1B, a positive perimental details areperimental provided details in (23 are), sections provided inthe (23 A), (at sectionsE ≈ 1.9 eV)the and A (at B (atE ≈E1.9≈ 2.1 eV) eV) and res- B (at EHall≈ 2.1 voltage eV) res- under R-LHall modulation voltage under for R-L a positive modulation for a positive 1.1 and 1.2). The usual1.1 and n-type 1.2). field The effectusual tran-n-type fieldonances effect of tran- monolayeronances MoS2 of(8 monolayer). MoS2 (8). bias is observed. Itbias is con issistent observed. with It the is con predic-sistent with the predic- sistor behavior is seensistor (26 behavior). We also is see seen that (26 the). We alsoBy see parking that the the laserBy spot parking at the the center laser of spot the at thetion center of a side-jump of the –tiondominated of a side-jump VHE (Eq.–dominated 1) (4, 6). VHE (Eq. 1) (4, 6). two-point (measuredtwo-point at Vx =0.5Vunderthege- (measured at Vx =0.5Vunderthege-device, we studieddevice, the Hall we response studied under the Hall on- responseThe sign under of on-the signalThe is sign reversed of the when signal the is exci-reversed when the exci- ometry shown in Fig.ometry 1B) and shown four-point in Fig. 1B) [mea- and four-pointresonance [mea- excitationresonance (centered excitation at E ≈ 1.9 (centered eV). To attationE ≈ 1.9 is eV). changed To totation L-R modulation is changed (dashed to L-R modulation red (dashed red sured by swappingsured the drain by swapping and the B the contacts drain andenhance the B contacts our detectionenhance sensitivity, our detection we modulated sensitivity,line). we modulated In contrast, noline). net Hall In contrast, voltage is no seen net when Hall voltage is seen when in Fig. 1B, and takingin Fig. into 1B, account and taking a geometric into accountthe a polarization geometric statethe polarization of the incident state light of the at incidentwe switch light to at a linearwe (s- switchp) modulation to a linear (Fig. (s-p) 2A, modulation (Fig. 2A, factor of ln2/p (27)]factor conductivities of ln2/p ( are27)] similar conductivities in 50 are kHz similar by use in of a50 photoelastic kHz by use modulator of a photoelastic and modulatordotted red line)and [measurementsdotted red line) on [measurements other mono- on other mono- magnitude, indicatingmagnitude, near-ohmic indicating contacts near-ohmic in our measured contacts in the our anomalousmeasured Hall the voltage anomalousVH with Hall a voltagelayerV devicesH with are a providedlayer devices in fig. are S4]. provided in fig. S4]. device (28). Althoughdevice the Ix (-V28x).characteristic Although the showsIx-Vx characteristiclock-in amplifier shows [(23lock-in), section amplifier 1.2]. Under [(23), quarter- section 1.2]. UnderTo study quarter- the polarizationTo study dependence the polarization care- dependence care- the presence of Schottkythe presence barriers of at Schottky small bias barrierswave at small modulation bias (Dlwave=1/4),thedegreeofexci- modulation (Dl =1/4),thedegreeofexci-fully, the anomalousfully, Hall the resistance anomalousRH HallVH resistance=Ix RH VH=Ix (Fig. 1C, inset), it has(Fig. little 1C, influence inset), it on has our little mea- influencetation on our ellipticity mea- cantation be continuously ellipticity can varied be continuously by as a function varied by of the angleas a functionq is shown of the in Fig.angle¼ 2Bq foris shown in Fig.¼ 2B for surements at high bias.surements A four-point at high carrier bias. A mo- four-pointchanging carrierq,theangleofincidenceofthelinearly mo- changing q,theangleofincidenceofthelinearlyboth the quarter- andboth half-wave the quarter- modulations. and half-wave We modulations. We 2 −1 −1 2 −1 −1 bility of 98 cm Vbilitys ofand 98 two-point cm V s carrierand two-pointpolarized carrier light withpolarized respect to light the fastwith axis respect of the to thesee fast that axis the of the Hall resistancesee that theRH Hallexhibits resistance a sine RH exhibits a sine mobility of 61 cm2 mobilityV−1 s−1 are of extracted 61 cm2 V− at1 s high−1 are extractedmodulator. at high On the othermodulator. hand, On half-wave the other modu- hand, half-wavedependence modu- on q underdependence quarter-wave on q under modulation. quarter-wave modulation. Vg,wherethesxx-VVg gdependence,wherethes becomesxx-Vg dependence linear lation becomes (Dl linear=1/2)allowsustomodulatelinearlation (Dl =1/2)allowsustomodulatelinearAmaximumHallresistanceof~2ohmsismea-AmaximumHallresistanceof~2ohmsismea- [(23), section 2.2]. [(23), section 2.2]. excitations betweenexcitations–q and q at between 100 kHz,–q whichand q at 100sured kHz, under which an excitationsured intensity under an of excitation ~150 mW intensitymm−2. of ~150 mW mm−2. In Fig. 1D, we examineIn Fig. the 1D, photoresponse we examine the of photoresponseis twice the fundamental of is twice frequency. the fundamental To indicate frequency.For To comparison, indicate zeroFor Hall comparison, resistance zero is observed Hall resistance is observed our device; this allowsour device;us to identify this allows the appro- us to identifythe special the appro- case of quarter-wavethe special case modulation of quarter-wave with modulationunder half-wave with modulation.under half-wave Our results modulation. are con- Our results are con- priate photon energypriate (E) photon for efficient energy injection (E) for efficientq =45°,inwhichthepolarizationismodulated injection q =45°,inwhichthepolarizationismodulatedsistent with recentsistent experimental with recent observations experimental observations of valley-polarized carriersof valley-polarizedDetecon of valley Hall effect (11, 29). Shown carriers in the (11, 29from). Shown right- in theto left- handed,from right- or q to= left-–45°, handed, in which or q =of–45°, a net in which valley polarization of a net valley under polarization the optical under the optical inset is the photocurrentinset isD theIx as photocurrent a function ofDVIxx as athe function polarization of Vx is modulatedthe polarization from left-is modulated to right- fromexcitation left- to right- of the Aexcitation resonance of with the A circularly resonance with circularly (at Vg =0V)underdifferentlaserexcitationin-(at Vg =0V)underdifferentlaserexcitationin-• Opcal pumping, electrical detecon handed, we use thehanded, notations we R-L use and the L-R notations below, R-Lpolarized and L-R below, light (10–polarized14). The sine light dependence (10–14). The of sine dependence of

Fig. 1. Monolayer MoSFig.2 1.Hall Monolayer MoS2 Hall bar device. (A)Schematicsofbar device. (A)Schematicsof the valley-dependentthe optical valley-dependent optical selection rules andselection the electrons rules and the electrons at the K and K′ valleysat the that K and K′ valleys that possess opposite Berrypossess opposite Berry ⇀ ⇀ curvatures W. The orangecurvatures W. The orange arrows represent thearrows clockwise represent the clockwise and counterclockwiseand hopping counterclockwise hopping motions of the K andmotions K′ of the K and K′ electrons. (B)Schematicofaelectrons. (B)Schematicofa photoinduced AHEphotoinduced driven by a AHE driven by a net valley polarizationnet (left) valley and polarization (left) and Mak et al, Science 344, 1489(2014) an image of the Hallan bar image device of the Hall bar device (right). In the schematic,(right). the In• the schematic,Faraday rotaon the intrinsic plus side-jumpintrinsic plus side-jump contribution as predictedcontribution by as predicted by Eq. 1 is shown. (C)Two-pointEq. 1 is shown. (C)Two-point

(dashed line, Vx = 0.5(dashed V) line, Vx = 0.5 V) and four-point (solidand line) four-point (solid line) conductivities of theconductivities device as a of the deviceΩ as a function of back gatefunction voltage ofV backg.K gate voltage Vg. K’ (Inset) Source-drain(Inset) bias ( Source-drainVx) bias (Vx) dependence of thedependence current along of the current along MoS2 the longitudinal channelthe longitudinal (Ix)at channel (Ix)at different back gatedifferent voltages backVg. gate voltages Vg. (D)Thechangeinconductivity(D)Thechangeinconductivity

Dsxx as a function ofDs incidentxx as a function of incident photon energy E underphoton laser energy illumination.E under The laser arrow illumination. indicates The the arrow excitation indicates energy the used excitation in most energy of the used measurements in most of the in this measurements paper, E ≈ 1.9 in eV. this (Inset) paper, E ≈ 1.9 eV. (Inset)

Source-drain bias (Source-drainVx)dependenceofthephotocurrent( bias (Vx)dependenceofthephotocurrent(DIx)atdifferentincidentlaserintensitiesDIx)atdifferentincidentlaserintensitiesP (Vg =0V). P (Vg =0V). J. Lee, et al, Nature Nanotech. 11, 421-425 (2016). 16 1490 27 JUNE 20141490• VOL 34427 JUNE ISSUE 2014 6191• VOL 344 ISSUE 6191 sciencemag.org SCIENCEsciencemag.org SCIENCE [email protected] Nov 7, 2017 Detecon of valley Hall effect

• Electrically pumped, electrically detected

hol hol µK µ e K’e K K’ net valley polarizaon is small electr electr on on

on gapped graphene and bilayer graphene Gorbachev, Science 346, 448 (2014) Sui et al, Nat. Phys. 11, 1027 (2015) Shimazaki et al, Nat. Phys. 11, 1032 (2015) 17 [email protected] Nov 7, 2017

LETTERS ValleyLETTERS filter and valley valve in graphene Valley filter and valley valve in graphene A. RYCERZ1,2,J.TWORZYDŁO3 AND C. W. J. BEENAKKER1* 1Instituut-Lorentz, Universiteit Leiden, PO Box 9506, 2300 RA Leiden, The Netherlands 1,2 3 1 2Marian SmoluchowskiA. RYCERZ Institute,J.TWORZYD of Physics, JagiellonianŁO AND University, C. W. Reymonta J. BEENAKKER 4, 30-059 Krak*ow,´ Poland 1 3Institute ofInstituut-Lorentz, Theoretical Physics, Universiteit Warsaw Leiden, University, PO Box 9506, Ho˙za 2300 69, RA00-681 Leiden, Warsaw, The Netherlands Poland 2Marian Smoluchowski Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Krakow,´ Poland *e-mail: [email protected] 3Institute of Theoretical Physics, Warsaw University, Ho˙za 69, 00-681 Warsaw, Poland *e-mail: [email protected] Electrical valleytronic device concepts

Published online: 18 February 2007; doi:10.1038/nphys547 A valley filter proposal uses a short and Published online: 18 February 2007; doi:10.1038/nphys547 Generang valley polarizaon narrow constricon with zigzag edges… –2, –1 2, 1, 0 –2, –1 2, 1, 0 Theis hard! potentialThe potential of graphene of graphene for carbon for carbon electronics electronics rests rests on on the the –2, –1 2, 1, 0 –2, –1 2, 1, 0 possibilitiespossibilities offered off byered its by unusual its unusual band band structure structure to to create create 1,2 devicesdevices that have that no have analogue no analogue in silicon-based in silicon-based electronics electronics1,2. . EE ConductionConduction and valence and valence bands bands in graphene in graphene form form conically conically shapedshaped valleys, valleys, touching touching at a point at a point called called the the Dirac Dirac point. point. There There E are twoare inequivalent two inequivalent Dirac Dirac points points in the in theBrillouin Brillouin zone, zone, related related EFF by time-reversalby time-reversal symmetry. symmetry. Intervalley Intervalley scattering scattering is suppressedis suppressed 3–5I wish I had a valley magnet… in purein samples pure samples3–5.Theindependenceanddegeneracyofthe.Theindependenceanddegeneracyofthe valley degree of freedom suggests that it might be used to valley degree of freedom suggests6 that it might be used to control an electronic device6 , in much the same way as the control an electronic device , in much7 the same way as the8 π ka π ka π ka electron spin is used in spintronics7 or quantum computing8 . π ka π ka π ka electronA spin key ingredient is used in for spintronics ‘valleytronics’or would quantum be a controllable computing way. A key ingredientof occupying for a ‘valleytronics’ single valley in would graphene, be a thereby controllable producing way a of occupyingvalley polarization. a single valley Here in we graphene, propose such thereby a valley producing filter, based a valley polarization.on a ballistic Here point we contact propose with such zigzag a valley edges. filter, The polarity based on a ballisticcan be inverted point contact by local with application zigzag of edges. a gate The voltage polarity to the point contact region. Two valley filters in series may function as can be inverted by local application of a gate voltage to the W W point contactan electrostatically region. Two controlled valley filters valley in valve, series representing may function a zero- as magnetic-field counterpart to the familiar spin valve. W W an electrostatically controlled9–15 valley valve, representing a zero- magnetic-fieldEarlier counterpart work on to the one-dimensional familiar spin (1D) valve. conduction in graphene ribbons (long and narrow ballistic strips) has shown that L Earlier work9–15 on one-dimensional (1D) conduction in they may support a propagating mode arbitrarily close to the Dirac L y graphenepoint, ribbons and that (long this and mode narrow lacks the ballistic valley strips) degeneracy has shownof modes that that a x they maypropagate support at a higher propagating energies. mode For armchair arbitrarily edges close of the to theribbon, Dirac this y point, and that this mode lacks the valley degeneracy of modes that lowest propagating mode is constructed from states in both valleys, U a x propagatebut at for higher zigzag energies. edges only For a armchairsingle valley edges contributes of the ribbon,9–13.Inaccord this lowest propagatingwith time-reversal mode symmetry, is constructed the mode from switches states fromin both one valleys, valley to U A Rycerz et al, Nat. Phys. 3, 172 (2007) 9–13 but forthe zigzag other edges on changing only a thesingle direction valley of contributes propagation. .Inaccord EF µ Here, we consider a 2D geometry consisting of a quantum point µ 0 withXiao et al, Phys. Rev. time-reversal symmetry,Le. 108, 196802 (2007) the mode switches from one valley to U contact (QPC) in a graphene sheet. A QPC is a short and narrow 0 the other on changing the direction of propagation. E 18 constriction with a quantized conductance G n 2e2 /h (ref. 16). F [email protected] Here, we consider a 2D geometry consisting of a quantum point µ 0 Nov 7, 2017 (The factor of two accounts for the spin degeneracy.)= × A current, I, µ U0 contactis (QPC) passed in through a graphene the QPC sheet. by application A QPC is aof short a voltage and di narrowfference, U 2 constrictionV, between with a the quantized wide regions conductance on oppositeG sidesn of2e the/h constriction(ref. 16). x = × L (The factor(see Fig. of two 1). The accounts orientation for the of thespin graphene degeneracy.) lattice A is suchcurrent, thatI the, Ls is passedconstriction through the haszigzag QPC byedges application along the direction of a voltage of current difference, flow. We U x demonstrate by numerical simulation that on the first conductance Figure 1 Schematic diagram of the valley filter. Middle panel: Honeycomb lattice V, between the wide regions on opposite sides of the constriction L (see Fig.plateau 1). The the orientation QPC produces of the a strong graphene polarization lattice of is the such valleys that in the the of carbon atoms in a strip containingLs a constriction with zigzag edges. Top panel: constrictionwide hasregions. zigzag Our edges finding along signifies the direction that the two of current valleys in flow. graphene We Dispersion relation in the wide and narrow regions. An electron in the first valley can be addressed individually as independent internal degrees of (modes n 0,1,2,...)istransmitted(filledcircle),whereasanelectroninthe demonstrate by numerical simulation that on the first conductance Figure 1 Schematic= diagram of the valley filter. Middle panel: Honeycomb lattice freedom of the conduction electrons. This is only possible in a 2D second valley (modes n 1, 2,...)isreflected(opencircle).Bottompanel: plateau the QPC produces a strong polarization of the valleys in the of carbon atoms in a strip= − containing− a constriction with zigzag edges. Top panel: geometry, because no well-separated valleys exist in 1D. Variation of the electrostatic potential along the strip, for the two cases of an abrupt Dispersion relation in the wide and narrow regions. An electron in the first valley wide regions.We Our show finding that the signifies polarization that of the this two valley valleys filter can in graphene be inverted and smooth potential barrier (solid and dashed lines). The polarity of the valley filter can be addressed individually as independent internal degrees of (modes n 0,1,2,...)istransmitted(filledcircle),whereasanelectroninthe by locally raising the Dirac point in the region of the constriction, switches when= the potential height, U0,intheconstrictioncrossestheFermi freedom of the conduction electrons. This is only possible in a 2D second valley (modes n 1, 2,...)isreflected(opencircle).Bottompanel: by means of a gate voltage, such that the Fermi level lies in the energy, EF. = − − geometry, because no well-separated valleys exist in 1D. Variation of the electrostatic potential along the strip, for the two cases of an abrupt We show that the polarization of this valley filter can be inverted and smooth potential barrier (solid and dashed lines). The polarity of the valley filter 172 nature physics VOL 3 MARCH 2007 www.nature.com/naturephysics by locally raising the Dirac point in the region of the constriction, switches when the potential height, U0,intheconstrictioncrossestheFermi

by means of a gate voltage, such that the Fermi level lies in the energy, EF.

Untitled-1172 1 nature physics VOL 3 MARCH 2007 www.nature.com/naturephysics13/2/07, 12:35:49 pm

Untitled-1 1 13/2/07, 12:35:49 pm Valley-coded electron highways and traffic control at a 4-way juncon

Quantum valley Hall effect Topological valleytronics

Ø Valley-momentum locked 1D channels Ø Valley valve Ø Conductance quanzaon at 4e2/h Ø A tunable electron beam splier

J. Li… JZ, Nature Nano. 11, 1060 (2016). J. Li…JZ, arXiv:1708.02311v1 (2017). 19 [email protected] Nov 7, 2017 R ESEARCH A RTICLE electron spins along the laser propagation Strikingly, the signal changes sign for the of the entire sample (Fig. 2, A and B). Here,

direction (19). The KR is detected with the two edges of the sample, indicating an instead of taking a full Bext scan, Bext is use of a balanced photodiode bridge with a accumulation of electron spins polarized in sinusoidally modulated at fB 0 3.3 Hz with noise equivalent power of 600 fW Hz–1/2 the z direction at x 0 –35 mm and in the –z an amplitude of 30 mT, and signal is in the difference channel. We apply a square directionþ at x 0 35 mm. This is a strong detected at f T 2f . The measured signal is þ E B wave voltage with a frequency fE 0 1.169 kHz signature of the spin Hall effect, as the spin then proportional to the second derivative of to the two contacts, producing an alternating polarization is expected to be out-of-plane a Bext scan around Bext 0 0 mT and can be electric field with amplitude E for lock-in and change sign for opposing edges (7–12). regarded as a measure of the spin density ns. detection. Measurements are done at a tem- A one-dimensional spatial profile of the spin The image shows polarization localized at perature T 0 30 K unless otherwise noted. accumulation across the channel is mapped the two edges of the sample and having op-

Detecting spin currents. An unstrained out by repeating Bext scans at each position posite sign, as discussed above. The polar- GaAs sample with a channel parallel to [110] (Fig. 1C). It is seen that A0, which is a mea- ization at the edges is uniform over a length with a width w 0 77 mm, a length l 0 300 mm, sure of the spin density, is at a maximum at of 150 mm but decreases near the contacts. and a mesa height h 0 2.3 mm was prepared the two edges and falls off rapidly with The latter is expected as unpolarized elec- (Fig. 1A) and was measured using a wave- distance from the edge, disappearing at the trons are injected at the contacts. length of 825 nm and an average incident center of the channel (Fig. 1D) as expected The origin of the observed spin Hall laser power of 130 mW. We took the origin for the spin Hall effect (10, 11). We note that effect is likely to be extrinsic, as the intrinsic of our coordinate system to be the center of equilibrium spin polarization due to current- effect is only expected in systems with spin the channel. In Fig. 1B, typical KR data for induced magnetic fields cannot explain this splitting that depends on electron wave 3 scans of Bext are shown. The red curve is spatial profile; moreover, such polarization vector k. Although k spin splitting in bulk taken at position x 0 –35 mm; the blue curve is estimated to be less than 10–6, which is GaAs (23) may give rise to an intrinsic spin is taken at x 0 35 mm, corresponding to the below our detection capability. Hall effect, this is unlikely because negligi- two edges of theþ channel. These curves can An interesting observation is that the width ble spin splitting has been observed in be understood as the projection of the spin of the Lorentzian becomes narrower as the unstrained n-GaAs (24). Measurements were polarization along the z axis, which dimin- distance from the edge increases, correspond- also performed on another sample with a ishes with an applied transverse magnetic ing to an apparent increase in the spin lifetime channel parallel to [110], and essentially the field because of spin precession; this is (Fig. 1E). It is possible that the finite time same behavior was reproduced (Fig. 3). known as the Hanle effect (8, 20). The data required for the spins to diffuse from the edge Effects of strain. The strained InGaAs sam-

are well fit to a Lorentzian function A0/ to the measurement position changes the ple, in contrast, offers the possibility of display- [(w t )2 1], where A is the peak KR, w 0 lineshape of B scans. Another conceivable ing the intrinsic spin Hall effect in addition to the L s þ 0 L ext gmBBext/I is the electron Larmor precession explanation is the actual change in ts for spins extrinsic effect. The lattice mismatch causes frequency, ts is the electron spin lifetime, g that have diffused away from the edge. Be- strain in the InGaAs layer (25), and partial strain is the electron g factor (21), mB is the Bohr cause these spins have scattered predomi- relaxation causes the in-plane strain to be aniso- magneton, and I is the Planck constant. nantly toward the center of the channel, spin tropic (26). Using reciprocal space mapping with From spin Hall effect to quantum spin Hall effectrelaxation due to the D’yakonov-Perel mecha- x-ray diffraction at room temperature, we deter- AB nism (20) may be affected. In Fig. 1G, Bext mined the in-plane strain along [110] and [110] ns (a.u.) Reflectivity (a.u.) scans at x 0 –35 mmforarangeofE are to be –0.24% and –0.60%, respectively, and the -2-1 021 1 2 3 4 5 shown. Increasing E leads to• larger spin accu- strain along [001] to be 0.13%. These strained mulation (Fig. 1H), but the polarizationCdTe/HgCdTe satu- layers/CdTe show QW electron spinþ precession at zero 150 rates because of shorter ts• forInAs larger/GaSbE (Fig. bilayer applied magnetic field when optically injected GaAs 1I). The suppression of ts with increasing E electron spins are dragged with a lateral electric is consistent with previous observations (22). field (24), which is due to an effective internal 100 The homogeneity of the effect is ad- magnetic field Bint perpendicular to both the dressed by taking a two-dimensional image growth direction and the electric field (Fig. 4A). A possible explanation for such behavior may be strain-induced k-linear spin splitting terms 50 Kerr rotation (µrad) in the Hamiltonian, which is expected to give

) -2 -101 2 m

µ rise to the intrinsic spin Hall effect (16).

( Δ > 0 Δ < 0 SOC ) 40 Δ > 0 n T AB The strained InGaAs sample was processed

o 0

i 20 m t ( i into a channel oriented along [110] with w 0 t

s 0 x e o

P 33 mm, l 0 300 mm, and h 0 0.9 mm. A laser B -20 -40 wavelength of 873 nm and an incident power -50 2 −

) E // [110] E // [110] of 130 mW were used for this measurement,

d 1 a r 0 and typical results are shown in Fig. 4B. Sur- µ ( prisingly, the spin polarization is suppressed at 0 -1 -100 A -2 CD B 0 0 mT, and we observe two peaks offset • ext -40 -200 20 40 -40 spin-momentum -20 04020 from Bext 0 0 mT. We attribute this behavior Spin-orbit coupling Position (µm) Position (µm) locked edge states to the presence of Bint. Because electron spins -150 induced band inversion Fig. 3. Crystal orientation dependence of the respond to the sum of Bext and Bint, the spin spin Hall effect in the unstrained GaAs sample -40 -20 0 20 40 -40 -20 02040 • ballisc conducon polarization is maximum at Bext 0 –Bint when Position (µm) Position (µm) with w 0 77 mm. (A and B) KR as a function of B cancels out B . The application of a x and B for E // [110]Konig and et al, Science 318, 766(2007) E // [110], ext int ext square-wave voltage causes the signal to arise Fig.Kato et al, Science 306, 1910 (2004) 2. (A and B) Two-dimensional images of respectively, with E 0 10 mVDu et al, PRL, 114, 096802 (2015). mm–1. A linear spin density n and reflectivity R, respectively, background has been subtracted from each B from both positive and negative electric-field s ext directions, resulting in a double-peak structure. for the unstrained GaAs sample measured at T 0 scan. (C and D) Spatial profile of A0 for E // 30 K and E 0 10 mV mm–1. [110] and E // [110], respectively. Although the spin polarization reverses direc- hps://www.sciencenews.org/arcle/physics-edge 20 [email protected] Nov 7, 2017 www.sciencemag.org SCIENCE VOL 306 10 DECEMBER 2004 1911 Inverng the band gap of bilayer graphene is easy

+ + Δ/2 - Δ E - Δ/2 - +

E -> -E Δ -> - Δ

21 [email protected] Nov 7, 2017 Quantum valley Hall effect in bilayer graphene

2 K K’ K and K’ 0

P otential (a.u.) -2 Δ < 0 Δ > 0 K’ x4 x4 K -200 -100 0 100 200 Distance (nm)

Theorecal proposal: Ø 4 pairs of counter-propagang metallic 1D modes in the juncon Ø Valley-momentum locked – “quantum valley Hall kink states”

Marn et al, Phys. Rev. Le. 100, 036804 (2008) 22 [email protected] Nov 7, 2017 PERSPECTIVES tecting groups or other synthetic manipu- lead to the development of therapeutic agents 3. J. T. Malinowski et al., Science 340, 180 (2013). lations), which helps reduce the number of with attenuated toxicity in mammalian cells. 4. A. D. Argoudelis, H. K. Jahnke, J. A. Fox, Antimicrob. Agents Chemother. 1962, 191 (1962). synthetic steps. Just as the synthesis reported by Malinowski 5. K. Otoguro et al., J. Antibiot. (Tokyo) 63, 381 (2010). The synthesis developed by Malinowski et al. will likely prove to be an enabling devel- 6. W. Lu, N. Roongsawang, T. Mahmud, Chem. Biol. 18, 425 et al. provides access to pactamycin in only opment in the story of pactamycin, it also (2011). 15 chemical steps (versus 32 steps previ- highlights the enabling power of symmetry as 7. M. Iwatsuki et al., J. Antibiot. 65, 169 (2012). 8. S. Hanessian et al., Angew. Chem. Int. Ed. 50, 3497 ously) and 1.9% overall yield. Notably, sev- a design element in rendering complex mol- (2011). eral late-stage intermediates used in the syn- ecules synthetically practical. 9. S. Hanessian, R. R. Vakiti, S. Dorich, S. Banerjee, B. thesis represent suitable precursors for the Deschênes-Simard, J. Org. Chem. 77, 9458 (2012). References preparation of analogs. Future investigations 1. D. E. Brodersen et al., Cell 103, 1143 (2000). of the biological effects of such analogs could 2. G. Dinos et al., Mol. Cell 13, 113 (2004). 10.1126/science.1236882

PHYSICS

Observation of a quantized resistance state The Complete Quantum Hall Trio in the absence of an external magnetic fi eld completes a trio of quantum Hall related effects. Seongshik Oh hen an electric current I flows cal insulator; the result confi rms the long- nal resistance of the sample reduces to zero. through a slab of conductor placed awaited quantum anomalous Hall effect As understanding of the QHE matured, W in an external magnetic field (QAHE), the fi nal member of the quantum questions arose as to whether such loss- H perpendicular to the flow direction, the Hall trio (see the fi gure). less edge channels could exist even in the on April 12, 2013 magnetic fi eld defl ects the current-carrying Soon after the discovery of the QHE, it absence of an external magnetic field. In charge particles toward the edge of the con- was realized that the quantization occurs 1988, it was shown theoretically that such an V ductor and a transverse voltage T develops when dissipationless (or lossless) one- edge channel can exist on a two-dimensional across the sample. This effect, discovered dimensional channels form around the lattice ( 4). Then, almost 20 years later, exper- by Edwin Hall in 1879 ( 1), is called the Hall edges of the sheet while the rest of the sam- imental demonstration of the presence of effect. Because the transverse resistance (or ple remains insulating and that the num- lossless edge channels in a HgTe/CdTe quan- V I Hall resistance) defined as T/ is propor- ber of these edge channels determines the tum well in the absence of an external mag- tional to H/n, where n is the sheet carrier den- integer value ν. In such a case, electrons netic fi eld was reported ( 5). However, due to sity of the sample, the Hall effect has been flowing on one side cannot be scattered the absence of a magnetic fi eld forcing the www.sciencemag.org widely used to quantify the carrier type (elec- backward because the backward channels current to fl ow one way or the other, there tron or hole), density, and mobilities of elec- exist only on the other side of the sample, existed both clockwise and counter-clock- tronic materials. However, in the 1980s it was which is separated by the insulating bulk in wise edge channels, whose direction was found that when the charge carriers are con- between, and whenever quantization occurs determined by the spin orientation (either up fi ned to a two-dimensional system (or sheet), in the transverse resistance, the longitudi- or down) of the occupying electrons, forced the Hall resistance becomes exactly quantized at h/(νe2), Downloaded from where h is the Planck con- Hall Spin Hall Anomalous Hall (1879) (2004) (1881) stant, e is the electron charge, Topological origin: Valley Chern number change and ν is a positive integer, Quantum Hall Quantum spin Hall Quantum anomalous Hall whenever H/n approaches (1980) (2007) (2013) Quantum spin Hall effect specifi c values ( 2). This phe- Quantum valley Hall effect K’ K’ nomenon, called the quantum K K Hall effect (QHE), always M requires an external magnetic H C = -1 C =±1 CK= +1 K Δ<0 fi eld. On page 167 this issue C=0 s Δ>0 3 et al ( ), Chang . have discov- CK’ = -1 CK’ = +1 ered that such exact quanti- zation in the transverse resis- tance can occur even without K: CK changes by +2 ’ an external magnetic fi eld on Quantum Hall Quantum spin Hall QuantuKm: C anK’om changes by -2alous Hall Cs changes by -1 a thin ferromagnetic topologi- Quantum Hall trio. Numbers in parentheses indicate the years of each discovery. H is the external ma2 modes per valley gnetic fi eld, and M is Cs changes by +1 the magnetization. For all three quantum Hall effects, electrons fl ow through the lossless edge channels, with the rest of the Department of Physics and Astron- system insulating. When there is a net forward fl ow1 mode per spin of electrons for Hall resistance measuremx2 for spin ent, (left) those extra electrons 4 modes per valley omy, and Institute for Advanced Mate- occupy only the left edge channels in the quantum Hall system regardless of their spins, (center) opposite-spin electrons occupy rials, Devices and Nanotechnology, Rutgers, The State University of New opposite sides in the quantum spin Hall systemØ , andspin-momentum locked (right) only spin-down electrons fl ow throØug hvalley-momentum locked the left edge in the quantum Jersey, Piscataway, NJ 08854, USA. anomalous Hall system. The locking schemes between spin and fl ow direction, and the number of edge channels depend on the E-mail: [email protected] material details, and only the simplest cases are illustrated here. 23 [email protected] Jung et al PRB 84, 075418 (2011), Li et al, PRB 82, 245404 (2010) Nov 7, 2017 www.sciencemag.org SCIENCE VOL 340 12 APRIL 2013 153 Published by AAAS w Band structure of a smooth juncon

+ w 3 - Zhenhua Qiao 2 w=60 nm Group, USTC 1 0 Δ -1

Potential (a.u) -2 Potenal (a.u .) -3 Si backgate -200 -100 0 100 200 Distance (nm) E Ø chiral in each valley

Ø ballisc conductance 4e2/h in the ’ Δ zigzag edge arfact absence of inter-valley scaering

Δ < 0 Δ > 0 - + K Kʹ

24 [email protected] J. Li et al, Nature Nano. 11, 1060 (2016). Zarenia, et al PRB, 84, 125451 (2011) Nov 7, 2017 “odd” vs “even” configuraon: built-in control

Δ > 0 Δ > 0 Δ < 0 Δ > 0 + + - +

3 3

2 2

1 1 0 Δ “even” 0 Δ “odd” -1 -1 Potential (a.u) Potential (a.u) -2 Potenal (a.u .) -2 Potenal (a.u .)

-3 -3 -200 -100 0 100 200 -200 -100 0 100 200 Distance (nm) Distance (nm)

Δ’

25 [email protected] Nov 7, 2017 Kink states in bilayer graphene: two helicies

- + + -

K K’ K K’ h=+1 h=-1

26 [email protected] Nov 7, 2017 A valley valve of kink states

Valve “on” state Valve “off” state A spin valve D D - + - + D D D D - + + -

S S hps://commons.wikimedia.org/wiki/File:Spin- valve_GMR.svg Valley index aligned Valley index an-aligned

27 [email protected] Nov 7, 2017 Outline

• The valley degree of freedom in hexagonal laces

Valley, Berry curvature, valley Hall effect and topological kink states

• Quantum valley Hall kink states in bilayer graphene Ø Precision lithography h 47e2 Ø Transport properes ) 6 Ω

Ø Valley valve and electron beam splier (k

ch 5 R

4

-60 -40 -20 0 20 40 V (V) • Summary Si

[email protected] Nov 7, 2017 A dual-split gated bilayer graphene device

E E h-BN

h-BN

SiO2

1. Use the four split gates to gap both sides 2. Measure transport along the juncon

3. Use the doped Si backgate to control EF in the juncon

29 [email protected] Nov 7, 2017 A high-quality GaAs 2D electron gas

GaAs capping layer

200-500 nm Al0.3Ga0.7As barrier Si δ-doping layer

30 nm GaAs quantum well 2DEG

Si δ-doping layer

Al0.3Ga0.7As barrier

GaAs/Al0.3Ga0.7As superlace

GaAs substrate Ji et al, Nature, 422, 415(2003) (Heiblum group)

μ ~ 3x107cm2V/s l ~ 0.1 mm

Ø The highest quality two-dimensional electron system Ø Devices dimensions cannot be made too small

30 [email protected] Nov 7, 2017 REPORTS

Fig. 1. Edge-contact. (A)Schematicoftheedge- contact fabrication process. (B)High-resolution bright-field STEM image showing details of the edge- contact geometry. The expanded region shows a magnified false-color EELS map (fig. S6) of the inter- face between the graphene edge and metal lead. (C) Two-terminal resistance versus channel length at fixed density, measured from a single graphene de- vice in the TLM geometry. Solid line is a linear fit to the data. Inset shows an optical image of a TLM device with edge-contacts. (D) Contact resistance calculated from the linear fit at multiple carrier densities for two separate devices. Error bars repre- sent uncertainty in the fitting. Inset shows resist- ance scaling with contact width measured from a separate device.

LETTERS NATURE PHYSICS DOI: 10.1038/NPHYS4140

a B = 0 QHE regime

dedrag −0.8 0.8 drag −0.8 0.8 Rxx (kΩ) Rxx (kΩ) Indirect Indirect 4 4 exciton exciton Direct 3 3 exciton 2 2 1 1 QWTunnel QW QWTunnel QW Van der Waals Transfer Method barrier barrier 0 0 top top ⌫ nature of the contact. The EELS map additionally ⌫ −1 DOI: 10.1038/NPHYS4140 −1 A LETTERS indicates that contact was made predominantly to NATURE PHYSICS thebc Cr adhesion layer. −2 −2 To characterize the quality of the edge-contact, d = 5 nm d = 5 nm a B = 0 we used the transfer-lengthQHE regime method (TLM). Mul- −3 −3 tiple two-terminal graphene devices consisting of Cr/Au T = 20 K T = 0.3 K auniform2-mmchannelwidthbutwithvaryingdedrag −0.8 0.8 B =drag 9 T −0.8 0.8 B = 15 T BLG RBNxx (kΩ) −4 Rxx (kΩ) −4 Indirect channel lengths wereIndirect fabricated, and their resist- 4 4 exciton ances were measuredexciton as a function of carrier den- BLG BN −4 −3 −2 −1201 34 −4 −3 −2 −1201 34 B D Direct sity n induced by a voltage applied to a siliconSiO 3 3 ⌫ ⌫ exciton 2 BN bot bot back gate.10 Figureµm 1C shows the resistance versus 2 FLG 2 channel length, measured at two different carrier densities. InTop the diffusiveBLG regime,Top gate where the chan- 1 1 QWTunnel QW QWTunnel QW barrier nel length remainsBot. BLG severalbarrierBot. times gate longer than the 0 0 top top ⌫ mean free path, the total resistance in a two- ⌫ −1 −1 Figureterminal 1 | Doublemeasurement bilayer can be written graphene. as R = a, Optically excited particle hole pairs combine to form, short-lived, spatially direct excitons. Electron–hole pairing bc2RC(W)+rL/W,whereRC is the contact re- −2 −2 sistance, L is the device length, W is the device d = 5 nm d = 5 nm C across a tunnel barrier prevents recombination,−3 T = 20 K leading to long-lived,− spatially3 T = 0.3 K indirect excitons. In the QHE regime at large magnetic field, spatially indirect width, and r is theCr/Au 2D channel resistivity; RC and excitonsr were extracted canBLG also asBN the result intercept from and slope coupling of a −4 betweenB = 9 T partially filled Landau− bands.4 B = 15 bT, Optical image of a double bilayer graphene device, with graphite contact linear fit toBLG the dataBN shown here for two separate −4 −3 −2 −1201 34 −4 −3 −2 −1201 34 andSiOdevices local (Fig. graphite 1D). RC was back remarkably gate. low, The reach- scale bar is 10 nm.⌫ c, Cartoon cross-section of our⌫ device construction (spatially indirect excitons are formed between the 2 BN bot bot ing ~150 ohm·mmforn-typecarriersathighden-FLG twosity. BLG This layers). value is ~25% FLG, lower than few-layer the best reported graphene. d,e, Longitudinal drag resistance for device 19 with a tunnel barrier thickness of d 5nm, as a function of filling Top BLG Top gate = Fig. 2. Polymer-free layer assembly. (A) Schematic of the van der WaalsBot. technique BLG forBot. polymer- gate factors.surfaced contacts, Measured without additional at B engineering9Tand T 20K. e, Measured at B 15T and T 0.3K. In e diverging response near zero density for each layer has been Wang et al, Science, 342, 614-617 (2013) B such as chemical (18)orelectrostatic(17)doping. free assembly of layered materials. ( ) OpticalColumbia groups image of a multilayered heterostructure using the Li et al, Nat. Phys. 13, 751 (2017) = = = = processDean et al, Nat. Nano. 5, 722-726 (2010) illustrated in (A). (C) AFM image of a large-area encapsulatedFigure graphene 1 | Double layer bilayershowing that graphene.removed it Becausea, Opticallyfor this clarity. value excited is obtained particle in a two-terminal hole pairs combine to form, short-lived, spatially direct excitons. Electron–hole pairing is pristine and completely free of wrinkles or bubbles except at its boundary. (D)High-resolutioncross- geometry, it includes the intrinsic limit set by the section ADF-STEM image of the device in (C). The BN-G-BN interfaceacross is found a tunnel to be pristine barrier and prevents free of quantum recombination, resistance of leading the channel, to long-lived, which can be spatially indirect excitons. In the QHE regime at large magnetic field, spatially indirect any impuritiesEnable heterostructures of different 2D materials… down to the atomic scale. excitons can also result froman coupling electron–holesubtracted between to yield an partially extrinsic asymmetry contact filled resistance Landau is bands. apparent.b, Optical imageIn the of a electron–electron double bilayer graphene device,double-well with graphite contact system by excitons generated (and then annihilated) and local graphite back gate.(e–e) The scale quadrant, bar is 10 nm. magnetodragc, Cartoon cross-section is observed of our device whenever construction there (spatially is partial indirect excitonsat are the formed contacts between (inset the Fig. 2e). Charge-neutral excitons feel no www.sciencemag.org twoSCIENCE BLG layers).VOL FLG,342 few-layer 1 NOVEMBER graphene. 2013 d,e, Longitudinal drag31 resistance615 for device 19 with a tunnel barrier thickness of d 5nm, as a function of filling [email protected] filling of both theNov 7, 2017 ⌫ 1 and 3 LLs [(1, 1), (1, 3)(3, 1) and (3, 3)], = Lorentz force even under very large B, and zero Hall resistance factors. d, Measured at B 9Tand T 20K. e, Measured= at B 15T and T 0.3K. In e diverging response near zero density for each layer has been =whereas= in the hole–hole= (h–h) quadrant= it is partially filled ⌫ 2 is expected4,15. Indeed, Fig. 2e shows vanishing counterflow Hall removed for clarity. = and 4 LLs [( 2, 2), ( 2, 4), ( 4, 2) and ( 4, 4)]. resistance when ⌫ 1. The dissipationless nature of the EC T = an electron–hole asymmetryBased is apparent. on recent In the understanding electron–electron ofdouble-well how the eightfold system by degeneracyexcitons generatedis (and revealed then annihilated) by simultaneous observation of zero longitudinal (e–e) quadrant, magnetodrag is observed whenever there is partial at the contacts (inset Fig. 2e). Charge-neutral excitonsCF feel no filling of both the ⌫ 1of and the 3 LLs ZLL [(1, in 1), ( BLG1, 3)(3, lifts 1) and at(3, large 3)], BLorentz(refs force 23,24), even we under can very assign large B, andresistance zero HallR resistancexx . Figure 2e also plots Hall resistance in the parallel = whereas in the hole–holea (h–h) spin, quadrant valley, it and is partially orbital filled index⌫ 2 tois the expected symmetry4,15. Indeed, broken Fig. 2e states shows vanishingflow configuration, counterflow Hall which is a linear combination of drag and = and 4 LLs [( 2, 2), ( 2, 4), ( 4, 2) and ( 4, 4)]. resistance when ⌫ 1. The dissipationless nature of the EC of each layer (see Supplementary Information), andT = observe that counterflow measurements. The Hall resistance in the parallel flow Based on recent understanding of how the eightfold degeneracy is revealed by simultaneous observation of zero longitudinalk ⌫ strong magnetodrag in Fig. 1e appears only whereCF both layers are geometry Rxy shows a prominent peak at T 1, approaching the of the ZLL in BLG lifts at large B (refs 23,24), we can assign resistance Rxx . Figure 2e also plots Hall resistance in the parallel 2 = a spin, valley, and orbitalin a index zero to orbital the symmetry state. Magnetodrag broken states flow due configuration, to momentum which or is energy a linear combinationquantized of value drag of and 2h/e . (This doubling of the quantization is due 19,25,26 of each layer (see Supplementarycoupling Information),is expected and observe to vanish that incounterflow the zero-temperature measurements. The limit, Hall resistanceto the in fact the parallel that current flow flows through the double BLG system twice, strong magnetodrag in Fig. 1e appears only where both layers are geometry Rk shows a prominent peak at ⌫T 1,k approaching the / / whereas the 0.3K response in Fig. 1e exceeds 1xy k in some regions, and=Rxy is defined as Vxy I instead of Vxy 2I.) The stark dierence in a zero orbital state. Magnetodrag due to momentum or energy quantized value of 2h/e2. (This doubling of the quantizationCF is due suggesting a dierent origin. One possibility is the formation of between R and Rk provides further evidence and confirmation the coupling19,25,26 is expected to vanish in the zero-temperature limit, to the fact that current flows through the double BLG systemxy twice,xy indirect excitons between the layers that are not yet phase coherent, origin of the ⌫T 1 state lies in the strong correlation and interlayer whereas the 0.3K response in Fig. 1e exceeds 1 k in some regions, and Rxyk is defined as Vxy /I instead of Vxy /2I.) The stark dierence CF 13 = suggesting a dierent origin.resembling One possibility the EC is precursor the formation reported of between in GaAsRxy doubleand Rxyk provides layers further. This evidencephase and coherence confirmation between the the two BLG layers. indirect excitons between the layers that are not yet phase coherent, origin of the ⌫ 1 state lies in the strong correlation and interlayer drag drive drag CF interpretation would suggest a selection rule whereT = EC formation is In Fig. 3a we plot the magnitude of Rxy , Rxy , Rxx and Rxy versus resembling the EC precursor reported in GaAs double layers13. This phase coherence between the two BLG layers. drive drag limited to the zero orbital ground states only. drag ddrive/`B.drag For deviceCF 37 (d 3.6nm), quantized Rxy and Rxy together interpretation would suggest a selection rule where EC formation is In Fig. 3a we plot the magnitudedrag of Rxy , Rxy , Rxx and Rxy versusCF = Figure 2a–c shows the longitudinal magnetodrag (R ), Hall withdrive zero-valueddrag R persist only over a narrow range, eectively limited to the zero orbital ground states only. d/`B. For device 37 (d 3.6nm),xx quantized Rxy and Rxy togetherxy drag drag driveCF = /` Figure 2a–c shows thedrag longitudinal (Rxy ), and magnetodrag drive layer (Rxx Hall), Hall conductancewith zero-valued ( xyRxy)persist for a only device over a narrowestablishing range, e bothectively an upper and lower critical value for d B. The drag drive /` drag (Rxy ), and drive layerwith Hall interlayer conductance separation ( xy ) ford a device3.6 nm,establishing measured both at anB upper18 and T lower and criticalupper value bound for d isB understood. The by the requirement to be in the so-called with interlayer separation d 3.6 nm, measured at B 18= T and upper bound is understood= by the requirement to be in the so-called T =20 mK (for simplicity= we focus our discussion on the e–e strongly interacting regime (that is, achieve a minimum eective T 20 mK (for simplicity we focus our discussion on the e–e strongly interacting regime (that is, achieve a minimum eective = = /` quadrant only, but a completequadrant mapping only, of but the ZLL a complete can be found mappinginterlayer of the interaction). ZLL can We be note found that theinterlayer critical value interaction).d/`B 0.6 We note that the critical value d B 0.6 ⇠ 4,14 ⇠ in the Supplementary Information).in the Supplementary A large response Information). is observed is A approximately large response 30% is that observed was reportedis for approximately GaAs4,14. Reducing 30% that was reported for GaAs . Reducing in Rdrag Rdrag and Rdrive followingdrag a diagonaldrag linedrive corresponding to the interlayer spacing from 3.6 nm to 2.5 nm results in a decrease xx xy xy in Rxx Rxy and Rxy following a diagonal line corresponding to the interlayer spacing from 3.6 nm to 2.5 nm results in a decrease total filling fraction ⌫T 1, 3 and 5 (⌫T ⌫top ⌫bot). Figure 2d shows of the lower critical d/`B (Fig. 3a). However, we note that this drag drive =total filling= fraction+ ⌫T 1, 3 and 5 (⌫T ⌫top ⌫bot). Figure 2d shows of the lower critical d/`B (Fig. 3a). However, we note that this Rxy and Rxy for varyingdrag magnetic fielddrive measured= along a line of boundary= + corresponds to approximately the same absolute magnetic drive varying drive layer density.RxyR andshowsRxy conventionalfor varying behaviour magnetic with field value measured of approximately along a 18 line T. This of mayboundary relate to the corresponds minimum to approximately the same absolute magnetic xy drive well-defined QHE plateauxvarying observed drive at layer⌫drive density.1 and 2, whileR theshowsmagnetic conventional field required behaviour to fully with lift thefield ZLL degeneracy value of approximately (set by 18 T. This may relate to the minimum = xy magnetodrag is near zero at this sample temperature over most of sample disorder, which is approximately the same between these two well-defined QHE plateaux observed at ⌫drive 1 and 2, while the magnetic field required to fully lift the ZLL degeneracy (set by the density range. However, when the drive and drag layer densities devices). Alternatively= this could be signal of a transition to a new, drive magnetodrag is near zero at this sample temperature over most of sample disorder, which is approximately the same between these two sum to ⌫T 1, Rxy deviates strongly from its single-layer value as yet unidentified, phase as d/`B tends towards zero. = the density range. However, when2 the drive and drag layer densities devices).CF Alternatively this could be signal of a transition to a new, and exhibits re-entrant behaviour with quantized magnitude h/e . Figure 3b shows the counterflow Hall resistance Rxy plotted as a drag drive ⌫ ⌫ At the same total filling,sumRxy totakes⌫T on this1, R samexy quantizeddeviates value. stronglyfunction from of its filling single-layer fractions top valueand bot. Theas EC yet state, unidentified, as evidenced phase as d/`B tends towards zero. drag = CF 2 CF The amplitude of Rxx andfirst exhibits rises dramatically re-entrant in the behaviour vicinity of withby a quantized zero-valued magnitudeRxy , again followsh/e a. diagonalFigure line corresponding 3b shows the counterflow Hall resistance R plotted as a ⌫ 1 and then dips rapidly to zero at exact filling. Quantization of to ⌫ 1. Along this diagonal the state is described by an interlayer xy T = drag T = ⌫ ⌫ both Rdriveand Rdrag at integerAt the total same filling, total concomitant filling, withRxy a localtakes ondensity this imbalance, same quantized which we parametrize value. asfunction1⌫ ⌫ of⌫ filling(1⌫ fractions0 top and bot. The EC state, as evidenced xy xy drag = bot top = CF zero-valued Rdrag, providesThe strong amplitude evidence of theR formationfirst rises of an dramaticallyonly for ⌫ in⌫ the1/2). vicinity To understand of by the a e zero-valuedect of this layerR , again follows a diagonal line corresponding xx xx top = bot = xy EC phase11. ⌫ imbalance, we examine the temperature dependence⌫ of the ⌫ 1 T 1 and then dips rapidly to zero at exact filling. Quantization of to T 1. AlongT this= diagonal the state is described by an interlayer Confirmation that a superfluid= drive phasedrag of charge carriers has state over a large range of 1⌫. = 1⌫ ⌫ ⌫ 1⌫ both Rxy and Rxy at integer total filling, concomitant with a localCF density imbalance, which we parametrize as bot top ( 0 truly formed is provided by magnetotransportdrag in the counterflow The minimum value of the Rxy shows activated behaviour with = = 9 zero-valued R , provides strong evidence of the formation of an only for ⌫ ⌫ 1/2). To understand the eect of this layer geometry , in which charge currentxx is carried through the varying temperature (Fig. 3c), allowing us to deduce antop associated= bot = EC phase11. imbalance, we examine the temperature dependence of the ⌫ 1 T = 752 Confirmation that a superfluid phaseNATURE of charge PHYSICS | carriers VOL 13 | AUGUST has 2017 |statewww.nature.com/naturephysics over a large range of 1⌫. CF truly formed© 2017 is provided Macmillan Publishers by magnetotransportLimited, part of Springer Nature. inAll rights the reserved. counterflow The minimum value of the Rxy shows activated behaviour with geometry9, in which charge current is carried through the varying temperature (Fig. 3c), allowing us to deduce an associated

752 NATURE PHYSICS | VOL 13 | AUGUST 2017 | www.nature.com/naturephysics

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. Generaon I: one kink channel device Layer by layer stacking of graphite/h-BN/bilayer graphene/h-BN

32 [email protected] Nov 7, 2017 2um

+ I+ V

I- V-

33 [email protected] Nov 7, 2017 Generaon II: 4-way juncon boom gate 3 4 pairs of split gates top gate

-Δ +Δ 2 4 +Δ -Δ

1 global Si back gate

Ø Dry van der Waals transfer Ø 1D side contact Ø 4 channels: 300 nm (L) x70 nm (W) 1μm 34 [email protected] Nov 7, 2017 Generaon II: 4-way juncon

BN bilayer graphene BN

[email protected] Nov 7, 2017 All the fun we (Jing) had since 2012…

graphite boom gates Au top gates

Jing Li 70 nm Au top gates

100 nm

I+ V+ 70 nm

100 nm

70 nm

I- V- 36 [email protected] Nov 7, 2017 Alignment of the top and boom gates is crical

2nd wring

1st wring

y 200nm

x 1μm

200nm Ø Center alignment beer than 10nm. Ø Dimension control beer than 5nm.

37 [email protected] Nov 7, 2017 Evidence of the kink states: generaon I

“even” 1MΩ “odd”

Rj

0.3 0.2 “even” 0.1 0 D (V/nm) -0.1 L -0.2 “odd” -0.3

-0.4 -0.2 0 0.2 0.4 J. Li… JZ, Nature Nano. 11, 1060 (2016). DR (V/nm) Kink states present only in the “+- “ and “- +” configuraons

Also Lee et al, Scienfic Reports 7, 6466(2017) (Hu-Jong Lee group) 38 [email protected] Nov 7, 2017 Generaon II: 4-way juncons

boom gate boom gate 3 3 access access top gate top gate

-Δ +Δ -Δ +Δ 2 4 2 4 +Δ -Δ doped doped

access access 1 1

• R13 can measure north, south or both channels - + • Rc ~ hundreds of Ω (metal interface + access region)

39 [email protected] Nov 7, 2017 Band structure of the kink states in a magnec field

h 2 12e EF h 8e2

h Δ’ 4e2

h 8e2 h 12e2

Ø Landau levels in the conducon and valence bands of the juncon Ø The increase of gap makes the kink states more robust

J. Li, … JZ, Nature Nano. 11, 1060 (2016). Zarenia et al, PRB 84, 125451 (2011) (Peeters group) 40 [email protected] Nov 7, 2017 Nearly ballisc conducon in individual channels

8k h 6T h 8T 2 2 +R 12e 7k 4T 4e C EF h 2T 2 6k 8e ) Ω 5k 0T h ( Ω ) 2 2-4 4e R R13-6 ( 4k h S D S D 3k 8e2 h 2k 2 -40 -20 0 20 40 60 12e VSi (V) EF

41 [email protected] Nov 7, 2017 Nearly ballisc conducon in individual channels

S D S D 8k 8 T 6T h 8T 100M 2 +R 7k 4T 4e C 6 T 2T 10M 6k ) ) Ω Ω 0T ( 1M

5k

( Ω ) even 4 T R 2-4

R R13-6 ( 4k 100k 2 T

S D 3k 10k 0 T S D

2k -40 -20 0 20 40 60 -60 -40 -20 0 20 40 VSi (V) VSi (V) EF Kink states + hopping in the gapped quadrants

42 [email protected] Nov 7, 2017 Nearly ballisc conducon in individual channels

8k 6T h 8T 2 +R 7k 4T 4e C

2T 6k ) Ω 5k 0T ( Ω ) S D 2-4

R R13-6 ( 4k

3k S D S D

2k -40 -20 0 20 40 60 magnec field suppresses VSi (V) hopping conducon EF

43 [email protected] Nov 7, 2017 Nearly ballisc conducon in individual channels

7200 L=300 nm T=1.5 K 7000

) 6800 Ω (

kink 6600 R h = 6.45 kΩ 6400 4e2

6200 0 2 4 6 8 10

B (T)

Rkink ~ 7 kΩ at zero magnec field 44 [email protected] Nov 7, 2017 A valley valve of kink states

Valve “on” state Valve “off” state D D - + - + D D D D - + + -

S S

Valley index aligned Valley index an-aligned

45 [email protected] Nov 7, 2017 Nano Letters LETTER

We model the case in which the leads are bilayer graphene ribbons with a single zero line at their centers, and the system is A valley valve and beam splier divided into four quadrants in which the sign of the potential difference can be varied independently as indicated schematically in Figure 1a. Both incoming and outgoing states in the leads fi D D Qiao et al, Nano Le. 11, 3453 (2011) therefore have de nite pseudospin labels. If pseudospin memory were perfect, injected electrons would travel following the continuation of the lead’s zero line to one of the reservoirs. Our explicit calculations are based on the π-orbital tight- - + - + binding model † † H γ c cj Uic ci 1 D D ∑ i, j i ∑ i D D ¼ÀÆi, jæ þ i ð Þ

Figure 1. (a) Model device with four regions that can be gated to where U is a π-orbital site energy and γ is either a nearest- positive or negative interlayer potential values. (b) Schematic represen- i i,j neighbor in-plane hopping amplitude with value t = 2.6 eV or a K’ K K K’ tation of the lattice geometry used in our numerical simulations. The - + + - horizontal and vertical axes are chosen to be aligned along the zigzag and vertical interlayer hopping amplitude with value t^ = 0.34 eV. armchair honeycomb lattice directions of both the device and the The trigonal warping γ3 0.1t term can play a role in the limit of 18∼ graphene bilayer reservoirs used in our four-probe NEGF calculations. vanishingly small gaps but is unimportant for our present † S S (c) One-dimensional band structure of a zigzag ribbon in which the discussion, as we show in the Supporting Information. Here ci interlayer potentials of eq 1 change sign at the ribbon center: Ui = (V(x) and ci are π-orbital creation and annihilation operators for site i. = (0.1t sgn(x). Two 1D modes traveling in each direction are spatially In most of our numerical simulations, we have considered a localized at the ribbon center. Right going states are labeled with letters bilayer graphene flake containing a total of 120 92 2 = 22080 A, B whereas left going states are labeled as C, D. Additional 1D channels (vertical, horizontal, layer) atomic sites in the central scattering appear in the gap that are localized near opposite edges of the ribbon. region, corresponding to a few hundreds of nm2 of flake area. We These states are plotted in black and are doubly degenerate due to create kink states by setting Ui f (0.1t so that the sum of site inversion symmetry across the ribbon. (d) The band structure in the ff ff armchair case has two pairs of oppositely propagating channels located energies in di erent layers is everywhere zero and the di erence is at the ribbon center but does not support edge states. The atomically (0.2t in the ( regions. The atomic scale variation of the scale sharp potential variation we used leads to a small avoided crossing potential difference is not physically realistic, of course, since gap Δ 0.0014t at the neutrality point. The size of this gap shrinks the sharpness of its spatial profile cannot exceed the greater of rapidly∼ when the potential variation becomes smoother. the physical gate separation and the vertical distance between bilayer and gate. In the Supporting Information we show that our results are not altered in any essential way as long as the distance The band structures of straight zigzag and armchair ribbons over which the potential difference shifts between positive and with a zero-line along the ribbon center are illustrated in panels c negative values is smaller than 100 nm. The potential differ- and d of Figure 1. The anticipated pair of kink states localized at ences open up gaps in the spectrum∼ so that the only states at the the sample center appears for both ribbon orientations. For the Fermi level of a nearly neutral bilayer are ribbon edge states and zigzag case in panel c, kink states appear near ribbon wavevectors Video courtesy of Ivar Marn kink states localized along zero lines. We label the four semi- k =2π/3a,4π/3a where a is the lattice constant of graphene, as infinite bilayer graphene reservoirs in our NEGF calculations up suggested by a bulk graphene band projection.13 Zigzag edges [email protected] Marn et al, PRL 100, 036804 (2008), Qiao et al, Nano Le. 11, 3453 (2011), PRL 112, 206601 (2014). Nov 7, 2017 46 (U), down (D), left (L), and right (R). support edge state channels localized at the ribbon/vacuum Our main numerical results, summarized in Figure 2, were edges in addition to the kink states, whereas armchairs ribbons obtained for model flakes with zigzag edges in the horizontal do not support edge states and all kink states appear near 1D direction and armchair edges in the vertical direction. By varying momentum k = 0. The close proximity of opposite-velocity kink the gating potentials, we can arrange to have vertical or horizontal states in both real space and 1D momentum space might suggest zero lines in the system, to have a single zero line that rotates by that the continuum model picture should fail badly when the 90° between zigzag and armchair directions, or to have two such zero-line direction is close to an armchair direction. We will show zero lines that intersect at the middle of the sample. Configura- that this is not the case. tions like this, in which the zero lines of interest do not intersect Kink states have definite chirality if they preserve their valley with the edge of the system, can be used to isolate kink state labels; states in one valley propagate along zero lines keeping conducting channels from edge state conducting channels. low-potential regions on the left, while states in the other valley We anticipate that disorder at the edges will tend to localize keep low-potential regions on the right. Zero lines intersect when edge state transport. For the devices that we have in mind, the potential difference landscape has a zero saddle point. For a increasing disorder at the edges may in fact be desirable in order general continuous potential-difference profile, a system can have to mitigate their possible role in transport. many zero lines, some of which are closed. When valley memory We study how controlling the potential-difference profile can is perfectly retained, only open paths connected to reservoir zero control transport properties by calculating the conductances lines are relevant for transport. Zero-line considerations are between probes for each gating geometry. The conductance therefore relevant to the analysis of transport in neutral systems G from the qth probe to the pth probe can be evaluated from with smooth random potential differences. In this paper, how- pq the Landauer B€uttiker formula ever, we concentrate on systems with simple gate-defined À potential-difference profiles designed to control current paths G e2=h Tr Γ GrΓ Ga 2 in bilayer graphene systems. pq ¼ð Þ ½ p q ŠðÞ

3454 dx.doi.org/10.1021/nl201941f |Nano Lett. 2011, 11, 3453–3459 !" " !" Magnec field: wave funcon control F = qv × B

- + hole electron

- + + - - + K K’ ε = 0 + - + - - + - +

+ - + - A tunable electron beam splier K K’ based on the chirality of the kink states K’ K ε < 0 ε > 0

Ren et al, arXiv: 1702.00089v1 (Qiao group) Zarenia, et al PRB, 84, 125451 (2011) (F. Peeters group) 47 [email protected] Nov 7, 2017 “On” State of the valley valve

9k kink regime North kink B=6T 8k D

7k - + ) 6k Ω ( Ω ) 5k 1-3 R R16-3 ( 4k - + 3k S 2k

-40 0 40 VSi (V)

48 [email protected] Nov 7, 2017 “On” State of the valley valve

9k kink regime North kink B=6T 8k South kink D

7k - + ) 6k Ω ( Ω )

5k 1-3 R R16-3 ( 4k - + 3k S 2k

-40 0 40 VSi (V)

49 [email protected] Nov 7, 2017 3

“On” State of the valley valve Δ Δ 2 4 Δ Δ

9k kink regime 1 North kink B=6T 8k South kink D North-south 7k - + ) 6k Ω ( Ω )

5k 1-3 R R16-3 ( 4k - + 3k S 2k

-40 0 40 VSi (V)

Ø Perfect transmission through the intersecon 50 [email protected] Nov 7, 2017 “On” State of the valley valve D Transmission coefficient τi of the juncon - + 1.0 τi + 0.8 - S

0.6 RC i τ

0.4 N N R para τ kink τi 0.2 RS S para τ kink 0.0 0 2 4 6 8 B (T) 51 [email protected] Nov 7, 2017 A reconfigurable waveguide

“right turn” 9 B= 5T - + 8 - -

) 7 Ω

(k 6

guide 5 “le turn” R 4 + +

3 + - -60 -40 -20 0 20 40 60 V (V) Si Kink states can go around a bend ! 52 [email protected] Nov 7, 2017 Valley valve and electron beam splier

100 B=8 T East West N 80

60 - +

40 E W

Current partition (%) 20 percentage current (%) North + - 0 -20 -10 0 10 V (V) Si S EF hole electron

• The valley valve works in the enre EF range

• EF controls spling rao between West and East terminals 53 [email protected] Nov 7, 2017 On/off rao of the valley valve

60 0T T=1.5 K 2T 50 4T 3 6T 9T 40 11T - +

30 (% )

2 4 3 I

20 8% + -

10 1 0 -40 -20 0 20 40 V (V) Si • ON/OFF rao 800% at B=0, more than 100 at high field.

54 [email protected] Nov 7, 2017 A tunable electron beam splier

100 100 8T 6T (% ) 3 2 4T

80 (% ) 3T 2 80 1T - + 2 60 4 60 + - 40 S 40

20 Normalized percentage current I

Tunable percentage current I 20 0 -20 0 20 0 2 4 6 8 V (V) B (T) Si • Current paron rao tunable from 0 to close to 100%. 55 [email protected] Ren et al, arXiv: 1702.00089v1 Nov 7, 2017 An S-Matrix model

16 B = 8 T

Chaoxing Liu 12 (Penn State) “two-terminal”

) R43 Ω 8

I

R (k 3 - + 4 2 4 S-matrix model “non-local” + - V 1 V + R12,43 0 Landauer Buker R12,43 -40 -20 0 20 40 V (V) Si Ø Excellent agreement between model and data 56 [email protected] Nov 7, 2017 Summary

Experimental realizaon of Valleytronics quantum valley Hall kink states

J. Li et al, Nature Nano. 11, 1060 (2016). J. Li, et al, arXiv:1708.02311v1 (2017).

• Valley-momentum locked topological channel • Valley valve • Gate-defined and scalable • Tunable beam splier

Outlook: • Larger on/off rao of the valley valve • Beam splier in the absence of a magnec field • Operaon at higher temperature

57 [email protected] Nov 7, 2017 Acknowledgement The Zhu lab Univ. Sci. Tech. China Jing Li

Zhenhua Qiao Ke Wang Yafei Ren PSU Hailong Fu (Eberly fellow)

• Quantum valley Hall kink states Chaoxing Liu Ruixing Zhang Peter Zhang • Quantum Hall and quantum spin Hall effect • Edge state tunneling and interferometry NIMS • Atomically thin 2D semiconductors

T. Taniguchi K. Watanabe [email protected] Nov 7, 2017 Thank you!

[email protected] Nov 7, 2017