Supplementary information for “Magnetic-gateable valley exciton emission”

Helena Braganca,1, 2 Hao Zeng,3, ∗ A. C. Dias,1 J. H. Correa,1 and Fanyao Qu1, †

1Instituto de Fısica, Universidade de Brasılia, Brasılia-DF 70919-970, Brazil2Departamento de Fısica, Universidade Federal de Ouro Preto, 35400-000, Ouro Preto, Minas Gerais, Brazil3Department of Physics, University at Buffalo, the State University of New York, Buffalo, NY 14260, USA

(Dated: May 11, 2020)

I. ELEVEN BANDS TB MODEL

As described in the main text, the effect of the tilted exchange field on the band structure of the monolayer MoS2 isdescribed through an eleven bands tight-binding (TB) model which accurately describe the electronic structure of theMoS2 monolayer1. Supplementary Figure 1 (a) shows a top view of the monolayer. The TB parameters are obtainedby fitting from density functional theory calculation, as shown in Supplementary Figure 1 (b). Panel (c) displays thefirst Brillouin zone and high-symmetry points Γ, K and M in reciprocal space.

Supplementary Figure 1: (a)A schematic of the top and side view of monolayer MoS2. Blue and yellow spheresdenote the metal (Mo) and chalcogen (S) atoms, respectively. The highlighted rhombus indicates the unit cell. (b)Band structure of monolayer MoS2 calculated by first principle DFT (red dots), using GGA-PBE exchangecorrelation functional and by the TB 11 bands model (blue solid line). The lowest conduction and highest valancebands calculated by DFT are well fitted by TB model. (c) First Brillouin zone and high-symmetry points Γ, K and

M in reciprocal space of the triangular lattice. ~b1 and ~b2 are the primitive lattice vectors.

II. EFFECTIVE FOUR BANDS ~k · ~p MODEL

To gain insights to understand our numerical results of valley exciton physics, we have also developed an effective~K.~p model 2, including the spin-orbit coupling and the exchange interaction terms. The total Hamiltonian Hτ =Hτo + Hτ

soc + Hτex in the basis {|ψc,↑〉, |ψc,↓〉, |ψv,↑〉, |ψv,↓〉}, where |ψn,s〉 represents the single-particle sz-spin state at

the conduction (n = c) or valence (n = v) band, can be describes as followsThe first term is the free Hamiltonian, i.e.,

Hτ0 =

[vF (τ σxkx + σyky) +

Eg

2σz

]⊗ 11s, (1)

where vF is the Fermi velocity, τ = ±1 indicate the K and K′ valleys respectively, ~k is the momentum measured fromK or K′, σi=x,y,z are the Pauli matrices, Eg is the energy gap, and 11s represents the 2 × 2 identity matrix in spinspace.

The second term describes the spin-orbit interaction, given by

Hτsoc = [λc(1 + σz)/2) + λv(1− σz)/2))⊗ τ sz, (2)

2

Ene

rgy(

eV)

−1.2

−1

−0.8

0.6

0.8

1

1.2

sz = +1sz = -1

K(a)

K'(b)

Ene

rgy(

eV)

−1.2

−1

−0.8

0.6

0.8

1

1.2

K (nm-1)−0.2 −0.1 0 0.1 0.2

K(c)

K (nm-1)−0.2 −0.1 0 0.1 0.2

K'(d)

Supplementary Figure 2: Band structure of MoS2 grown on ferromagnetic substrate around K and K′ valleys forexchange values J=0 (upper pannels) and J=20 meV (lower pannels). Blue and red curves correspond to spin-upand down respectively. λc = 10.5 meV, λv = 79 meV.

where λc and λv are the spin-orbit interaction intensity for conduction and valence bands, respectively. Note thatλc = ∆c/2 in the main text, and s refers to the spin Pauli matrix.

Finally, the third term represents the exchange field interaction induced by magnetic proximity effect in MoS2

deposited on a ferromagnetic substrate

Hex = J [(1 + σz)/2) + (1− σz)/2)]⊗ (~s · m), (3)

where m is a unit vector indicating the magnetization direction, ~s = (sx, sy, sz) are Pauli matrices of spin, and J isthe exchange parameter. The product ~s · m can be written in matrix form as

~s · m =

(cos θ sin θe−iφ

sin θeiφ − cos θ

), (4)

therefore, the matrix form of the total Hamiltonian is given by

H =

Eg/2 + λcτ − J cos θ −J sin θe−iφ vfk− 0

−J sin θeiφ Eg/2− λcτ + J cos θ 0 vfk−vfk+ 0 −Eg/2 + λvτ − J cos θ −J sin θe−iφ

0 vfk+ −J sin θeiφ −Eg/2− λvτ + J cos θ

. (5)

Where, θ is the angle between the magnetization direction and the normal of the MoS2 monolayer (z-axis), and φ isthe azimuthal angle of the magnetization in xy plane. Furthermore, k+ = kx + iky and k− = kx − iky.

A. Switching of conduction band ground state characteristics

Let us first consider the case in which m = z (out-of-plane magnetization, θ = 0o). In this case, the bands withdifferent spin components are decoupled, then we can separate the Hamiltonian for each spin and valley,

3

Hτ,s =

(Eg/2 + λcτs− J vfk−

vfk+ −Eg/2 + λvτs− J

). (6)

We use s to represent z component. Diagonalizing Eq.(6), the valley-spin dependent dispersion relation is given by

Eτ,s =λvτs+ λcτs− 2sJ

2±

√[Eg + λcτs− λvτs

2

]2+ (vfk)2, (7)

where +(−) correspond to the conduction (valence) band. Then, the dispersion relation of conduction bands at Kpoint (k = 0, τ = 1) is given by

EcK,↑ = λc − J +Eg2

(8)

for spin-up and

EcK,↓ = −λc + J +Eg2

(9)

for spin-down. The Zeeman splitting for the conduction bands in K valley is equal to

∆EcK = EcK,↑ − EcK,↓ = 2(λc − J). (10)

It is noticed that spin-up and spin-down CB energy levels switch at the critical value J⊥ = λc, in agreement withthe numerical results presented in the main text. Supplementary Figure 2 shows the band structure around K-point(left pannels) and K′-point (right pannels). Notice that in the case of J = 0, time-reversal symmetry is preserved.However, in the case of J = 20 meV, the order of spin-up and spin-down states in conduction band is reversed.

An analogous analysis can be performed for the K′ valley (τ = −1), in which the conduction band energies aregiven by

EcK′,↑ = −λc − J +Eg2

(11)

for spin-up, and

EcK′,↓ = λc + J +Eg2

(12)

for spin-down. And the Zeeman splitting is given by

∆EcK′ = EcK′,↑ − EcK′,↓ = −2(λc + J). (13)

Notice that an increase of J leads to large Zeeman splitting between spin-up and spin-down states in K′ valley , asshown in the left panels of Supplementary Figure 2.

B. Spin mixing induced by a tilted exchange field

We now turn to the case in which the magnetization m points in an arbitrary direction. At K (τ = 1) or K′

(τ = −1) points (~k = 0), the matrix in Eq. 5 can be separated into two blocks corresponding to the valence andconduction bands. The Hamiltonian of the conduction band block, is given by

Hcb =

(Eg/2 + λcτ − Jc cos θ −Jc sin θe−iφ

−Jc sin θeiφ Eg/2− λcτ + Jc cos θ

). (14)

Then the energy spectrum can be easily derived as

Ecτ =Eg2±√

(λcτ − J⊥)2 + J2‖ , (15)

4

with J⊥ = J cos θ and J‖ = J sin θ. The corresponding eigenstate for the higher conduction band is

|Ψ〉 =1√

2E′1(Ec1,τ − Eg/2)

(−J‖e−iφE′1

), (16)

where E′1 = Ec1,τ − Eg/2 − λcτ + J⊥. With wave-function in hand, we can calculate the expectation value of sz asfollows

〈sz〉 = 〈Ψ|sz|Ψ〉 =τλc − J⊥√

(τλc − J⊥)2 + J2‖

. (17)

Then after some algebraic calculations we obtain

J‖〈sz〉 = (τλc − J⊥)

[1 +

(τλc − J⊥

J‖

)2]−1/2

≈ (τλc − J⊥)− 1

2

(λc − J⊥)3

J2‖. (18)

In the first order approximation in J‖ and J⊥, we finally get

J⊥ = −〈sz〉J‖ + τλc. (19)

It indicates that the lines with fixed 〈sz〉 are linear in the (J⊥, J‖) plane, with the slope determined by -〈sz〉, crossingthe J‖ axis at J⊥ = +(−)λc at K (K′) valley. An analogous analyses for the lower conduction bands leads to otherwiseidentical results except with a slope +〈sz〉. These results obtained with the effective low energy Hamiltonian agreeswell with the ones presented in Fig. 3 of the main text, obtained by tight-binding. We also note that the maximumspin mixing, 〈sz〉 = 0, is obtained for J⊥ = τλc, for any J‖ 6= 0.

C. Exciton eigenproblem

The calculation of the exciton binding energy via Bethe-Salpeter equation is described in the Methods Section ofthe main text. The calculated bright and dark exciton energies in K and K′ valleys are shown in SupplementaryFigure 3 for J = 25 meV, pointing in different directions (θ represents the angle between the exchange field and thenormal of the monolayer). The values without exchange field are also shown, for comparison.

J=0

X'b

X'b

X'b

X'b

X'd

X'd

X'd

X'd

Ener

gy (e

V)

1.35

1.4

1.45

1.5

J=0J=25θ=0o

J=25θ=45o

J=25θ=90o

J=25θ=90o

J=25θ=45o

J=25θ=0o

Xb

Xb

Xb

Xb

Xd

Xd

XdXd

K K'

Supplementary Figure 3: Energy of the bright Xb and dark Xd excitons in K (left) and K′ (right) valleys for J=0and J=25 meV, and different values of θ, the angle between the field and the normal of the monolayer.

5

III. VALLEY SELECTIVE SWITCHING OF EXCITON GROUND STATE

J (meV)0 5 10 15 20

(b) K'

Ener

gy se

para

tion Δ

E bd (

meV

)

0

20

40

J (meV)0 5 10 15 20

(a) K

θ

θ

Supplementary Figure 4: Energy separation between dark and bright exciton states at (a) K and (b) K′ points as afunction of J. The arrows indicate the direction of of increasing angle θ from 0o to 90o with step 15o.

As described in the main text, the conduction band crossing have important consequences to the exciton valleydynamics. In particular, it affects the alignment between bright and dark exciton states. Supplementary Figure 4shows the intra-valley exciton energy splitting calculated via BSE, i.e., the energy difference between dark and brightstates, as a function of the exchange field amplitude, J, for different angles θ of magnetization. At J=0, the intra-valley splitting of 10 meV is determined by the spin orbit splitting and the energy difference between bright and darkbinding energies. For θ = 0o, the intra-valley splitting decreases and increases linearly as a function of J in K andK′ valley, respectively, and the ground state switches from bright to dark excitons in the K valley. At θ 6= 0 thein-plane exchange field components leads to a non linear dependence of the energy splitting on J. The result is thatthe bright-dark crossing is replaced by an anticrossing for large θ in K valley; in K′ valley, the energy splitting stillincreases monotonically.

IV. EXCITON DYNAMICS AND VALLEY POLARIZATION OF BRIGHTENED DARK EXCITONS

In the main text, we focused on bright and dark exciton PL intensity in the K valley for clarity. For completeness,in Supplementary Figure 5 we present the PL intensity and the exciton lifetime of bright and dark excitons in bothvalleys.

6

(c) J⟂=15 meV

J//(meV)0 5 10 15 20 25 30

3456

(g) (e)

Exci

ton

lifet

ime

(ps)

101

102

103

0 5 10 15 20 25 30

3456

(d)

0 5 10 15 20 25 30

3456

(f)

(b) J⟂=7 meV

PL in

tens

ity0

0.2

0.4

0.6

0.8

1

3456

XK',b

XK',d

XK',b

XK',d

(a)J⟂=4 meV

Supplementary Figure 5: Room temperature optical properties of the bright XK,B (XK′,B) and dark XK,D (XK′,D)exciton states at K (K′) valley of a MoS2 monolayer under tilted exchange fields. (a)-(c) PL intensity as a functionof the in-plane exchange field, J‖, for J⊥ = 4 meV (left column), J⊥ = 7 meV (middle column), and J⊥ = 15 meV(right column). Blue, red, yellow and green lines represent XK,B , XK′,D, XK,B , and XK′,D, respectively. (d)-(f)corresponding exciton lifetime as a function of J‖.

∗ haozeng@buffalo.edu† fanyao@unb.br1 A. C. Dias, F. Qu, D. L. Azevedo, and J. Fu, Phys. Rev. B 98, 075202 (2018).2 A. Kormanyos, G. Burkard, M. Gmitra, J. Fabian, V. Zolyomi, N. D. Drummond, and V. Falko, 2D Materials 2, 022001

(2015).