Momentum dependence of spin–orbit interaction effects in single-layer and multi-layer transition metal dichalcogenides

TLDR: In this article, the effect of spin-orbit coupling (SOC) on the band structure of single-layer and bulk TMDs, including explicitly the role of the chalcogen orbitals and their hybridization with the transition metal atoms, was studied.

Abstract: One of the main characteristics of the new family of two-dimensional crystals of semiconducting transition metal dichalcogenides (TMDs) is the strong spin–orbit interaction, which makes them very promising for future applications in spintronics and valleytronics devices. Here we present a detailed study of the effect of spin–orbit coupling (SOC) on the band structure of single-layer and bulk TMDs, including explicitly the role of the chalcogen orbitals and their hybridization with the transition metal atoms. To this aim, we combine density functional theory (DFT) calculations with a Slater–Koster tight-binding (TB) model. Whereas most of the previous TB models have been restricted to the K and K’ points of the Brillouin zone (BZ), here we consider the effect of SOC in the whole BZ, and the results are compared to the band structure obtained by DFT methods. The TB model is used to analyze the effect of SOC in the band structure, considering separately the contributions from the transition metal and the chalcogen atoms. Finally, we present a scenario where, in the case of strong SOC, the spin/orbital/valley entanglement at the minimum of the conduction band at Q can be probed and be of experimental interest in the most common cases of electron-doping reported for this family of compounds.

Figures

TABLE II: Lattice parameters used for DFT calculation for singlelayer, bilayer and bulk MoS2 and WS2 systems, as taken from Refs. 42 and 43, respectively. a represents the M -M atomic distance, u the internal vertical distance between the M plane and the X plane, and c′ the distance between the M layers. In bulk systems the z-axis lattice parameter is given by c = 2c′. All values are in Å units.

FIG. 6: Band structure of bilayer MoS2 and WS2 obtained from DFT calculations. The combined eUect of inter-layer hopping and spinorbit interaction drives the minimum of the conduction band to the Q point, and the maximum of the valence band to the Γ point (see text) .

FIG. 2: Tight-binding band structure of single-layer MoS2 (a) and WS2 (b) including SO interaction. Red dashed lines corresponds to the TB bands including the whole SO coupling terms. Black solid lines correspond to the TB band structure including only the spinconserving terms of the SO coupling.

FIG. 5: DFT band structure of the four compounds MoS2, WS2, MoSe2 and WSe2. Black circles show the results when the SOC on all the atoms are included. Red (blue) lines correspond to the removal of the SOC on the chalcogen (transition metal) atoms.

FIG. 3: Fermi surfaces of MoS2, obtained from the TB band structure. Panels (a) and (c) correspond to single-layer and panels (b) and (d) to the bulk. Top panels represent hole-doped systems, with the Fermi energy in the valence band (atEF = −1.134 eV), whereas bottom panels represent electron-doped systems, with the Fermi energy in the conduction band (EF = 0.95 eV). Energies are measured with respect to the zero of the TB Hamiltonian. The hexagonal 2D BZ is shown in (c) by the black solid lines. In panels (a) and (c), solid blue and dashed blue lines correspond to Fermi surfaces with main Sz =↑ and Sz =↓ polarization, respectively. Solid black lines indicate pockets which are degenerate in spin, like the central pocket in (a) (around the Γ point) and all the pockets in the Fermi surfaces of the bulk compound [panels (b) and (d)].

Full text

Thisistheacceptedversionofthearticle:
RoldánR.,López-SanchoM.P.,GuineaF.,CappellutiE.,
Silva-GuillénJ.A.,OrdejónP..Momentumdependenceof
spin-orbitinteractioneffectsinsingle-layerandmulti-layer
transitionmetaldichalcogenides.2DMaterials,(2014).1.
034003:-.10.1088/2053-1583/1/3/034003.
Availableat:
https://dx.doi.org/10.1088/2053-1583/1/3/034003

Momentum dependence of spin-orbit interaction eects in single-layer and multi-layer
transition metal dichalcogenides
R. Roldán, M.P. López-Sancho, F. Guinea
Instituto de Ciencia de Materiales de Madrid, CSIC, c/ Sor Juana Ines de la Cruz 3, 28049 Cantoblanco, Madrid, Spain
E. Cappelluti
Istituto de Sistemi Complessi, U.O.S. Sapienza, CNR, v. dei Taurini 19, 00185 Roma, Italy
J.A. Silva-Guillén, P. Ordejón
ICN2 - Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, 08193 Bellaterra, Spain and
CSIC - Consejo Superior de Investigaciones Ciaenticas, ICN2 Building, 08193 Bellaterra, Spain
(Dated: February 21, 2018)
One of the main characteristics of the new family of two-dimensional crystals of semiconducting transition
metal dichalcogenides (TMD) is the strong spin-orbit interaction, which makes them very promising for future
applications in spintronics and valleytronics devices. Here we present a detailed study of the eect of spin-
orbit coupling (SOC) on the band structure of single-layer and bulk TMDs, including explicitly the role of
the chalcogen orbitals and their hybridization with the transition metal atoms. To this aim, we combine
density functional theory (DFT) calculations with a Slater-Koster tight-binding model. Whereas most of the
previous tight-binding models have been restricted to the K and K’ p oints of the Brillouin zone (BZ), here we
consider the eect of SOC in the whole BZ, and the results are compared to the band structure obtained by
DFT methods. The tight-binding mo del is used to analyze the eect of SOC in the band structure, considering
separately the contributions from the transition metal and the chalcogen atoms. Finally, we present a scenario
where, in the case of strong SOC, the spin/orbital/valley entanglement at the minimum of the conduction band
at Q can be probed and be of experimental interest in the most common cases of electron-doping reported for
this family of compounds.
I. INTRODUCTION
Transition metal dichalcogenides have emerged as a new
family of layered materials with a number of remarkable
electrical and optical properties.
1
Among them, single lay-
ers of the semiconducting compounds of the group-VIB MX
2
(where M = Mo, W and X = S, Se) are of special interest
because they have a direct band gap in the visible range of
the spectrum,
2
which is located in the K and K’ points of the
hexagonal BZ.
3
The absence of inversion symmetry in single
layer samples lifts the spin degeneracy of the energy bands
in the presence of SOC.
4
Interestingly, the spin splitting in
inequivalent valleys must be opposite, as imposed by time
reversal symmetry. This leads to the so called spin-valley
coupling,
5
which has been studied theoretically
6–10
and ob-
served experimentally.
11–16
Although the SOC splitting of the
bands is particularly large in the valence band (∼ 150 meV
for MoS
2
and ∼ 400 meV for WS
2
), a nite SOC splitting of
the conduction band is also allowed by symmetry,
17
as con-
rmed by recent density functional theory calculations.
18–24
In addition, interlayer coupling plays here also a fundamen-
tal role. Indeed, the band structure dramatically changes
from single-layer to multi-layer samples, involving a tran-
sition from a direct gap for single-layer samples to an in-
direct gap for multi-layer samples,
3
as it has been observed
experimentally.
2,25–27
Both numerical r st-principles techniques and analytical
approaches have been employed to investigate the role of
the SOC in these materials. Within this context, the SOC
has been mainly included in tight-binding (TB) models valid
only in the low-energ y range, where the presence of the p-
orbitals of the chalcogen atoms has been integrated out in
an eective model (Refs. 5,28–32). Alternatively, DFT calcu-
lations can provide a more compelling description, but their
complexity hampers the extraction of a simple model of the
SOC. From a more general point of view, nally, most of the
recent works on the eects of SO C in TMDs have been fo-
cused on single-layer samples, whereas fewer investigations
have been devoted to the eect of SOC on the band structure
of multi-layer and bulk samples. In particular, a complete TB
model that can account for the eect of SOC in the whole BZ,
including explicitly the p-orbitals of the chalcogen atoms, is
lacking. Such a TB model is especially useful to study cases
where DFT methods result too challenging computationally,
as the eect of disorder, inhomogeneous strain, strong many-
body interactions, etc.
In this paper we use a combination of TB and DFT calcu-
lations to provide a complete TB mo del, in the whole BZ, of
the eects of SOC on the band structure of single-layer and
multi-layer TMD taking explicitly into account the p-orbitals
of the chalcogen atoms, and the atomic spin-orbit interaction
on both the metal and chalcogen orbitals. The bands ob-
tained from the TB model are compared to the correspond-
ing DFT band structure for single layer and bulk MoS
2
and
WS
2
. By considering the main orbital contribution at each
relevant point of the BZ, we analyze the origin and main
features of the SOC eects at the dierent band edges. Such
model provides a useful base not only for the analytical in-
vestigation of the role of the SOC in the presence of local
strain tuning the M -X distance, but also for the investiga-
tion of the microscopical relevant spin-orbit processes. In
particular, we show that the terms associated to second or-

2
der spin-ip processes of the SOC can be safely neglected
for most of the cases of experimental interest. The TB model
developed here is especially useful to analyze the eect of
SOC at the so-called Q point of the BZ, which corresponds
to the absolute minimum of the conduction band of multi-
layer samples. We nally discuss also the peculiarities of the
SOC in bilayer MX
2
, for which the spin-valley-layer coupling
could be exploit for future valleytronics applications.
The paper is organized as follows. In Section II we present
the model for the single layer and bulk cases. The compar-
ison between TB and DFT band structures considering the
SOC eects, is illustrated in Section III for MoS
2
and WS
2
.
Results are presented and discussed in Section IV. Finally
the main ndings are summarized and some conclusions are
given in Section V.
II. SPIN-ORBIT INTERACTION AND THE
TIGHT-BINDING HAMILTONIAN
In this section we present the analytical structure of the
TB Hamiltonians for single-layer and bulk TMD MX
2
com-
pounds including the SO interaction. Sp ecic parameters for
realistic materials will be provided in the next section, as well
as a discussion of the physical consequences of the SOC.
A. Single-layer case
The TMD MX
2
are composed, in their bulk conguration,
of two-dimensional X −M −X layers stacked on top of each
other, coupled by weak van der Waals forces. The M atoms
are ordered in a triangular lattice, each of them bonded to
six X atoms locate d in the top and bottom layers, forming
a sandwiche d material. Our starting point will be a 11-band
TB spinless model which, for the single-layer, considers the
ve d orbitals of the metal atom M and the three p orbitals
for each of the two chalcogen atoms X in the top and bottom
layer.
3
We can introduce a Hilbert base dened by the 11-fold
vector:
φ
†
i
= (p
†
i,x,t
, p
†
i,y,t
, p
†
i,z,t
, d
†
i,3z
2
−r
2
, d
†
i,x
2
−y
2
,
d
†
i,xy
, d
†
i,xz
, d
†
i,yz
, p
†
i,x,b
, p
†
i,y,b
, p
†
i,z,b
), (1)
where d
†
i,α
creates an electron in the orbital α of the M atom
in the i-unit cell, p
†
i,α,t
creates an electron in the orbital α of
the top (t) layer atom X in the i-unit cell, and p
†
i,α,b
creates
an electron in the orbital α of the bottom (b) layer atom X
in the i-unit cell. After an appropriate unitary transforma-
tion, the spinless (sl) representation of the single-layer (1L)
Hamiltonian can be expressed in the block form
ˆ
H
sl
1L
(k) =

ˆ
H
E
ˆ
0
6×5
ˆ
0
5×6
ˆ
H
O

, (2)
where
ˆ
H
E
and
ˆ
H
O
are a 6 × 6 and 5 × 5 blocks with even
(E) and odd (O) parity respe ctively upon the mirror inversion
z → −z, and
ˆ
0
m×n
denotes m × n zero matrices.
3
In p ar-
ticular,
ˆ
H
E
is built from hybridizations of the d
xy
, d
x
2
−y
2
,
d
3z
2
−r
2
orbitals of the metal M with the symmetric (anti-
symmetric) combinations of the p
x
, p
y
(p
z
) orbitals of the
top and bottom chalcogen atoms X. On the other hand, the
odd block,
ˆ
H
O
, is made by hybridizations of the d
xz
and d
yz
orbitals of M with the antisymmetric (symmetric) combina-
tions of the p
x
, p
y
(p
z
) orbitals of the X atom in the top
and bottom layers. Explicit expressions for all the matrix
elements in terms of the Slater-Koster parameters were ob-
tained in Ref. 3, and we notice that the 6 × 6 even block
ˆ
H
E
contains the relevant orbital contribution for the states of the
upper valence band and the lower conduction band.
In the context of the present TB model, we include the
SOC term in the Hamiltonian by means of a pure atomic
spin-orbit interaction acting on both the metal and chalco-
gen atoms. Explicitly we consider here the SOC given by:
ˆ
H
SO
=
X
a
λ
a
¯h
ˆ
L
a
·
ˆ
S
a
, (3)
where λ
a
, the intra-atomic SOC constant, depends on the
specic atom (a = M, X).
ˆ
L
a
is the atomic orbital an-
gular momentum operator and
ˆ
S
a
is the electronic spin
operator.
33–35
It is convenient to use the representation
ˆ
H
SO
=
X
a
λ
a
¯h
ˆ
L
+
a
ˆ
S
−
a
+
ˆ
L
−
a
ˆ
S
+
a
2
+
ˆ
L
z
a
ˆ
S
z
a
!
, (4)
where (omitting now for simplicity the atomic index a):
ˆ
S
+
=

0 1
0 0

,
ˆ
S
−
=

0 0
1 0

,
ˆ
S
z
=
1
2

1 0
0 −1

.
(5)
In a similar way, the orbital angular momentum operator
ˆ
L
acts on the states |l, mi as
ˆ
L
±
|l, mi = ¯h
p
l(l + 1) − m(m ± 1) |l, m ± 1i,
ˆ
L
z
|l, mi = ¯hm |l, mi, (6)
where l refers to the orbital momentum quantum number
and m to its z component.
We choose the orbital basis set in the following manner:
|p
z
i = |1, 0i
|p
x
i = −
1
√
2
[|1, 1i−|1, −1i]
|p
y
i =
i
√
2
[|1, 1i+ |1, −1i]
|d
3z
2
−r
2
i = |2, 0i
|d
xz
i = −
1
√
2
[|2, 1i−|2, −1i]
|d
yz
i =
i
√
2
[|2, 1i+ |2, −1i]
|d
x
2
−y
2
i =
1
√
2
[|2, 2i+ |2, −2i]
|d
xy
i = −
i
√
2
[|2, 2i−|2, −2i] (7)

3
We further simplify the problem by introducing the afore-
mentioned symmetric (S) and antisymmetric (A) combina-
tion of the p orbitals of the top (t) and bottom (b) X layers:
|p
α,S
i =
1
√
2
[|p
α,t
i + |p
α,b
i],
|p
α,A
i =
1
√
2
[|p
α,t
i − |p
α,b
i]. (8)
The total Hamiltonian, including the SO interaction for
the single-layer, can be now written as
ˆ
H
1L
(k) =
ˆ
H
sl
1L
(k) ⊗ 1
2
+
ˆ
H
SO
1L
, (9)
where the SOC term
ˆ
H
SO
1L
is
ˆ
H
SO
1L
=

ˆ
M
↑↑
ˆ
M
↑↓
ˆ
M
↓↑
ˆ
M
↓↓

, (10)
and where
ˆ
M
σσ
=

ˆ
M
σσ
EE
ˆ
0
6×5
ˆ
0
5×6
ˆ
M
σσ
OO

, (11)
and
ˆ
M
σ¯σ
=

ˆ
0
6×6
ˆ
M
σ¯σ
EO
ˆ
M
σ¯σ
OE
ˆ
0
5×5

. (12)
Here we have chosen the spin notation ¯σ =↓ (¯σ =↑) when
σ =↑ ( σ =↓).
The dierent blocks
ˆ
M
σσ
EE
,
ˆ
M
σσ
OO
,
ˆ
M
σ¯σ
EO
,
ˆ
M
σ¯σ
OE
, that con-
stitute the above 22 × 22 matrix, are explicitly reported in
the Appendix A. We notice here that, in the most general
case, the SO interaction couples the E and O sectors of the
22 × 22 TB matrix. Such mixing arises in particular from
the spin-ip/spin-orbital processes associate d with the trans-
verse quantum uctuation described by the rst two terms
of Eq. (4). The eective relevance of these terms can now
be directly investigated in a simple way, pointing out the ad-
vantages of a TB model with respect to rst-principles calcu-
lations. The explicit analysis of this issue is discussed in Sec-
tion III. We anticipate here that the eects of the o-diagonal
spin-ip terms result to be negligible for all the cases of inter-
est here. This is essentially due to the fact that such processes
involve virtual transitions towards high-order energy states.
17
At a very high degree of accuracy, we are thus justied in ne-
glecting the spin-ip terms and retaining in (4) only the spin-
conserving terms ∝ λ
a
ˆ
L
z
a
ˆ
S
z
a
. An immediate consequence of
that is that the even and odd sectors of the Hamiltonian re-
main uncoupled, allowing us to restrict our analysis, for the
low-energy states of the valence and conduction bands, only
to the E se ctor.
B. Bulk case
Once introduced the TB model for a single-layer in the
presence of SOC, it is quite straightforward to construct a
corresponding theory for the bulk and bilayer systems by in-
cluding the relevant inter-layer hopping terms in the Hamil-
tonian. Considering that the unit cell is now doubled, we can
thus write the Hamiltonian for bulk MX
2
in the presence of
SOC in the matrix form:
ˆ
H
Bulk
(k) =
ˆ
H
sl
Bulk
(k) ⊗ 1
2
+
ˆ
H
SO
Bulk
, (13)
which is a 44 ×44 matrix due to the doubling of the unit cell
with respect to the single-layer case discussed in Sec. II A.
Here
ˆ
H
sl
Bulk
(k) represents the spinless Hamiltonian for
the bulk system,
ˆ
H
sl
Bulk
(k) =
ˆ
H
sl
1
ˆ
H
⊥,Bulk
ˆ
H
†
⊥,Bulk
ˆ
H
sl
2
!
, (14)
where
ˆ
H
sl
i
describes the spinless Hamiltonian (i.e. in the ab-
sence of SOC) for the layer i = 1, 2, while
ˆ
H
⊥,Bulk
accounts
for the 11 × 11 Hamiltonian describing interlayer hopping
between X atoms beloging to dierent layers. We remind
that
ˆ
H
sl
2
is related to
ˆ
H
sl
1
through the following relation dic-
tated by the lattice structure:
3
H
sl
2,α,β
(k
x
, k
y
) = P
α
P
β
H
sl
1,α,β
(k
x
, −k
y
), (15)
where P
α
= +(−)1 if the orbital α has even (odd) symmetry
with respect to y → −y. Furthermore, the (spin-diagonal)
interlayer term
ˆ
H
⊥,Bulk
can be written as:
ˆ
H
⊥,Bulk
(k) =

ˆ
I
E
cos ζ
ˆ
I
EO
sin ζ
−
ˆ
I
T
EO
sin ζ
ˆ
I
O
cos ζ

, (16)
where ζ = k
z
c/2 (c being the vertical size of the unit cell
in the bulk system), and where the matrices
ˆ
I
E
,
ˆ
I
O
and
ˆ
I
EO
describe the inter-layer hopping between the p orbitals of
the adjacent chalcogen atoms. One can notice that interlayer
hopping leads, for an arbitrary wave-vector k, to a mixture of
the E and O sectors of the Hamiltonian, which is accounted
for by the term
ˆ
I
EO
in (16).
3
The analy sis is however simpli-
ed at specic high-symmetry points of the BZ, as we discuss
below.
50
Finally
ˆ
H
SO
Bulk
in Eq. (13) accounts for the spin-orbit cou-
pling in the bulk system, and it can be written as:
ˆ
H
SO
Bulk
=




ˆ
M
↑↑
0
ˆ
M
↑↓
0
0
ˆ
M
↑↑
0
ˆ
M
↑↓
ˆ
M
↓↑
0
ˆ
M
↓↓
0
0
ˆ
M
↓↑
0
ˆ
M
↓↓




, (17)
where both the spin-diagonal (
ˆ
M
σσ
) and spin-ip (
ˆ
M
σ¯σ
)
processes induced by the atomic spin-orbit interaction are
present.
Eqs. (13)-(17) provide the general basic framework for a
deeper analysis in more specic cases. In particular, as al-
ready mentioned above, the spin-ip terms triggered by SOC
can be substantially neglected for all the cases of interest
without loosing accuracy. The total Hamiltonian (13) can
thus be divided in two 22 × 22 blo cks
ˆ
H
σσ
Bulk
(k) related by

4
the symmetry
ˆ
H
↑↑
Bulk
(k) =
ˆ
H
↓↓
Bulk
(−k). Further simplica-
tions are available at specic symmetry points of the BZ.
More specically, we can notice that for k
z
= 0 the E and
O sectors remain uncoupled. Focusing, at low-energies for
the conduction and valence bands, only on the E sector, we
can write
ˆ
H
Bulk,E
(k, k
z
= 0) =
ˆ
H
sl
Bulk,E
(k) +
ˆ
H
SO
Bulk,E
, (18)
where
ˆ
H
sl
Bulk,E
(k) =




ˆ
H
E,1
ˆ
I
E
0 0
ˆ
I
†
E
ˆ
H
E,2
0 0
0 0
ˆ
H
E,1
ˆ
I
E
0 0
ˆ
I
†
E
ˆ
H
E,2




, (19)
and
ˆ
H
SO
Bulk,E
=




ˆ
M
↑↑
EE
0 0 0
0
ˆ
M
↑↑
EE
0 0
0 0
ˆ
M
↓↓
EE
0
0 0 0
ˆ
M
↓↓
EE




, (20)
where the explicit expression of each block Hamiltonian is
also reported in Appendix A.
C. Bilayer
The Hamiltonian for the bilayer can also be derived in a
very similar form as in the bulk case. In particular, we can
write:
ˆ
H
2L
(k) =
ˆ
H
sl
2L
(k) +
ˆ
H
SO
2L
. (21)
Since we are considering intrinsic SOC, thus it is not af-
fected by the interlayer coupling. Therefore we have
ˆ
H
SO
2L
=
ˆ
H
SO
Bulk
, where
ˆ
H
SO
Bulk
is dened in Eq. (20).
On the other hand, similar to the bulk case in Eq. (14), the
spinless tight-binding term
ˆ
H
sl
2L
(k) for the bilayer case can
be written as:
ˆ
H
sl
2L
(k) =
ˆ
H
sl
1
ˆ
H
⊥,2L
ˆ
H
†
⊥,2L
ˆ
H
sl
2
!
, (22)
where now
ˆ
H
⊥,2L
(k) =
1
2

ˆ
I
E
ˆ
I
EO
−
ˆ
I
T
EO
ˆ
I
O

. (23)
Note that Eq. (23) can be obtained as limiting case of Eq. (16)
by setting ζ = π/4, corresponding to the eective uncou-
pling of bilayer blocks.
III. TIGHT-BINDING PARAMETERS AND COMPARISON
WITH DFT CALCULATIONS
After having developed a suitable tight-binding model for
single and multi-layer MX
2
compounds, we comp are in this
MoS
2
WS
2
SOC λ
Mo
0.075 0.215
λ
S
0.052 0.057
Crystal Fields ∆
0
-1.512 -1.550
∆
1
0.419 0.851
∆
2
-3.025 -3.090
∆
p
-1.276 -1.176
∆
z
-8.236 -7.836
Intralayer Mo-S V
pdσ
-2.619 -2.619
V
pdπ
-1.396 -1.396
Intralayer Mo-Mo V
ddσ
-0.933 -0.983
V
ddπ
-0.478 -0.478
V
ddδ
-0.442 -0.442
Intralayer S-S V
ppσ
0.696 0.696
V
ppπ
0.278 0.278
Interlayer S-S U
ppσ
-0.774 -0.774
U
ppπ
0.123 0.123
TABLE I: Spin-orbit coupling λ
α
and tight-binding parameters for
single-layer MoS
2
and WS
2
(∆
α
, V
α
) as obtained by tting the low
energy conduction and valence bands. Also shown are the inter-
layer hopping parameters U
α
relevant for bulk compounds. The
Slater-Koster parameters for MoS
2
are taken from Ref. 3, and the
SOC ter ms from Ref. 10 and 24. All hopping terms V
α
, U
α
, crystal
elds ∆
α
, and spin-orbit coupling λ
a
are in units of eV.
section the band structure obtained by the TB model to the
corresponding band structure obtained from DFT methods.
An appropriate set of tight-binding parameters can be de-
rived by tting the low-energy dispersion of the conduction
and valence bands of these compounds in the whole BZ, in-
cluding the secondary minimum of the conduction band at
the Q point, along the Γ-K line. The crystal eld ∆
1
is ob-
tained by xing the minimum at K of the electronic bands
belonging to the odd block to the same energy of the DFT
calculations. The only left unknown parameters are thus
the atomic spin-orbit constants λ
M
and λ
X
for the transi-
tion metal and for the chalcogen atom, respectively. We take
the corresponding values from Ref. 10 and 24, and we list
the full set of TB parameters for MoS
2
and WS
2
in Table I.
Therefore, we can compare the resulting band structure for
the full tight-binding model in the presence of SOC, with cor-
responding rst-principles results including also spin-orbit
interaction.
DFT calculations were performed using the Siesta
code.
36,37
The spin-orbit interaction is treated as in Ref.
38. We use the exchange-correlation potential of Ceperley-
Alder
39
as parametrized by Perdew and Zunger.
40
We use
also a split-valence double-ζ basis set including polarization
functions.
41
The energy cuto and the Brillouin zone sam-
pling were chosen to converge the total energy. Lattice pa-
rameters for MoS
2
and WS
2
were chosen according to their
experimental values, as reported in Refs. 42 and 43, and they

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Momentum dependence of spin-orbit interaction euects in single-layer and multi-layer transition metal dichalcogenides" ?

In this paper, the authors studied the role of the SO associated to the chalcogen atom in the band structure of TMD and found that, for the cases of interest, spin-Wip processes are negligible in the SOC Hamiltonian. 

Q2. What is the eUect of SOC in bilayer TMD?

Whereas for single-layerMX2 inversion asymmetry leads to spin-valley coupling, the band edges of bilayer TMD are spin degenerate. 

Q3. What is the effect of the layer polarization on the spin-valley coupling?

although inversion symmetry forces each Fermi pocket to be spin degenerate, the layer polarization makes that each layer contributes with opposite spin in alternating valleys. 

Q4. Why is the eUect of SOC in bilayer MX2 so?

since inter-layer hopping conserves the spin, the spin physics can be exploited in bilayer MX2 due to spin-valley-layer coupling. 

Q5. What is the structure of the TMDMX2?

The TMDMX2 are composed, in their bulk conVguration, of two-dimensionalX−M−X layers stacked on top of each other, coupled by weak van der Waals forces. 

Q6. What is the main orbital character of the conduction bands at the Q point?

On the microscopic ground, the authors can notice that the main orbital character of the conduction bands at the Q point is due to a roughly equal distribution of the dx2−y2 and dxy orbitals of the transition metal M , and of the px and py orbitals of the chalcogen atomX . 

Q7. What is the effect of the inter-layer hopping on the spin/layer/valley?

the SOC for the TMD families with stronger spin-orbit interaction, likeWS2 andWSe2, can be larger than the inter-layer hopping, enhancing the spin/layer/valley en-tanglement. 

Q8. How is the crystal Veld 1 obtained?

The crystal Veld ∆1 is obtained by Vxing the minimum at K of the electronic bands belonging to the odd block to the same energy of the DFT calculations. 

Q9. What is the role of the SOC on the entanglement of the valence?

The role of the SOC on the spin-orbital-valley entanglement at the band edge at K of the single-layer and bilayer compounds has been previously discussed in the literature, using mainly low-energy eUective Hamiltonians focused on the role of the transition metalM d-orbitals and of their corresponding spin-orbit coupling. 

Q10. What is the limiting case of Eq. (16)?

(23)Note that Eq. (23) can be obtained as limiting case of Eq. (16) by setting ζ = π/4, corresponding to the eUective uncoupling of bilayer blocks. 

Q11. How can the authors turn on the SOC on a particular species?

In the DFT calculation, the authors can also turn on and oU the SOC on a particular species, by removing the SO component of the pseudopotential. 

Q12. What is the common doping scenario for electron-doped samples?

This spin-valley coupling scenario resembles that of single-layer and bilayer MX2 discussed in the literature, but for electron-doped samples, which is the kind of doping most commonly reported for those materi-10als. 

Q13. What is the promising state for tuning the spin/orbital coupling in these materials?

Red dashed lines in (b) correspond to λMo = 0.075 eV and λS = 0.minima of the conduction band at the Q point as the most promising states for tuning the spin/orbital/valley entanglement in these materials by means of strain engineering47 or (in multilayer systems) by means of electric Velds. 

Q14. What are the experimental bands for bulk samples?

The TB band structure for bulk samples, shown in Fig. 1(c) and (d), have been obtained by adding only two extra Slater-Koster parameters, Uppσ and Uppπ , which account for inter-layer hopping between p orbitals of the adjacent chalcogen atoms of diUerent layers.