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Hybrid k.p tight-binding model for subbands and infraredintersubband optics in few-layer films of transition-metaldichalcogenides: MoS2, MoSe2, WS2, and WSe2DOI:10.1103/PhysRevB.98.035411

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Citation for published version (APA):Ruiz-Tijerina, D., Danovich, M., Yelgel, C., Zolyomi, V., & Fal'ko, V. (2018). Hybrid k.p tight-binding model forsubbands and infrared intersubband optics in few-layer films of transition-metal dichalcogenides: MoS2, MoSe2,WS2, and WSe2. Physical Review B, 98(3), [035411]. https://doi.org/10.1103/PhysRevB.98.035411

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Hybrid k � p-tight binding model for subbands and infrared intersubband optics infew-layer �lms of transition metal dichalcogenides: MoS2, MoSe2, WS2, and WSe2

David A. Ruiz-Tijerina,1 Mark Danovich,1 Celal Yelgel,1 Viktor Z�olyomi,1 and Vladimir I. Fal’ko11National Graphene Institute, University of Manchester, Booth St E, Manchester M13 9PL, UK

(Dated: July 4, 2018)

We present a density functional theory parametrized hybrid k�p tight binding model for electronicproperties of atomically thin �lms of transition-metal dichalcogenides, 2H-MX2 (M=Mo, W; X=S,Se). We use this model to analyze intersubband transitions in p- and n-doped 2H�MX2 �lms andpredict the line shapes of the intersubband excitations, determined by the subband-dependent two-dimensional electron and hole masses, as well as excitation lifetimes due to emission and absorptionof optical phonons. We �nd that the intersubband spectra of atomically thin �lms of the 2H-MX2family with thicknesses of N = 2 to 7 layers densely cover the infrared spectral range of wavelengthsbetween 2 and 30 �m. The detailed analysis presented in this paper shows that for thin n-doped�lms, the electronic dispersion and spin-valley degeneracy of the lowest-energy subbands oscillatebetween odd and even number of layers, which may also o�er interesting opportunities for quantumHall e�ect studies in these systems.

I. INTRODUCTION

The 2H-MX2 transition metal dichalcogenide com-pounds (M=Mo, W; X=S, Se) are layered materials,where chalcogens and metal atoms form covalent bondswithin two-dimensional (2D) layers with hexagonal lat-tice structure, and neighboring layers couple weaklythrough electrical quadrupole and van der Waals interac-tions. This feature of chemical bonding makes atomicallythin �lms of MX2 su�ciently stable for extensive exper-imental studies aimed at their implementation in variousoptoelectronic devices [1{3]. In those recent studies, theclosest attention has been paid to the inter-band opti-cal properties of the monolayer transition-metal dichalco-genide (TMD) crystals [4{7], due to their direct band gap[8], valley-spin coupling [9{11], and long spin and valleymemory of photo-excited carriers [12], spiced up by theBerry curvature e�ects for electrons and excitons in thesetwo-dimensional semiconductors [13, 14]. This is becausein monolayer MoS2;MoSe2;WS2, and WSe2 the valenceand conduction band edges both appear at the Brillouinzone (BZ) corners K and K 0, where the electronic Blochstates carry intrinsic angular momentum.

Thicker crystals of 2H-MX2 quickly lose the directband gap property upon increasing the �lm thicknessto two or three layers [15{22]. Density functional the-ory (DFT) of few-layer transition metal dichalcogenidespredicts [18, 20] that for holes the band edge relocates tothe � point, whereas for electrons it appears at six pointssituated somewhere near the Q points at the middle ofeach �K segment (see inset in Fig. 1). This has beendemonstrated by studies of Shubnikov{de Haas oscilla-tions in n-doped MoS2 [23]. While the indirect characterof few-layer TMD band structures suppresses their inter-band photo response, the multiplicity of subbands njN(1 � n � N) at the conduction and valence band edges ofthe N -layer crystal, open a new avenue for optical studiesof atomically thin TMD �lms.

Here, we analyze theoretically intersubband transitions

FIG. 1. Energy spacings between the �rst two conduc-tion (�lled symbols, solid lines) and valence (empty symbols,dashed lines) subbands 1jN and 2jN as a function of num-ber of layers N of the four TMDs, corresponding to n andp doping, respectively, for 2 � N � 7. Both axes are in logscale, showing the approximate quadratic dependence of thespacings on the number of layers. The left vertical axis showsthe wavelength � in �m, corresponding to the energy spacingsshown along the right vertical axis in meV. The inset showsthe building block of 2H-MX2 bulk crystals, composed of twomonolayers with metal atoms in the middle and chalcogens inthe outer sublayers of each monolayer, and the Brillouin zonewith the � point and Q valleys highlighted, corresponding tothe conduction and valence band edges.

in few-layer MoS2;MoSe2;WS2 and WSe2, and show thatthe absorption/emission spectra of the primary transi-

2

tions in p- and n-doped crystals (Fig. 1) densely coverthe infrared (IR) spectrum down to the far-infrared range(FIR) of photon energies. The analysis of subbandproperties of few-layer 2H-MX2 presented in this paperis based on the hybrid k�p theory tight-binding model(HkpTB) approach, recently applied to the description ofmultilayer �lms of post-transitional-metal chalcogenides(such as InSe and GaSe) [24, 25]. This approach consistsof minimal k �p theory Hamiltonians for 2H-MX2 mono-layers [26, 27], supplemented by a k � p expansion of theinterlayer hopping near the relevant point (here, � or Q)in the BZ, with all parameters �tted to DFT calculatedfew-layer dispersions, and kz dispersions in bulk crystals.

First, in Section II we describe the lattice structure anddiscuss symmetries of few-layer 2D crystals of 2H-MX2,especially the di�erence between �lms with odd and evennumbers of layers and the corresponding degeneracies intheir band structures. The DFT-parametrized HkpTBmodels for few-layer TMDs are formulated in SectionsIII and IV for the valence band edge (holes) near the� point and for conduction band (electrons) near the Qpoints, respectively.

In the case of n-doped �lms, our models predict amulti-valley subband structure with two valley triadsconnected by time-reversal symmetry, each consisting ofthree valleys related by C3 rotations. We �nd that thelowest-energy subbands alternate between spin-split andspin-degenerate with number of layers N , consequence ofthe �h mirror and full inversion symmetries of �lms withodd and even N , respectively. These band structure fea-tures open a variety of new possibilities for studies ofquantum Hall physics in multi-layer TMD �lms.

We calculate subband energies, dispersions, and wavefunctions of electron/hole subbands in N -layer crystalsof all four 2H-MX2 compounds, and optical oscillatorstrengths for radiative intersubband transitions. In par-ticular, we take into account n- and p-type doping usinga self-consistent analysis of charge and potential distri-butions across the �lm. Then, we analyze the inelas-tic broadening due to optical phonon emission, and theresulting spectral line shapes of IR/FIR absorption byp- and n-doped 2H-MX2 �lms. We �nd that the inter-subband relaxation rates, determined by electron-phononinteractions, are much slower (one to two orders of mag-nitude) than the intra-subband relaxation in the samematerials [28, 29], and also these are an order of mag-nitude slower than intersubband relaxation of electronsand holes in III-V semiconductor quantum wells [30].Also, in Section III B we show that the di�erence be-tween the two-dimensional masses m1jN and m2jN ofelectrons and holes in consecutive subbands leads to anadditional temperature-dependent broadening of the in-tersubband transitions, � �

���1� m1jNm2jN

���maxfkBT; �F g,which appears to be the dominant intrinsic broadeninge�ect for the IR/FIR absorption by 2H-MX2 �lms atroom temperature.

FIG. 2. (a) Crystal structure of a bilayer 2H-stacked TMD(MX2), the building block of multilayer TMDs, viewed fromthe side and (b) top. M and X represent the metal andchalcogen atoms, respectively. (c) Brillouin zones of the twomonolayers in the 2H-stacked bilayer rotated by 180� relativeto each other, and schematics of the band dispersions at thevalence band � point and conduction band Q point, includingthe six symmetry-related Q valleys and spin-orbit splitting.(d){(f) DFT band structures of monolayer, bilayer, and tri-layer WS2, showing the transition from direct-gap (K � K)monolayer, to indirect-gap (� � Q) multilayer semiconduc-tor. At the � point, we label the di�erent valence bands inthe monolayer and the irreducible representations of the D3hpoint group [26]. The bilayer and trilayer valence subbandsare further labeled according to the subscript notation njN ,where n is the subband number and N the number of lay-ers. For odd number of layers (N = 1; 3), where spin-orbitsplitting is present, the conduction subbands near the Q pointare colored according to the spin projection quantum number,with red (blue) corresponding to s =" (s =#), giving a totalof 2N spin-polarized states. Kramer’s doublets are given byEs(k) = E�s(�k). For N = 2 all bands are spin degenerate,resulting in N doubly degenerate states, Es(k) = E�s(k).

II. MULTILAYERS OF HEXAGONALTRANSITION METAL DICHALCOGENIDES:

OVERVIEW

The lattice structure of monolayer TMDsMX2 (M = Mo;W; X = S;Se) contains two hexago-nal sublattices of metal and chalcogen atoms in its unitcell, as shown in Fig. 2(b). The chalcogens form twosublayers, one above and one below the metal sublayer,

3

forming a trigonal prismatic structure with the metalatom connected to three chalcogens above and below.The monolayer point-group symmetry is D3h, consistingof C3 rotations, �v in-plane mirror reections, and �hout-of-plane mirror reections. The most common bulkallotrope for these transition-metal dichalcogenides has2H stacking [31], built by adding subsequent layers ro-tated by 180� with respect to the center of the hexagon,resulting in a structure where the chalcogen atoms fromone layer are directly above or below metal atoms inthe other layer [see Figs. 2(a) and 2(b)]. The interlayerdistance c2 , with c the out-of-plane lattice constant, isshown in Fig. 2(a), and Fig. 2(b) shows the in-planelattice constant a. The resulting 3D layered crystal hasa bipartite structure with two monolayers in the unitcell, belonging to the space group P63=mmc.

Multilayer TMDs with an even number of layers be-long to the point group D3d, which contains spatial inver-sion (r ! �r) but lacks out-of-plane mirror symmetry.The combination of spatial inversion and time-reversalsymmetry prescribed at zero magnetic �eld results in aconstraint on the spin splitting of the electronic statesfor even number of layers, Es(k) = E�s(k), where s isthe spin projection quantum number, such that all statesthroughout the BZ must be spin degenerate. Similarly tothe monolayer case, multilayer �lms with an odd numberof layers belong to the point group D3h, which containsthe z ! �z mirror symmetry �h but lacks spatial inver-sion symmetry. Therefore, s is a good quantum numberfor which spin degeneracy (present in �lms with evennumber of layers) can be lifted by spin-orbit (SO) cou-pling. While the SO splitting is absent for bands basedon pz and dz2 orbitals at the � point, it is substantialnear the Q points, leading to the alternation of subbandproperties. That is, the subbands are spin degenerate foreven numbers of layers, resulting in a six-fold degeneracyof dispersion along the �K line. For odd number of lay-ers, subband spectra are three-fold degenerate, but withEs(k) = E�s(�k).

In Figs. 2(d){(f) we show how the DFT calculatedband structure of WS2, representative of all four TMDs,evolves from monolayer to trilayer [15, 17{22] (DFT bandstructures of all four TMDs can be found in Ref. [32]).The DFT calculations were performed using a plane-wavebasis within the local density approximation (LDA), withthe quantum espresso [33] plane-wave self-consistent�eld (PWSCF) ab initio package. We considered thePerdew-Zunger exchange correlation scheme [34], withfully-relativistic norm-conserving pseudo-potentials, in-cluding non-collinear corrections. Pseudopotentials forMo, W, S, and Se atoms were generated using atomiccode ld1.x of the PWSCF package [35]. The cuto� en-ergy in the plane-wave expansion was set to 60 Ry, andthe BZ sampling of electronic states was approximatedusing a Monkhorst-Pack uniform k grid of 24 � 24 � 1for all structures [36]. We adopted a Methfessel-Paxtonsmearing [37] of 0:005 Ry and set the total energy conver-gence to less than 10�6 eV in all calculations. Spin-orbit

coupling was included in all electronic band structurecalculations. To eliminate spurious interactions betweenadjacent supercells, a 20��A vacuum bu�er space was in-serted in the out-of-plane direction. We used experimen-tal values for the interlayer separations [38{41] and LDAoptimized in-plane lattice constants for all four TMDs[32].

Using WS2 as an example, Fig. 2 illustrates that amonolayer MX2 has a direct band gap at the K point ofthe BZ. The z ! �z mirror symmetry and lack of inver-sion symmetry result in SO-split conduction and valencebands, classi�ed by their spin projection quantum num-ber [Fig. 2(d)]. The large SO splitting at the valenceband (VB) K point and conduction band (CB) Q pointresults from their metal dxy and dx2�y2 orbital composi-tions. This is in contrast to the CB K point, which is pri-marily made of metal dz2 orbitals, resulting in weaker SOsplitting [5, 27]. In a 2H-MX2 bilayer, the combinationof spatial inversion and time reversal symmetry forbidsSO splitting, resulting in two spin-degenerate subbands(four bands in total) in the CB and VB, split by the in-terlayer coupling [Fig. 2(e)]. Additionally, the interlayercoupling shifts the band edges to the � point (VB) andin the vicinity of the Q point (CB), making indirect gapsemiconductors.

In the trilayer 2H-MX2, the valence and conductionband edges remain at the � and near Q points. As shownin Fig. 2(f), the CB subbands are split by SO coupling atthe Q point due to the lack of spatial inversion symmetryin the case of odd numbers of layers. The resulting spec-trum consists of two SO-split subbands in the middle,and two pairs of nearly spin-degenerate subbands aboveand below (see Appendix C for details). For the valencesubbands, however, SO splitting is forbidden exactly atthe � point, due to it being its own time reversal coun-terpart, resulting in three nearly spin degenerate sub-bands [exact degeneracy for even N , and spin-splittingEs(k)�E�s(k) / k3 for odd N ]. This trend, which con-sists of the alternation of SO-split (for odd N) and spin-degenerate (for even N) subbands persists for TMD �lmswith a larger number of layers, and all the same featuresare present in the spectra of all four 2H-MX2 shown inRef. [32]. Finally, we note that the in-plane (2D) carrierdispersions in di�erent subbands njN (both on the VBand CB sides) are di�erent, which a�ects the intersub-band absorption line shapes, as we discuss in Secs. IIIand IV.

III. HOLE SUBBANDS IN p-DOPEDFEW-LAYER TMDS

Figure 2(d) shows the monolayer valence bands rele-vant for the multilayer description, based on symmetryand energy considerations. The v and w valence bandsare non-degenerate at the � point, with the v band com-posed of the metal dz2 orbital and chalcogen pz orbitals,whereas the w band is composed of metal and chalcogen

4

!M

K

AL

H

FIG. 3. Bulk dispersion of 2H-stacked WS2 along the �A line.The DFT data (points) are well �tted by the two-band modelEq. (5) (solid lines). The gray points correspond to the v1and v3 valence bands. Inset: the �rst Brillouin zone of bulkTMDs.

pz orbitals. Bands v1 and v3 belong to two-dimensionalirreducible representations (Irreps), with the v1 bandcomposed of chalcogen px; py and metal dxz; dyz orbitals,and the v3-band formed by chalcogen px; py and metaldxy; dx2�y2 orbitals [5, 26, 27]. In the multilayer case, thew and v bands strongly repel as the w band gets closer inenergy to the v band. The v1 and v3 bands, on the otherhand, are weakly split with a narrow spread due to theirorbital characters, and are pushed downwards relative tothe v band edge. The two-dimensional Irreps of v1 andv3 allow their coupling with the VB being only throughSO interactions (see Appendix B). These features involv-ing the symmetry, orbital composition, and proximity of

the valence bands, supported by our numerical calcula-tions, indicate that the VB is most strongly hybridizedwith the w band, while the other valence bands v1, v3,provide corrections in second-order perturbation theoryto the model parameters through the action of SO cou-pling (see Appendix B). Additionally, as pointed out inSec. II, in two-dimensional 2H-MX2 crystals the CB andVB are almost spin degenerate at the � point, despitethe fact that atomic SO coupling in TMD compoundsis strong. Therefore, to describe the valence subbandsat the � point, we construct a spinless two-band modelincluding the v and w bands, �tting the band parame-ters and interlayer hopping terms to the DFT calculatedband structure, where SO coupling is included implicitly.As indicated in Fig. 2(d), these bands belong to the A01and A002 Irreps of the D3h group of the � point [26, 27],respectively. Therefore, bands v and w are, respectively,even and odd under �h transformations, and do not mixin the monolayer case. However, in multilayers, bandmixing across consecutive layers is allowed by symmetry.

A. HkpTB for the �-point valence band edge

The monolayer dispersions of the valence bands � = vand w, are described by isotropic parabolic dispersionswith band-dependent e�ective masses

E�(k) = E0� �~2k2

2m�: (1)

To construct the multilayer Hamiltonian, we includesymmetry-constrained interlayer couplings, given to low-est orders in k by

t�(k) = t(0)� + t(2)� k

2; tvw(k) = t(0)vw + t(2)vwk

2; (2)

where tv and tw are interlayer intra-band hopping terms,and tvw couples di�erent bands in two consecutive layers.

The multilayer Hamiltonian is given by

ĤN�(k) =X

s=";#

X

�=v;w

dN=2eX

n=1

hE�(k) + 2�� + 2��(k)

ihayns�(k)ans�(k) + �(N2 � n)b

yns�(k)bns�(k)

i

�X

s=";#

X

�=v;w

h�� + ��(k)

i hay1s�(k)a1s�(k) +

�1� #N2

�byN=2;s;�(k)bN=2;s;�(k) +

#N2 ay(N+1)=2;s;�(k)a(N+1)=2;s;�(k)

i

+X

s=";#

X

�=v;w

dN=2eX

n=1

t�(k)�(N2 � n)hayns�(k)bns�(k) + H.c.

i+dN=2e�1X

n=1

t�(k)hayn+1;s;�(k)bns�(k) + H.c.

i!

+X

s=";#

X

�=v;w

dN=2eX

n=1

tvw(k)�(N2 � n)haynsv(k)bnsw(k)� a

ynsw(k)bnv(k) + H.c.

i

+X

s=";#

X

�=v;w

dN=2e�1X

n=1

tvw(k)hayn+1;s;w(k)bnsv(k)� a

yn+1;s;v(k)bnsw(k) + H.c.

i

+X

s=";#

X

�=v;w

dN=2eX

n=1

hU2n�1ayns�(k)ans�(k) + U2n�(N2 � n)b

yns�(k)bns�(k)

i;

(3)

5

where we have de�ned #N � 1 � (�1)N . a(y)ns�(k) and

b(y)ns�(k) annihilate (create) a band-� electron with spinprojection s and in-plane wave vector k in the odd andeven layers of the nth unit cell, respectively. Addi-tional model parameters include the on-site energy cor-rections �v and �w, and k-dependent corrections of theform ��(k) = ��k2, for the v and w bands, respec-tively, which take into account both the pseudo-interlayerpotentials, as well as the spin-ip-induced interband-interlayer hopping (Appendix B). For odd N the systemhas bN=2c complete unit cells and a truncated last unitcell n = dN=2e, where bAc and dAe are the oor andceiling functions, respectively. This case is considered inEq. (3) through the Heaviside function �(N2 � n), whichremoves the operators b(y)N=2;�(k) when N is odd. Theminus sign in the last row of Eq. (3) for the interbandinterlayer hoppings (tvw) is due to the opposite parity un-der z ! �z of the v and w bands, described in Sec. III.Finally, we include on-site potential energy shifts Un, for1 � n � N , to take into account the e�ects of an elec-tric �eld applied along the TMD �lm’s z axis. Underexperimental conditions, such an electric �eld may origi-

nate from a negative back gate, which in addition dopesthe system with a �nite hole density �h. This density isrelated to the potential pro�le fUng as [42],

Un = U1 + edNX

m=2

Em�1;m; n > 1;

Em�1;m =2e"0

NX

l=m

�l;

(4)

where e is the (positive) fundamental charge, d = c=2 isthe interlayer distance, and �l is the electrostatically in-duced density of holes in layer l, such that �h =

PNl=1 �l.

The factor of 2 in the second equation comes from thespin degeneracy at the � point. The potential pro�leand hole density must be determined self-consistently tosatisfy Eq. (4).

Setting Un = 0, we obtain the model parameters in Eq.(3) for each 2H-MX2 by �tting the results of numericaldiagonalization of Eq. (3) to DFT calculations of bulkand few-layer dispersions. For example, the DFT bulkkz dispersion of WS2 is shown in Fig. 3 for the �A cutthrough the 3D BZ. The solid lines in Fig. 3 correspondto the bands of the bipartite Bloch Hamiltonian

H�(k; kz) =

0

BB@

Ev(k) + 2�v + 2�v(k) 0 2tv(k) cos (kzc2 ) 2itvw(k) sin (kzc2 )

0 Ew(k) + 2�w + 2�w(k) �2itvw(k) sin (kzc2 ) 2tw(k) cos (kzc2 )

2tv(k) cos (kzc2 ) 2itvw(k) sin (kzc2 ) Ev(k) + 2�v + 2�v(k) 0

�2itvw(k) sin (kzc2 ) 2tw(k) cos (kzc2 ) 0 Ew(k) + 2�w + 2�w(k)

1

CCA ; (5)

obtained from the model Eq. (3). Equation (5) is writtenin the basis of the v and w bands of layers one and two ofthe bulk 2H crystal unit cell. The �tted parameters forthe four TMDs are given in Tables I and II, and samplecomparisons between the HkpTB model and DFT resultsfor WS2, representative of all four TMDs, are shown inFig. 4. Detailed comparisons for few-layer �lms of allfour materials are available in Ref. [32].

Noting that the bulk VB edge is located at the � point(Fig. 3), the dispersion near the band edge can be ob-tained from Eq. (5) as

E�(kz;k) � �~2k2z2mv;z

�~2k2

2mv;xy

�1 + �k2z

�; (6)

where the bulk parameters are given in terms of theHkpTB model parameters,

m�1v;z =~2

2d2

4t(0)vw

2

�E+ t(0)v

!

(7a)

is the out-of-plane bulk e�ective mass, with d = c=2 theinterlayer distance and �E = Ev � Ew � 2t

(0)v + 2t(0)w +

2�v�2�w the bulk gap between the topmost v and lowestw bands at the � point.

m�1v;xy =�1 +

4mv~2

�t(2)v � �v

��m�1v ; (7b)

is the in-plane bulk e�ective mass, and

� = �~�2mvd2

[1 + 4mv~2 (t(2)v � �v)]

�2t(2)v +

4~2t2vw�E2

mw �mvmvmw

+16tvw

t(2)vw�E

+tvw

�E2(t(2)v � t

(2)w + �w � �v)

!)

;

(7c)

is an anisotropic non-linearity factor. These parametervalues, obtained by �tting to DFT calculations, can befound in Table I.

Then, the subband energies and dispersions in TMD�lms with N � 1 can be analyzed by quantizing holestates with dispersions described by Eq. (6) in a slab ofthickness L = Nd. When doing so, one has to comple-ment Eq. (6) with the general Dirichlet-Neumann bound-ary condition for the standing waves of holes at both �lm

6

TABLE I. Model parameters �tted to DFT data for the mono-layer valence bands Ev(k) and Ew(k), and bulk valence banddispersion for the four TMDs. The monolayer parametersinclude the band edges energy di�erence E0v � E0w, and thee�ective masses mv;mw given in terms of the free electronmass m0. The 3D bulk parameters include the out-of-planeand in-plane e�ective masses mv;z;mv;xy, respectively, andthe in-plane dispersion non-linearity parameter �.

E0v � E0w [eV] mv [m0] mw [m0]mv;z [m0] mv;xy [m0] � [�A2]

MoS2 1.75 3.726 0.3041.04 0.693 -5.24

MoSe2 1.56 5.575 0.5051.42 0.786 -5.99

WS2 2.08 2.885 0.3530.840 0.615 -5.86

WSe2 1.81 3.420 0.7601.08 0.700 -5.45

TABLE II. Model parameters �tted to DFT data for the va-lence band interlayer hopping terms tv(k), tw(k) and tvw(k).�v, �w, �v, and �w are the on-site energy o�sets due to thepseudo-interlayer potential and spin-ip coupling terms.

t(0)v [eV] t(0)w [eV] t(2)v [eV�A2] t(2)w [eV�A

2]t(0)vw [eV] t(2)vw [eV�A2] �v [meV] �w [meV]

�v [eV�A2] �w [eV�A2]MoS2 -0.333 0.592 1.744 2.684

0.432 -1.206 -62.18 -41.43-0.351 6.770

MoSe2 -0.307 0.657 1.830 2.6260.453 -1.140 -29.13 -10.85

-0.261 2.736WS2 -0.322 0.574 1.718 3.205

0.404 -1.226 -36.98 -48.89-0.614 5.834

WSe2 -0.291 0.649 1.814 -1.3820.4309 -0.049 -25.27 -69.93

-0.519 0.192

surfaces

[��d@z (z) + (z)]z=�L2 = 0; (8)

where the � correspond to the top and bottom layers, re-spectively, and � is a dimensionless parameter. Assuminga solution of the form (z) = ueikzz + ve�ikzz, one �ndsfrom Eq. (8) that kz in Eq. (6) obeys

Lkz + 2 arctan(�kzd) = �n; (9)

where the integer n is the subband index. For largenumber of layers and for subbands near the band edge,kz � 1L �

1d , and arctan(�kzd) � �kzd, so that we can

approximate

kz ��n

d(N + 2�); (10)

FIG. 4. HkpTB model dispersions (solid lines) �tted to DFTresults for the WS2 valence subbands near the � point, rep-resentative of all four TMDs. Results are shown for numberof layers N = 3 to 6. Fittings for all four TMDs can be foundin Ref. [32].

leading to the subband energies and dispersions

En�N jN (k) = �~2

2mv;z�2n2

d2(N + 2�)2

�~2k2

2mv;xy

�1 +

��2n2

d2(N + 2�)2

�:

(11)

The large-N asymptotics of the separation between thelowest two subbands, jE1jN � E2jN j, was used to deter-mine the value of the boundary parameter � for holesin each TMD, resulting in � � 0 for MoS2 and MoSe2,� = 0:11 for WS2, and � = 0:007 for WSe2, using thedispersions and lowest intersubband splittings shown inFigs. 4 and 5(a). The good agreement between the fullHkpTB model and the asymptotic analysis shown in Fig.5(a) enables us to describe the main intersubband tran-sition 1jN ! 2jN in p-doped N -layer 2H-MX2 as

jE1jN � E2jN j =3�2~2

2mv;zd2(N + 2�)2: (12)

Furthermore, the hole subband e�ective masses

m�1njN = m�1v;xy

�1 +

��2n2

d2(N + 2�)2

�; (13)

obtained from Eq. (11), describe well the subband de-pendence of the in-plane masses, as seen in Fig. 5(a).

Fig. 5(b) shows the e�ects of a doping-induced poten-tial pro�le in the �lm on the �rst subband transitionenergy jE2jN � E1jN j, as a function of the total gate-induced hole density �h. Results are shown for all fourTMDs with �lm thicknesses of N = 2 to 6 layers, up tomoderate doping levels. Following a slight decrease forweak doping, the transition energies grow monotonically,but with only a weak blue shift.

7

FIG. 5. (a) Energy spacings between the �rst and nth valencesubband (n = 2 to 5) for the four TMDs, as a function of thenumber of layers N . The � parameter corresponding to eachTMD is given in each panel. The solid lines for the �rst twotransitions are obtained using Eq. (11), showing a good �t be-tween the model and DFT. (b) 1jN ! 2jN transition energyas a function of the hole density �h. Each curve correspondsto the value of N indicated on the right.

B. Selection rules for intersubband transitions, anddispersion-induced line broadening

Next, we use the model developed above for the de-scription of hole subbands to study intersubband opticaltransitions, electron-phonon relaxation, and absorptionline shapes of IR/FIR light.

The optical transition amplitude between two givensubbands n and n0 is determined by the out-of-planedipole moment

dn;n0

z (k) = ehn;kjzjn0;ki

= eNX

j=1

X

�=v;w

zjC�n;j;�(k)Cn0;j;�(k);(14)

where N is the total number of layers, zj denotes the z

d z/e

[ß A]

MoS2 MoSe2

WS2 WSe2

N

1 ! 21 ! 3

FIG. 6. Out-of-plane dipole moment matrix elements for theVB subbands, for the �rst two intersubband transitions 1! 2(solid) and 1! 3 (dashed).

coordinate of layer j, and Cn;j;�(k) are the componentsof the nth subband eigenstate. The calculated dipole mo-ment matrix element for the �rst two intersubband tran-sitions is plotted in Fig. 6 as a function of the number oflayers. The selection rules for intersubband transitionsdriven by out-of-plane polarized light are determined bythe odd parity of z under both spatial inversion and mir-ror reection (�h). The subband states for even and oddnumber of layers also have a de�nite parity under spatialinversion and mirror reection, respectively, due to thecrystal’s symmetry. Therefore, intersubband transitionsbetween same parity subbands are forbidden, as shown inFig. 6 for the �rst two intersubband transitions. All thismakes the 1jN ! 2jN transition the dominant feature inthe IR/FIR absorption by thin TMD �lms.

The intersubband absorption line shape is a�ected bythe di�erence between the e�ective masses of subbands1jN and 2jN . The lighter in-plane hole mass in the initialstate (1jN subband) as compared to the �nal state (2jNsubband) spreads the absorption spectrum toward lowerenergies. Heavy p-doping of the TMD �lm or Boltzmanndistribution of the holes in the case of light p-doping,sets the lower limit for the line width of the 1jN ! 2jNabsorption line, which we call the density of states (DOS)broadening

� =�

1�m1jNm2jN

�maxf�F ; kBTg log 2: (15)

Here, kB is the Boltzmann constant, T is the tempera-ture, and m1; m2 are the e�ective masses of the �rst andsecond subbands. This limit for the line width is illus-trated in Fig. 7(a) for N -layer 2H-MX2 at room tem-perature. We note here that DOS broadening is similarto the inhomogeneous broadening, in the sense that itcan be overcome by placing the TMD �lm inside an opti-

8

! !

(a)

HP

LO

ZO

MoS2MoSe2WS2WSe2

N

Total

(a)

(b)

(c)

(d)

(b)

(c)

(d)

(e)

MoS2

MoSe2WS2WSe2

N

HP

LO

ZO

Total

MoS2MoSe2WS2WSe2

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(b)AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBahXkoignorevFY0dhCG8pmO22XbjZhdyOU0J/gxYOKV/+RN/+N2zYHbX0w8Hhvhpl5YSK4Nq777RRWVtfWN4qbpa3tnd298v7Bo45TxdBnsYhVK6QaBZfoG24EthKFNAoFNsPRzdRvPqHSPJYPZpxgENGB5H3OqLHSfTU87ZYrbs2dgSwTLycVyNHolr86vZilEUrDBNW67bmJCTKqDGcCJ6VOqjGhbEQH2LZU0gh1kM1OnZATq/RIP1a2pCEz9fdERiOtx1FoOyNqhnrRm4r/ee3U9C+DjMskNSjZfFE/FcTEZPo36XGFzIixJZQpbm8lbEgVZcamU7IheIsvLxP/rHZV8+7OK/XrPI0iHMExVMGDC6jDLTTABwYDeIZXeHOE8+K8Ox/z1oKTzxzCHzifP/hKjR8=AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBahXkoignorevFY0dhCG8pmO22XbjZhdyOU0J/gxYOKV/+RN/+N2zYHbX0w8Hhvhpl5YSK4Nq777RRWVtfWN4qbpa3tnd298v7Bo45TxdBnsYhVK6QaBZfoG24EthKFNAoFNsPRzdRvPqHSPJYPZpxgENGB5H3OqLHSfTU87ZYrbs2dgSwTLycVyNHolr86vZilEUrDBNW67bmJCTKqDGcCJ6VOqjGhbEQH2LZU0gh1kM1OnZATq/RIP1a2pCEz9fdERiOtx1FoOyNqhnrRm4r/ee3U9C+DjMskNSjZfFE/FcTEZPo36XGFzIixJZQpbm8lbEgVZcamU7IheIsvLxP/rHZV8+7OK/XrPI0iHMExVMGDC6jDLTTABwYDeIZXeHOE8+K8Ox/z1oKTzxzCHzifP/hKjR8=AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBahXkoignorevFY0dhCG8pmO22XbjZhdyOU0J/gxYOKV/+RN/+N2zYHbX0w8Hhvhpl5YSK4Nq777RRWVtfWN4qbpa3tnd298v7Bo45TxdBnsYhVK6QaBZfoG24EthKFNAoFNsPRzdRvPqHSPJYPZpxgENGB5H3OqLHSfTU87ZYrbs2dgSwTLycVyNHolr86vZilEUrDBNW67bmJCTKqDGcCJ6VOqjGhbEQH2LZU0gh1kM1OnZATq/RIP1a2pCEz9fdERiOtx1FoOyNqhnrRm4r/ee3U9C+DjMskNSjZfFE/FcTEZPo36XGFzIixJZQpbm8lbEgVZcamU7IheIsvLxP/rHZV8+7OK/XrPI0iHMExVMGDC6jDLTTABwYDeIZXeHOE8+K8Ox/z1oKTzxzCHzifP/hKjR8=AAAB6XicbVBNS8NAEJ3Ur1q/qh69LBahXkoignorevFY0dhCG8pmO22XbjZhdyOU0J/gxYOKV/+RN/+N2zYHbX0w8Hhvhpl5YSK4Nq777RRWVtfWN4qbpa3tnd298v7Bo45TxdBnsYhVK6QaBZfoG24EthKFNAoFNsPRzdRvPqHSPJYPZpxgENGB5H3OqLHSfTU87ZYrbs2dgSwTLycVyNHolr86vZilEUrDBNW67bmJCTKqDGcCJ6VOqjGhbEQH2LZU0gh1kM1OnZATq/RIP1a2pCEz9fdERiOtx1FoOyNqhnrRm4r/ee3U9C+DjMskNSjZfFE/FcTEZPo36XGFzIixJZQpbm8lbEgVZcamU7IheIsvLxP/rHZV8+7OK/XrPI0iHMExVMGDC6jDLTTABwYDeIZXeHOE8+K8Ox/z1oKTzxzCHzifP/hKjR8=

![m

eV]

N

FIG. 7. (a) Absorption line widths for VB subbands at roomtemperature (300 K) as a function of number of layers for thefour TMDs, considering only DOS broadening. (b) Phonon-induced broadening at room temperature (T = 300 K) due tointersubband emission and intrasubband absorption of opticalphonons modes (top to bottom) HP, LO and ZO, for the fourTMDs as a function of number of layers N , with the combinedbroadening shown in the bottom panel.

cal resonator that would select intersubband modes withparticular values of in-plane momentum k.

C. Broadening due to electron-phonon intra- andintersubband relaxation

In contrast to the elastic DOS broadening, phonon-induced intra- and intersubband relaxation broaden theabsorption line in a way that cannot be avoided by aclever choice of the electromagnetic environment. Be-low we consider emission and absorption of homopo-lar (HP), longitudinal (LO) and out-of-plane (ZO) op-tical phonons, which we assume to be dispersionless.This choice is motivated by the fact that these are thestrongest coupled modes in TMDs, as established by ear-lier studies [28, 43]. Also, we take phonon modes of

few-layer �lms as independent and degenerate. This ap-proximation is justi�ed by the fact that splittings due tohybridization between layers are much smaller than themonolayer phonon frequency [44].

The hole-phonon couplings for a phonon in mode � =HP; LO, or ZO in layer j, interacting with a hole in layeri, are given by (see Appendices D and E)

gj;iHP(q) = �ij

s~

2�!HPDv; (16a)

gj;iLO(q) =

s~

2�MrM !LO

2�ie2Z(�1)j

A(1 + r�q)e�qdji�jj; (16b)

gj;iZO(q) =

s~

2�MrM !ZO

2�e2ZzA

e�qdji�jji� jji� jj

; (16c)

where !� denotes the corresponding phonon frequency; �is the mass density of the material; Dv is the deformationpotential in the valence band; A is the unit cell area; Mand Mr are the total unit-cell mass and reduced mass ofthe metal and two chalcogens, respectively; Z and Zz arethe in-plane and out-of-plane Born e�ective charges, re-spectively; and r� is the screening length in the material.The various parameters taken from Refs. 28, 43, 45, and46 are given in Table III.

The phonon-induced broadening is determined by thelifetime of the hole in the excited subband state, whichincludes contributions from intersubband relaxation dueto emission (low and high temperature) and intrasub-band absorption (high temperature) [Fig. 7(b)]. We notethat intrasubband emission contributions are thermallyactivated, since they require carriers to be thermally ex-cited to energies higher than the corresponding phononenergy. The typical energy of thermally distributed carri-ers at room temperature is 12kBT � 13 meV, whereas thephonon energies are of order 30{50 meV (Table III), mak-ing this process irrelevant. Similarly, the process involv-ing intersubband absorption from the second subband tothe third is suppressed by the larger intersubband spac-ings, as compared to the phonon energies and the �rstintersubband spacings for N

9

TABLE III. Electron-phonon coupling parameters for LO, HP, and ZO phonon modes. !HP, !LO, and !ZO are the HP, LO,and ZO mode energies; � is the mass density; Dv; Dc are the valence and conduction deformation potentials; Z; Zz are thein-plane and out-of-plane Born e�ective charges; r� is the screening length; Mr=M is the ratio of the reduced mass of the metaland chalcogens to the total unit-cell mass; and A is the unit-cell area.

~!HP [meV] ~!LO [meV] ~!ZO [meV] � [g/cm2] Dv [eV/�A] Dc [eV/�A] Z Zz r� [�A] Mr=M A [�A2]MoS2 51 49 59 3:1� 10�7 3.5 7.1 1.08 0.1 41 0.24 8.65MoSe2 30 37 44 4:5� 10�7 3.8 7.8 1.8 0.15 52 0.249 9.37WS2 52 44 55 4:8� 10�7 1.5 3.4 0.47 0.07 38 0.29 8.65WSe2 31 31 39 6:1� 10�7 2.2 2.7 1.08 0.12 45 0.25 9.37

the Bose-Einstein distribution for a phonon in mode � attemperature T . The �rst term in curly brackets describesintersubband phonon emission, whereas the second termdescribes intrasubband phonon absorption.

The resulting phonon-induced broadenings at roomtemperature are shown in Fig. 7(b). The main contri-bution comes from intersubband relaxation, with intra-subband absorption suppressed by the phonon occupa-tion number. The intersubband LO phonon contributiondominates the broadening due to the strong couplingattributed to the large in-plane Born e�ective charge,and the long-range nature of the coupling. The reducedbroadening for N = 2 is due to the large intersubbandspacing, which suppresses intersubband relaxation, andthe fact that the second subband is almost at, whichsuppresses intrasubband absorption. The peaks in thebroadenings for certain numbers of layers correspond tonear resonances between the phonon energies and the in-tersubband spacings. Phonon broadening is seen to bemost detrimental for MoSe2 in particular, and in gen-eral for all TMDs with seven or eight layers. Beyondthis number of layers, the phonon energies become largerthan the intersubband spacings, thus preventing inter-subband relaxation, however, intrasubband absorption isstill present and dominates for N > 7. Finally, we notethat the broadening values are found to be smaller thanthose observed in III-V quantum wells [30], implying aweaker detrimental e�ect on the absorption/emission lineshape in these materials.

D. Room-temperature absorption spectrum inp-doped TMD �lms

The cumulative e�ect of inelastic (e-ph) and elastic(DOS) broadening of the intersubband 1jN ! 2jN ab-sorption spectra of lightly p-doped TMD �lms is de-scribed by

I(~!) =4�~jEz(~!)j

2X

k

��d1;2z (k)��2 fT (k)

�

=�

[E1(k)� E2(k)� ~!]2 + 2;

(18)

where fT (k) is the Fermi function for hole occupation inthe lowest subband corresponding to hole density nh, andtemperature T (we assume that all higher-energy hole

! ! [eV]

I[a

.u.]

MoS2 MoSe2

WS2 WSe2

N = 2

N = 3

N = 4

N = 5

FIG. 8. Optical absorption lines for N = 2 to 5 layers oflightly p-doped MoS2; MoSe2; WS2, and WSe2 at room tem-perature (T = 300 K).

subbands are empty). The resulting absorption spec-tra at room temperature for the four TMDs with dif-ferent number of layers are shown in Fig. 8. The spectrashow the combination of DOS broadening, which pro-duces a tail towards lower photon energies, with thephonon-induced broadening, most relevant for N > 2,which gives a small tail towards higher energies, mak-ing the lines more symmetric and reducing their ampli-tudes. The smaller phonon couplings in WS2 result intall, narrow, and asymmetric line shapes, with intensityincreasing with the number of layers, reecting the grow-ing dipole matrix element. This is in contrast to MoSe2,where the larger phonon-induced broadening results insmaller and more symmetric peaks for N > 2.

10

TABLE IV. Monolayer and bulk conduction band parameters �tted to DFT calculations of the four TMDs. The e�ectivemasses are given in terms of the free electron mass m0. The band-edge energy E0 is given relative to the valence band edgeat the � point, and 2�0 is the spin-orbit splitting at the Q point. The monolayer parameters include the e�ective masses inthe x and y directions for the spin split bands, and the band minima o�sets q# and q". The conduction band bulk dispersionparameters include the in-plane e�ective masses mc;x;mc;y, and out-of-plane mass mc;z; band minima o�sets �0 and �, andin-plane dispersion non-linearity parameters �x and �y.

mx;" [m0] my;" [m0] q" [10�3�A�1] mx;# [m0] my;# [m0] q# [10�3�A

�1] E0 [eV] 2�0 [meV]mc;z [m0] mc;x [m0] mc;y [m0] �x [�A2] �y [�A2] �0 [�A�1] � [10�4 �A]

MoS2 0.595 1.035 20.49 0.666 1.105 7.16 1.994 67.00.525 0.550 0.735 -3.90 -7.94 0.0456 -1.3

MoSe2 0.583 1.060 54.21 0.518 1.106 26.93 1.891 21.00.500 0.510 0.760 -4.65 -4.26 0.0663 0.32

WS2 0.529 0.722 13.74 0.763 0.892 -20.65 2.059 2540.510 0.528 0.596 -4.30 -4.12 0.0344 0.53

WSe2 0.468 0.753 49.63 0.676 0.908 1.88 1.94 2140.466 0.479 0.608 -4.19 -5.80 0.0599 -0.9

IV. ELECTRON SUBBANDS IN n-DOPEDFEW-LAYER TMDS

A. HkpTB for the conduction band near the Qpoint

The conduction band edges in monolayer MoS2,MoSe2, WS2, and WSe2 are located at the K points,but accompanied by local dispersion minima that appearnear the six inequivalent points �Q, �C3Q and �C23Q,where � = �1, and Q = 2�3a x̂ is the midpoint between �and K (a is the lattice constant). For a given value of � ,there are three valleys connected by C3 rotations aboutthe BZ center [Fig. 2(c)], such that we need only describethe dispersion near the two points �Q, which are relatedby time reversal.

For spin projection s, the monolayer dispersion near

the �Q valley is given by [27]

E�s (k) =~2(kx � q�s )2

2m�x;s+

~2k2y2m�y;s

+ E0 + �s�0; (19)

where m�x;s and m�y;s are e�ective masses; E0 is a con-stant energy shift; 2�0 is the spin-orbit splitting betweenthe spin-up and down band edges; and q�s is the band-edge momentum relative to the valley along the x̂ axis.From time-reversal symmetry we obtain the dispersionfor the opposite valley as E�s (k) = E

���s (�k), which re-

quires (� = x; y) m��;s = m���;�s and q�s = �q

���s .

As described in Sec. II, the 2H-stacked bilayer consistsof subsequent layers rotated by 180� with respect to eachother. In reciprocal space, this means that a conduction-band state of spin projection s and momentum �Q + kof the �rst layer will hybridize with its in-plane inversionpartner of spin s and momentum ��Q � k in the sec-ond one [Fig. 2(c)]. The multilayer Hamiltonian for theconduction subbands about �Q is given by

H�NQ(k) =dN=2eX

n=1

X

s=";#

�E�s (k) +

��n;1 + �n;dN=2e

��E� hayn�s(k)an�s(k) + �(N2 � n)b

yn;��;�s(�k)bn;��;�s(�k)

i

+dN=2eX

n=1

X

s=";#

t� (k)�(N2 � n)hbyn;��;s(�k)an�s(k) + H.c.

i+dN=2e�1X

n=1

X

s=";#

t�� (k)hbyn;��;s(�k)an+1;�;s(k) + H.c.

i

+dN=2e�1X

n=1

X

s=";#

t0[ayn+1;�;s(k)an�s(k) + byn+1;��;s(�k)bn;��;s(�k)]

+dN=2eX

n=1

X

s=";#

hU2n�1 ayn�s(k)an�s(k) + U2n�(N2 � n)b

yn;��;�s(k)bn;��;�s(k)

i;

(20)

where a(y)n;�;s(k) and b(y)n;�;s(k) annihilate (create) electrons of spin projection s, in-plane wave vector k and valley

11

!M

K

AL

H

Q

FIG. 9. Bulk dispersion of 2H-stacked WS2 along the Bril-louin zone path QA, de�ned by kx = ky = 0 and kz 2 [0; �=c],where kx and ky are measured relative to the Q point, asshown in the inset. DFT data (points) are well �tted by themodel Eq. (23) (solid line). Inset: the �rst Brillouin zone ofbulk TMDs.

quantum number � , on the odd and even layers of thenth bulk unit cell. The alternation of spin indices andhopping terms are a result of 2H stacking. The model isparameterized by the terms in t� (k) given in Eq. (21),the interlayer pseudo-potential �E, implemented as anon-site energy shift at the boundary layers, and the next-nearest-neighbor hopping t0 included to improve the �t-ting to DFT bands. The interlayer hopping has the form(see Appendix A)

t� (k) = t0 + �t1kx + iu1ky + t2k2x + u2k2y; (21)

up to second order in the in-plane crystal momentum.Given the lack of �h symmetry for even N , the spinprojection s is, strictly speaking, not a good quantumnumber, and spin mixing is allowed. This is discussed inAppendix A. However, using an expansion about the Qpoint in our DFT results shows that spin mixing is muchweaker [47] than t� (k), and can be neglected. We alsofound u1 to be several orders of magnitude smaller thant1; as a result, we consider t� (k) to be real.

Finally, in the last term of Eq. (20) we take intoaccount electrostatic doping e�ects through the layer-dependent potential energy Un (1 � n � N) [42]:

Un = U1 + edNX

m=2

Em�1;m; n > 1;

Em�1;m =3e"0

NX

l=m

X

�=�1

X

s=";#

�s�l :

(22)

Here, �s�l is the electron (number) density induced inlayer l belonging to the spin-s subbands in valley � . Thefactor of 3 in the second equation comes from the valley

FIG. 10. HkpTB model dispersions (solid lines) �tted to DFTresults for the WS2 conduction subbands (points) near the Qpoint (kx = 0), representative of all four TMDs. Resultsare shown for number of layers N = 3 to 6. For odd N ,line and point colors indicate subbands with di�erent spinprojections, with blue (red) corresponding to spin down (spinup). For even N , spin-up and -down subbands are degenerate,and shown with black dots. Fittings for all four TMDs canbe found in Ref. [32].

TABLE V. Model parameters �tted to DFT data for the con-duction band interlayer hopping terms. �E is an energy o�setfor the �rst and last layers of the structure that accounts forsurface e�ects.

t0 [eV] t1 [eV�A] t2 [eV�A2]

t0 [meV] u2 [eV�A2] �E [meV]

MoS2 0.203 0.213 0.041912.7 -0.662 8.90

MoSe2 0.215 0.180 -0.14520.5 -0.447 -4.29

WS2 0.210 0.233 -0.1235.24 -0.864 -3.95

WSe2 0.211 0.209 0.2319.54 -0.797 4.21

degeneracy, which is preserved even in the presence of anout-of-plane electric �eld.

In the bulk limit, and in the absence of external electric�elds (Un = 0), we have the bipartite Hamiltonian

H�Q(k; kz) ="0(k; kz)s0�0 + ��(k)s3�3

+ 2t� (k) cos�kzc2

�s0�1;

"0(k; kz) =E+" (k) + E

+# (k)

2+ 2t0 cos (kzc);

�(k) =E+" (k)� E

+# (k)

2;

(23)

where �(k) is the k-dependent monolayer spin-orbit

12

splitting for wave vector k measured relative to the Qpoint; si and �i (i = 0 to 3) are Pauli matrices acting onthe spin and layer degrees of freedom, respectively, ands0 and �0 are the identity in their corresponding sub-spaces. The model parameters for the four TMDs were�tted to the DFT-calculated monolayer and 3D bulk dis-persions, and are presented in Tables IV and V. A samplebulk �tting is shown in Fig. 9 for WS2, along the pathde�ned by kx = ky = 0 (Q point) and kz 2 [0; �=c], withthe solid line corresponding to the model Eq. (23). Asample comparison between our HkpTB model to DFTresults for WS2 few-layer structures, representative of allfour TMDs, is shown in Fig. 10. Detailed comparisonsfor few-layer �lms of all four materials are available inRef. [32].

As discussed in Sec. II, the global symmetry alterna-tion between �h for odd N , and spatial inversion sym-metry for even N , results in the striking qualitative dif-ferences between the cases with even and odd number oflayers in Fig. 10. The two-fold spin degeneracy observedfor even N is a consequence of spatial inversion and timereversal symmetry, resulting in E�s (k) = E��s(k). Bycontrast, �h mirror symmetry for N odd makes s a goodquantum number, while the lack of inversion symmetryallows for spin-orbit splitting. Notice also that the twomiddle spin-split subbands remain �xed for all odd val-ues of N , while the rest of the bands are nearly spin-degenerate. As discussed in Appendix C, these featurescan be traced back to the SO splitting in the monolayercase, and the particular form of Hamiltonians H�NQ(k)for odd N .

Expanding the lowest eigenvalue of Eq. (23) for valley� about kz = 0, corresponding to the bulk conductionband edge (Fig. 9), the dispersion can be written as

E�Q(k; kz) �~2

2mc;x(kx � � [�0 � �k2z ])

2 �1 + �xk2z�

+~2k2y2mc;y

�1 + �yk2z

�+

~2k2z2mc;z

+ E0Q;(24)

where mc;z is the out-of-plane bulk e�ective mass;mc;x; mc;y are the in-plane e�ective masses in the xand y directions, respectively; �x; �y are anisotropic non-linearity factors; �0 and � account for the band minimumo�set from the Q point along the x̂ direction; and E0Q isa constant energy shift. These constants are related toour HkpTB model parameters through the expressions[see Eq. (19)]

mc;x =2m�x;"m

�x;#

m�x;" +m�x;#; (25a)

mc;y =2m�y;"m

�y;#

m�y;" +m�y;#; (25b)

mc;z =~2

8d2

�t20

4�0

�1�

t21t21 + 2t0t1

�� t0

+t21 + 2t0t2

4�0

��0 +

t0t1t21 + 2t0t2

�2#�1;

(25c)

�0 =m+x;"q

+# +m

+x;#q

+"

m+x;" +m+x;#

; (25d)

� =2mc;xd2

~2t21 + 2t0t2

�0

��0 +

t0t1t21 + 2t0t2

�; (25e)

�x =2mc;xd2

~2t21 + 2t0t2

�0; (25f)

�y =4mc;yd2

~2t0u2�0

; (25g)

E0Q = E0 � 2�0 +~2

4(q�" + q

�# )

2

m�x;" +m�x;#

+ 2t0: (25h)

Similarly to the subbands on the valence-band side(Sec. III), the conduction subbands in TMD �lms withN � 1 can be analyzed by quantizing the electron statesin a slab of �nite thickness L = Nd, with dispersions de-scribed by Eq. (24). However, note that the coe�cients ofEqs. (25a){(25h) are independent of spin projection andvalley, and thus not representative of the odd N case.This is a consequence of the explicit inversion symmetryof the bulk model (23). Nonetheless, the SO splittingresulting from the lack of inversion symmetry and thepresence of �h symmetry in a system with odd N , canbe introduced through the TMD quantum well boundaryconditions.

The unit cell for 2H crystals contains two layers, whichbelow we label A and B [see Eq. (20)]. For odd N , in-version symmetry is broken in opposite ways for the twolayers in the unit cell, given that, as discussed in Sec.I, they are rotated by 180� with respect to each other.This results in di�erent boundary conditions for electronsat a given termination of the TMD �lm, depending onwhether the �nal layer is of type A or B. This generalizesthe boundary conditions used for the valence band at the� point [Eq. (8)] to

[� (�0 + s��1) d@z �s (z) + �s (z)]z=�L2

= 0; (26a)

for the boundary at z = �L=2 when the �lm terminateson an A layer, and

[� (�0 � s��1) d@z �s (z) + �s (z)]z=�L2

= 0; (26b)

when the �nal layer at position z = �L=2 is of type B.Here, �0 ; �1 � N are dimensionless parameters. This

13

results in spin- and valley-dependent quantization condi-tions

ks;�z;njN ��n

d[N + 2�0 + s��1#N ]; (27)

where #N � 1� (�1)N gives 0 for even N and 2 for oddN . Overall, the low-energy spectrum of a thin �lm hasthe form

Es;�n�N jN (k) =~2

2mc;z�2n2

d2[N + 2�0 + s��1#N ]2

+~2

2ms;�c;x;njN(kx � �s;�njN )

2 +~2k2y

2ms;�c;y;njN+ E0Q;

(28)

where the subband in-plane e�ective masses in the � =x; y directions arehms;��;njN

i�1� m�1c;�

�1 +

���2n2

d2[N + 2�0 + s��1#N ]2

�;

(29a)and the momentum o�set from the Q point is given by

�s;�njN � ��0 +�n2�2 �

d2 [N + 2�0 + s��1#N ]2 : (29b)

As in the monolayer case, the low-energy subband dis-persions described by Eq. (28) near the six valleys at BZpoints �Q, �C3Q, and �C23Q, can be divided into twotriads related by time-reversal symmetry, with quantumnumbers � = �1. The three valleys for a given � are con-nected by C3 rotations, as sketched in the inset of Fig.2. As a consequence, for odd number of layers, whereinversion symmetry is broken and SO splitting is param-eterized by �1, the spin and valley degrees of freedom ofthe bottom subband are locked, and the low-energy stateshave valley degeneracy of godd = 6. Conversely, for evennumber of layers the bottom subbands are spin degen-erate, giving a total degeneracy of geven = 12. Theselarge subband degeneracies and multi-valley structures,together with the anisotropic dispersions found withineach valley, may have important implications for thetransport and quantum Hall properties of n-doped mul-tilayer TMDs [48].

Both inversion and �h symmetry are broken when anelectric �eld is applied along the ẑ axis of the �lm, leadingto a modulation of the spin splittings in the case of oddN , and lifting of the spin degeneracy for even N . As a�rst approximation, this e�ect can be introduced into thelowest subband dispersion by substituting

E0Q ! E0Q(�e) + s��(�e) (30)

in Eq. (28) for n = 1, where �e = 3Ps;�;n �

s�n is the

total electron density induced by the �eld. These spin-dependent shifts of the band edges, abbreviated Es1jN �Es+1jN (�

s+1jN ; 0), parametrize the symmetry breaking by the

potential pro�le. As an example, representative of allfour TMDs, Figure 11 shows this splitting for the low-est conduction subband of �ve- and six-layer WSe2, in

the case where the �eld is induced by a single positiveback gate near layer one. The corresponding inducedelectron densities and potential pro�les were determinedself-consistently to satisfy Eq. (22), and chosen inside arange that is easily accessible to experiments.

In the case of evenN we can interpret the spin-splittingstrength in the weak doping regime in terms of the spa-tial charge distributions of the lowest-energy spin-up and-down subband states. The left insets in Fig. 11(b) showthe spread of these two states along the �lm’s ẑ axisin the zero-doping limit (�e ! 0). The two distribu-tions are asymmetric with respect to the middle of the�lm, and related to each other by a z ! �z mirroroperation, resulting in opposite electric dipole momentsh�"Ei = �h�

#Ei about the middle plane of the multilayer

structure. The splitting in the zero-doping limit can beapproximated as �(�e ! 0) � �(h�"Ei � h�

#Ei)Ez > 0,

where Ez > 0 is the electric �eld in the ẑ direction. Thisis shown in Fig. 11(b) for electron densities between 0and 0:2 � 1012 cm�2. This linear splitting contributionis weak: even at small doping, it is overcome by non-linear screening e�ects, which produce a large negativesplitting � < 0 [Fig. 11(c)] and quickly deplete the spin-down subband, as shown in the right insets of Fig. 11(b).For odd N , where in the zero-doping limit the states ofboth spin band edges are symmetric about the middlelayer of the �lm [see insets of Fig. 11(a)], spin splittingis determined entirely by non-linear e�ects. This anal-ysis for the �e ! 0 limit can be generalized to all evenand odd values of N for all four TMDs, as shown in Fig.11(d), where we show that, in all four cases, the spin-resolved electric dipole moments alternate between �niteand zero for even and odd N , respectively. Also, notethat according to Eq. (30), the sign of the spin splittingis inverted for opposite valleys.

B. Intersubband transitions and dispersion-inducedline broadening in n-doped N-layer TMDs

Numerically diagonalizing the HkpTB Hamiltonian inEq. (20) with the parameters of Tables IV and V, weobtain the energy spacings between the �rst and nextfew subbands of TMD �lms shown in Fig. 12. Using Eq.(28), we estimate the separation between the lowest twosubbands of a given spin projection s as

Es;�2jN � Es;�1jN �

15�4~2�2

2mc;xd4(N + 2�0)4+

3�2~2

2mc;zd2(N + 2�0)2

� #Ns��16�2~2

2mc;zd2(N + 2�0)3:

(31)

Similarly, we estimate the splitting between the lowestsubbands of opposite spin as

E�s;�1jN � Es;�1jN � s��1#N

2�2~2

2mc;zd2(N + 2�0)3: (32)

14

FIG. 11. Spin-up and -down subband edge energies Es1jN atvalley � = +, measured with respect to the Fermi level, for(a) �ve-layer and (b) six-layer WSe2, as a function of electrondensity �e. The potential pro�les across the TMD �lm weredetermined self-consistently for a temperature of T = 10 K fora single (positive) back gate. In both panels, left and rightinsets show the spin-up and -down charge distributions for�e ! 0 and for �nite �e, respectively. (c) Corresponding gate-doping-induced spin splittings at valley � = +. (d) Imbalancebetween the ẑ-axis electric dipole moments of the spin-up and-down subband edges for �e ! 0.

FIG. 12. (a) Energy spacings between the �rst and nth con-duction subbands (n = 2 to 5) for the four TMDs, as functionsof the number of layers N . The solid lines in each panel cor-responds to Eq. (31) for the main transition between the �rstand second subbands using the DFT bulk parameters, show-ing good agreement between the HkpTB model and DFT.Parameters �0 and �1, �tted for N � 4, are given for thefour TMDs in their corresponding panels. Blue (red) pointsand solid lines in each panel represent spin-down (-up) po-larized subband spacings and �ttings. Black points and solidline correspond to subband spacings for even N layers, wheresubbands are spin degenerate. (b) Lowest spin-up and -downsubband transition as a function of total electron doping, for�lm thicknesses N = 2 to 6 layers. The number of layers Ncorresponding to each pair of curves is indicated on the right.For two layers, the doping level shifts the spin-up (spin-down)transition energy upward (downward) in energy, following alinear trend within for moderate doping levels. For N > 2non-linear e�ects begin to appear already for weak doping,and the splitting of the spin-up and -down transitions is in-verted with respect to N = 2.

We used Eqs. (31) and (32) to determine the bound-ary parameters �0 and �1 for each of the consideredTMDs (MoS2: �0 = 0:82; �1 = �0:016, MoSe2: �0 =0:76; �1 = �0:0055, WS2: �0 = 0:80; �1 = �0:0031,WSe2: �0 = 0:72; �1 = �0:028). The results are shown

15

with solid lines in Fig. 12. In the absence of an electric�eld, the energy splitting between the lowest two spin po-larized transitions is of order few meV for the four TMDs.However, this splitting is enhanced by electron doping,as shown in Fig. 12(b) for two- to six-layer �lms of allfour TMDs. For N = 2, the spin-up (-down) transitionenergy grows (decreases) linearly with the doping levelfor small electron densities. The opposite trend is foundfor N = 3 to 6, where the spin-up subband transitionappears at lower energy, and non-linear e�ects begin toappear already for low electron doping. We conclude thatthe electron doping level can be used as an additional tun-able parameter to modify the energies and degeneraciesof the lowest optical transitions in 2H-TMDs.

Next, we use the model developed above for electronsubbands to study intersubband optical transitions, in-tersubband electron-phonon relaxation, and the inter-subband absorption line shapes for IR/FIR light. Asdiscussed in Sec. III B, the optical transition amplitudebetween two given subbands n; n0 is determined by theout-of-plane dipole moment

dn;n0

�;s;z(k) =ehn; s; �;kjzjn0; s; �;ki

= eNX

j=1

zjC�;s�n;j (k)C�;sn0;j(k);

(33)

where N is the total number of layers, zj denotes the zcoordinate of layer j, and C�;sn;j(k) are the componentsof the nth subband eigenstate of spin projection s andvalley quantum number � . The calculated dipole momentmatrix element as a function of number of layers for the�rst two intersubband transitions is plotted in Fig. 13.

Similarly to the valence subbands case, optical tran-sitions in �lms with odd number of layers N are al-lowed only between states with opposite-parity subbandindices, corresponding to opposite parity under �h trans-formation. The spin-orbit splitting present for odd N re-sults in a spin selection rule, allowing transitions only be-tween subbands with the same out-of-plane spin projec-tion s. For even N , where �h symmetry is absent, transi-tions between subbands with same-parity indices are al-lowed. This is in contrast to the VB at the � point, and isa consequence of the multiple-valley structure of the CB,which makes it possible to form degenerate even and odd(under inversion) combinations of states, giving a �nitedipole moment, as shown in Fig. 13 for the �rst two in-tersubband transitions, considering both spin-down and-up polarized subbands. This makes 1jN ! 2jN transi-tion the dominant feature in the IR/FIR absorption bythin n-doped TMD �lms.

Similarly to the holes in p-doped TMDs, the line shapeof the electron intersubband absorption in n-doped �lmsis also a�ected by the di�erence between the e�ectivemasses of subbands 1jN and 2jN . However, in contrast tothe case of holes, for electrons the line shapes depend alsoon the relative in-plane wave vectors of the conductionsubband minima, as well as the anisotropic subband dis-persions. The resulting broadening for N -layer 2H-MX2

FIG. 13. Out-of-plane dipole moment matrix elements for the�rst two conduction intersubband transitions, 1 ! 2 (solidline) and 1! 3 (dashed line). Transitions between spin-down(-up) subbands are shown in blue (red).

�lms at room temperature, obtained numerically fromthe calculated line shapes, is shown in Fig. 14(a). Ourcalculations show that the aforementioned DOS broaden-ing factors result in a typically larger broadening, whichspreads the absorption spectrum towards both lower andhigher energies from the main transition.

C. Electron-phonon relaxation androom-temperature absorption spectra in n-doped

TMD �lms

The phonon-induced broadening for conduction sub-bands m and n, generated by intrasubband and inter-subband relaxation, is accounted for by

�;sn;m = 2�X

�;q;j

�����X

i

gj;i� (q)C�;s�n;i (�

�;sm x̂+ q)C

�;sm;i(�

�;sm x̂)

�����

2

� f[1 + nT (~!�)]� [Em(��;sm x̂)� En(��;sm x̂+ q)� ~!�]

+�nmnT (~!�)� [Em(��;sm x̂)� Em(��;sm x̂+ q) + ~!�]g ;

(34)

where ��;sm is the subband edge o�set from the Q pointof subband m with spin projection s, and gj;i� are theelectron-phonon couplings for the three phonon modes�= HP, LO, and ZO given in Eq. (16), with Dv replacedby Dc for the HP phonon. The �rst term in Eq. (34)describes intersubband relaxation due to phonon emis-sion, whereas the second describes intrasubband phononabsorption in the excited subband. The phonon in-duced broadening for the four TMDs obtained usingthe electron-phonon coupling parameters in Table IIIare shown in Fig. 14(b). The dominant contributioncomes from intersubband relaxation due to HP and LO

16

FIG. 14. (a) Absorption line widths for CB subbands atroom temperature (T = 300 K) as a function of number oflayers for the four TMDs, considering only DOS broaden-ing. (b) Phonon-induced broadening at room temperature(T = 300 K) due to intersubband emission and intrasubbandabsorption of optical phonons in modes (top to bottom) HP,LO and ZO. Results are shown for the four TMDs as a func-tion of number of layers N , with the total broadening shownin the bottom panel.

phonon modes, with a smaller contribution from the ther-mally suppressed intrasubband absorption. The large HPphonon deformation potential at the Q point, as com-pared to its value at the � point [45], in particular forMoS2 and MoSe2 (see Table III), results in a large con-tribution to the broadening. Additional di�erences be-tween the phonon-induced broadenings for the conduc-tion and valence subbands originate from the di�erentintersubband spacings as a function of number of lay-ers (Figs. 5 and 12), and di�erent dispersions (Figs. 4and 10). As in the valence subbands case, the phononbroadening is most signi�cant for MoSe2 due to strongerelectron-phonon coupling.

FIG. 15. Optical absorption lines for N = 2-5 layers of lightlyn-doped MoS2;MoSe2;WS2, and WSe2, taking into accountintrinsic broadening at room temperature (T = 300 K). Inthe case of odd N , lines corresponding to di�erent subbandspin projections are summed.

The absorption spectra of n-doped TMD �lms calcu-lated using Fermi’s golden rule, as in Eq. (18), and takingthe discussions of Secs. IV A, IV B, and IV C into account,are shown in Fig. 15 for N = 2 to 5 layers. The predictedabsorption spectra for the four TMDs are seen to be moresymmetric than those for the holes, primarily due to thee�ect of di�erent dispersions in consecutive subbands,here aggravated by the shifts ��;snjN and the BZ positionof the subbands minima, in addition to the di�erence be-tween the in-plane subband e�ective masses. The largeSO splitting between the middle two subbands for N = 3results in two distinct lines, whereas for N = 5 the spin-polarized subbands are nearly degenerate, resulting inthe overlap of the two lines and giving a combined linewith twice the amplitude.

V. CONCLUSIONS

We have presented hybrid k�p tight-binding modelsfor the conduction and valence band edges of multilayerTMDs, capable of reproducing the rich low-energy sub-band dispersions, and allowing us to describe the inter-subband optical transitions when coupled to out-of-planepolarized light. In particular, we �nd the following:

� The subbands at the CB edge are found near theQ valleys of the Brillouin zone, whereas the valenceband edge is found at the � point. The main dif-ferences between the two sets of subbands are dueto the signi�cant spin-orbit splitting, multi-valleystructure, and anisotropic dispersions of the con-duction subbands, by contrast to the valence sub-bands. These di�erences manifest themselves in

17

the absorption line shapes and additional selectionrules, particularly for odd number of layers, wherespin-orbit splitting is present.

� The four studied TMDs were found to have mainintersubband transition energies for the conduc-tion and valence subbands, which densely cover thespectrum range of wavelengths from � = 2 �m to30 �m (~! = 40 to 700 meV), for N = 2 to 7 lay-ers. This allows tailoring structures of a speci�cmaterial, appropriate type of doping, and numberof layers for a particular device application, fromIR to the THz range.

� Two contributions to the absorption line-shapebroadening are identi�ed. The �rst, broadeningdue to intersubband phonon relaxation, is found toproduce a meV limit to the intersubband linewidth.This is in contrast to III-V quantum wells, wherephonon broadening is found to be more damagingto the intersubband transition line quality factor[30]. A second, elastic contribution to the linebroadening caused by the di�erent 2D masses ofcarriers in consecutive subbands yields a thermalbroadening of the order of kBT . Similarly to inho-mogeneous broadening, this e�ect can be reducedby coupling the transition in the �lm to a standingwave of light in a high-Q resonator.

Finally, we propose a speci�c design of van der Waalsmultilayer structure utilizing the intersubband transi-tions in atomically-thin �lms of TMDs. The sketchin Fig. 16 depicts the band con�guration of a few-layer transition-metal dichalcogenide �lm, encapsulatedby hexagonal boron nitride (hBN) and placed betweentwo graphene electrodes. Applying a bias (and possiblyalso gate) voltage between the two electrodes results ina shift of the Dirac points relative to each other, and al-lows for the alignment of the Dirac point of the \top"graphene electrode with the lower-energy subband in theTMD, while keeping the Fermi level in graphene abovethe higher-energy subband. The carriers can then tun-nel from the graphene electrode into the higher-energysubband. Once in the excited subband state, the carriercan undergo an intersubband transition, emitting lightpolarized in the out-of-plane direction, followed by tun-neling to the second graphene electrode from the bottomsubband state.

A potentially more favorable realization of the aboveprocess which avoids carrier loss directly from the sec-ond subband, or carrier tunneling into the bottom sub-band, involves using ABC-stacked few-layer graphene.The band structure of ABC few-layer graphene has a VanHove singularity in its density of states at the edge be-tween conduction and valence bands. Aligning the VanHove singularities of two such electrodes with the sec-ond and �rst subbands, respectively, would enable oneto achieve preferential injection and extraction of carri-ers into/from the TMD �lm, thus o�ering a new way toproduce functional optical �ber cables.

FIG. 16. Proposed device application for intersubband tran-sitions in few-layer TMDs. (a) Few-layer TMDs encapsulatedbetween two hexagonal boron nitride (hBN) crystals and twographene (G) electrodes with an applied bias voltage betweenthem. The applied bias voltage allows to realize light emis-sion through intersubband transitions in the few-layer TMDsystem, by carriers tunneling between the two graphene elec-trodes. (b) An alternative realization using few-layer ABC-stacked graphene instead of monolayer, utilizing the Van Hovesingularity in the density of states. The bias voltage alignsthe Van Hove singularities near the second and �rst subbands,making the desired emission process more favorable.

The proposed \LEGO"-type design of IR/THz emit-ting materials has potential for implementation as partof a composite optical �ber, where the coupling to theout-of-plane polarized photon would be supported by thewave-guide mode.

ACKNOWLEDGMENTS

The authors acknowledge funding by the EuropeanUnion’s Graphene Flagship Project, ERC Synergy GrantHetero 2D, EPSRC Doctoral Training Centre, GrapheneNOWNANO, and the Lloyd Register Foundation Nan-otechnology grant, as well as support from the N8 Po-laris service and the use of the ARCHER supercomputer

18

(RAP Project e547) and the Tianhe-2 supercomputer(Guangzhou, China). The authors would like to thankF. Koppens, K. Novoselov, R. Gorbachev, M. Potemski,R. Curry, C. Kocabas, S. Magorrian and A. Ceferino forfruitful discussions.

Appendix A: Symmetry constraints for the bilayerHamiltonians

For N = 2 (bilayer), the Hamiltonian must be invari-ant under spatial inversion P, x! �x mirror symmetryD(�v), and time reversal T . For the conduction-bandmodel about the Q point, this gives the conditions [49]

PH�2 (k)P�1 = H��2 (�k) (A1a)

D(�v)H�2 (kx; ky)D�1(�v) = H��2 (�kx; ky) (A1b)

T H�2 (k)T�1 = H��2 (�k): (A1c)

We have P = �1s0, D(�v) = �0s1 and T2 = �i�0s2C,where �j (sj) are Pauli matrices acting on the layer (spin)subspace, and C represents complex conjugation. As a re-sult, the most general valley-spin structure for the bilayerHamiltonian at the �Q valley is

H�2 (k) =3X

i;j=0

A�ij(k)�isj ; (A2)

where the symmetry constraints (A1a)-(A1c) require that

A�i0(kx; ky) = A��i0 (�kx; ky) = A

�i0(kx;�ky); i = 0; 1;

(A3a)

A�20(kx; ky) = A��20 (�kx; ky) = �A

�20(kx;�ky); (A3b)

A�31(kx; ky) = A��31 (�kx; ky) = �A

�31(kx;�ky); (A3c)

A�32(kx; ky) = �A��32 (�kx; ky) = A

�32(kx;�ky); (A3d)

A�33(kx; ky) = �A��33 (�kx; ky) = A

�33(kx;�ky): (A3e)

One can check that, for N = 2, Eq. (20) corresponds

to A�00(k) =E+" (k)+E

+# (k)

2 , A�33(k) = �

E+" (k)�E+# (k)

2 ,A�10(k) = t0 + �t1kx + t2k2x + u2k2y and A�20(k) = �u1ky,and that these terms meet the symmetry requirements.Furthermore, we carried out �ttings to DFT data us-ing the additional spin-orbit terms A�32(k) = �kx andA�31(k) = �ky. The �ttings give j�j; j�j; ju1j � jt1j;hence, we conclude that these terms can be neglected.

For the interlayer hopping used in the HkpTB modelfor the valence band near �, setting N = 2 in Eq. (3)and using a basis ordering similar to that of Eq. (5), we

have T = �0�0C, P = �1�3 and D(�v) = �0�0, where�i act on the band (v and w) subspace. The symmetryconditions require

Re t�(k) = Re t�(�k) = Re t�(�kx; ky); (A4a)

Im t�(k) = �Im t�(�k) = Im t�(�kx; ky); (A4b)

tvw(k) = tvw(�k) = tvw(�kx; ky); tvw 2 R: (A4c)

Appendix B: Spin-orbit-coupling induced interbandcoupling at the � point valence bands

Here, we analyze the role of spin-orbit coupling andcoupling to distant bands in determining parameters forthe valence-band HkpTB model.

The spin-orbit coupling is given by ĤSO = �L̂�Ŝ, whereL̂ and Ŝ are the orbital and spin angular momentumoperators. This can also be written in terms of the ladderoperators L� = Lx � iLy and S� = (Sx � iSy)=2 as

ĤSO = �(LzSz + L+S� + L�S+); (B1)

whereL�S� describe a spin ip with correspondingchange in orbital angular momentum projection. Theseterms couple the v and w bands with the bands v1 andv3 [Fig. 2(d)], which in the absence of SO coupling aredoubly degenerate. Band v1 (E00 Irrep of C3d) has ba-sis functions which are odd under z ! �z (metal d�1orbitals being the dominant component [27], as well aschalcogen p�1). Band v3 (E0 Irrep of C3d) has basisfunctions even under z ! �z (metal d�2 orbitals andchalcogen p�1). Including SO coupling results in thesplitting of these bands into new bands denoted by theorbital and spin angular momentum projections alongẑ, v1(�3=2); v3(�3=2), and v1(�1=2); v3(�1=2), corre-sponding to total angular momentum projections of Jz =�3=2 and Jz = �1=2, respectively.

The v-band belongs to the one-dimensional A01 Irrep(even under z ! �z, with metal d0 dominant and chalco-gen p0), and has Lz = 0 and sz = �1=2. Similarly, thew-band belongs to the one-dimensional A002 Irrep, and isdominated by the odd (under z ! �z) chalcogen p0 or-bitals, giving two states with Lz = 0 and sz = �1=2.

Therefore, in the bilayer, where z ! �z symmetry isbroken, the v and w-bands can couple to v1 and v3 bands,with the appropriate spin-ip terms. In the second-orderperturbation theory, this coupling produces correctionsto the on-site energy

�� =X

Lz;szi=1;3�=v;w

jhvi(Lz; sz)j�L�S�j�(Lz = 0; sz = �1=2)ij2

E� � Evi(Lz;sz):

(B2)Note that these corrections are the same for both spincomponents of the v or w bands, with only one of theterms L�S� contributing for a given spin state.

19

An additional SO induced interband coupling with aspin-ip may be present in the multilayer case, a�ectingthe interlayer coupling

Ĥ 0SO = �ẑ � (k� S) = i�(S�k+ � S+k�); (B3)

where the pre-factor � is related to the gradient of theinterlayer pseudo-potential � / @zV , and we de�nedk� = kx � iky. In contrast to the previous coupling,this coupling has a k-dependence, which a�ects the dis-persions. The coupling in Eq. (B3) is odd under spatialinversion. Due to the 2H-stacked bilayer having spatialinversion symmetry, the coupling is non-zero only be-tween di�erent bands in the two layers. In second-orderperturbation theory, we get a nominal rede�nition of the2D mass used in the HkpTB model, by adding the term

��(k) =X

vi�=v;w

jhvij�S�k�j�ij2

E� � Evi= ��k2; (B4)

with �� a �tting parameter.

Appendix C: Spin-split bands at the Brillouin zoneedge for odd number of layers

The e�ective Q-point Hamiltonians H�NQ(k) for N oddcan be split into two decoupled blocks of di�erent spinprojection as H�NQ(k) = diagfh

�;"N (k); h

�;#N (k)g, where

the blocks have the alternating N �N matrix form

h�;sN (k) =

0

BBBBBB@

"0(k) + s��(k) t� (k) 0 0 � � � 0t�� (k) "0(k)� s��(k) t�� (k) 0 � � � 0

0 t� (k) "0(k) + s��(k) t� (k) � � � 00 0 t�� (k) "0(k)� s��(k) � � � 0...

......

.... . . t� (k)

0 0 0 � � � t�� (k) "0(k) + s��(k)

1

CCCCCCA; (C1)

and we have de�ned

"0(k) =E+" (k)+E

+# (k)

2 ; (C2a)

�(k) =E+" (k)�E

+# (k)

2 : (C2b)

De�ning the even-dimensional (N � 1)� (N � 1) matrix

~h�;sN�1(k) =

0

BBBB@

"0(k)� s��(k) t�� (k) 0 � � � 0t� (k) "0(k) + s��(k) t� (k) � � � 0

0 t�� (k) "0(k)� s��(k) � � � 0...

......

. . . t� (k)0 0 � � � t�� (k) "0(k) + s��(k)

1

CCCCA; (C3)

the eigenvalues " of (C1) are given by a secular equation

det f"� h�;sN g = ["� "0(k)� s��(k)] det f"� ~h�;sN�1g

� jt� (k)j2 det f"� h�;sN�2g

= ["� "0(k)� s��(k)] det f"� ~h�;sN�1g

� jt� (k)j2�

["� "0(k)� s��(k)] det f"� ~h�;sN�3g

� jt� (k)j2 det f"� h�;sN�4g

�= � � � :

(C4)

Using the fact that det f"� h�;s1 g = "� "0(k)� s��(k),we can continue expanding Eq. (C4) to obtain

det f"� h�;sN g = ["� "0(k)� s��(k)]

�

N�32X

m=0

(�1)m jt� (k)j2m det f"� ~hN�(2m+1)g

+ (�1)N�1

2 jt� (k)jN�1

!

;

(C5)

which explicitly shows that [" � "0(k) � s��(k)] is anoverall factor, and thus " = "0(k) + s��(k) � "�s (k) is

20

always an eigenvalue, regardless of the (odd) value of N .For a given � , the di�erent s quantum numbers give twospin-split monolayer dispersions "�s (k) about the �Cm3 Q(m = 0; 1; 2) points, corresponding to the features ob-served in Fig. 10. The fact that this prediction is veri�edin the DFT band structures clearly con�rms the validityof our hybrid model.

For large odd N , nearly spin-degenerate bands growdenser on either side of the spin-split bands "�s (k) withoutcrossing them, as shown in Fig. 17(a). The reason for thisbecomes clear when we take the bulk limit, and �nd thatthe spin-split states form the band edges around a centralgap in the subband structure. This is shown in Fig. 17(b).Indeed, in the limit of large N the Hamiltonian (C1)corresponds to the bulk Hamiltonian at kz = �=c, since"0(k) = "0(k; kz = �c ) [see Eq. (23)].

FIG. 17. (a) Subband structure of 101-layer WS2 near �Q(kx = 0), along the �K line. Spin-up (-down) bands areshown with solid (dashed) curves. The spin-split bands "�s (k),pinned in the middle of the odd N subband structure, areshown in blue and red. (b) Bulk band structure for WS2along the 3D BZ line indicated in the inset of Fig. 9. Blueand red dots mark the position of the spin-split bands in theBrillouin zone.

Appendix D: Electron-phonon coupling for LOphonon in multilayer system

In this appendix, we derive the expression used for theelectron-phonon coupling with LO phonon in a multilayersystem. As described in the text, we treat the LO phononin each layer as independent and degenerate. However, in

the LO phonon case, the generated electrostatic potentialdue LO phonon in one layer interacts with the electronsin all the other layers in the system, following similarsteps as in Ref. [50].

Within a monolayer, the LO phonon-induced in-planepolarization is given by its in-plane Fourier component,

Pq(z) =eZ�(q)A

uq�(z); (D1)

where Z is the Born e�ective charge on the metal andchalcogens, A is the unit-cell area, uq =

q~

2MrNcell!LOê

is the phonon-induced atomic displacement in the direc-tion connecting the metal and chalcogens in the unit cell,withMr the reduced mass of the metal and chalcogens, Nthe number of unit cells in the sample, and !LO the LOphonon frequency. �(q) is the dielectric function char-acterizing the response of the material to the phononinduced electric �eld.

The induced charge density in the layer is given by� = �r �P, with the Fourier component

�q = �iq � Pq: (D2)

The potential resulting from the charge distributionis given by Poisson’s equation r2� = �4��. Fourier-transforming in three dimensions gives,

�q(k) =�4�ieZ�(q)A

q � uqq2 + k2

; (D3)

where k is the Fourier parameter in the z direction. In-verse Fourier transforming in k gives the z dependenceof the potential with in-plane Fourier component q

�q(z) = �i2�eZuq�(q)A

e�qjzj: (D4)

The electron-phonon coupling for an electron localizedin an isolated monolayer is given by g(q) = e�q(0) =�i 2�e

2Zuq�A . This form of the coupling is similar to the

form derived in Refs. [28 and 43], where the polarizabilityof a two-dimensional dielectric was taken into accountby the replacement �(q)! 1 + r�q, with r� the screeninglength in the material. For multilayer 2H-stacked TMDs,as the polarization in subsequent layers alternates in itssign, the resulting electrostatic potential also alternatesin its sign.

Appendix E: Electron-phonon coupling for ZOphonon in multilayer system

The atomic vibrations for the ZO optical phonon moderesult in a polarization in the out-of-plane direction dueto the opposite motions of the metal and two chalcogens,and the �nite Born e�ective charges in the z-direction.The interaction energy in the multilayer system between

21

charges and phonon-induced out-of-plane polarizations inall layers is given by [51]

Eint =X

n;m

Zd2rd2r0

�n(r)Pz;m(r0)d(n�m)�r3

;

�r = [(r� r0)2 + d2(n�m)2]1=2;

(E1)

where �n(r) is the charge density on layer n, Pz;m(r0) isthe out of plane polarization in layer m caused by theZO optical phonon, d the interlayer separation, and wesum over all layer pairs. Fourier transforming the chargedensity and polarization in the in-plane momentum com-ponents gives

�n(r) =Z

d2q(2�)2

eiq�r�m(q); (E2)

and similarly for the polarization. The interaction energythen takes the form,

Eint =X

n;m

Zd2rd2r0

d(n�m)�r3

�Zd2qd2q0

(2�)4eiq�reiq

0�r0�n(q)Pz;m(q0):

(E3)

De�ning the new variables ~r = r � r0, R = r + r0, andintegrating over R gives �q;�q0 ,

Ee-ph =X

n;m

Zd2~r

Zd2q

(2�)2d(n�m)

�~r3eiq�~r

� ��n(q)Pz;m(q);

(E4)

where in the last row we used ��n(q) = �n(�q), since thedensity is real. Carrying out the integration over ~r gives

Ee-ph =0X

n;m

2�(n�m)jn�mj

Zd2q

(2�)2e�qdjn�mj��n(q)Pz;m(q);

(E5)where the prime over the sum means that the summationexcludes the term with n = m. Quantizing the phononpolarization and the carrier density gives

��n(q) = eX

k

cyk;nck+q;n;

Pz;m(q) =eZzA

r~

2NcellMr!(a�q;m + ayq;m);

(E6)

where ck;n (cyk;n) is the annihilation (creation) operatorfor an electron in state k in layer n, and aq;m (ayq;m) isthe annihilation (creation) operator for a phonon with in-plane wave vector q. The phonon-induced polarizationis given, similarly to the LO phonon case, by the Borne�ective charge and the phonon displacement.

The electron-phonon interaction Hamiltonian is thengiven by

He-ph =2�e2ZzA

r~

2NcellMr!ZO

0X

n;m

X

k;q

n�mjn�mj

e�qdjn�m