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transition metal dichalcogenides ( tmdcs ) are layered materials with the chemical composition mx@xmath1 , where m  is a transition metal element ( such as molybdenum or tungsten ) , and x  is a chalcogen ( sulfur , selenium , or tellurium ) . the interest to semiconducting tmdcs has been sparked by the recent discovery of the monolayer mos@xmath1 being a direct - gap semiconductor , in contrast to its bulk indirect - gap counterpart @xcite . atomically thin tmdc monolayers can be extracted from bulk crystals by exfoliation , similarly to graphene @xcite , or grown by molecular beam epitaxy or chemical vapor deposition @xcite . the optical gap in the visible light range @xcite and tightly bound excitons @xcite , make them quite promising for optical applications @xcite . a unique feature of tmdcs is the so - called spin - valley locking @xcite . the conduction and valence band extrema are located at the two inequivalent @xmath2 points ( valleys ) of the hexagonal first brillouin zone . absence of inversion symmetry and strong spin - orbit interaction , originating from @xmath3-orbitals of the metal atoms , leads to spin splitting of the bands . the sign of the splitting is opposite in the two valleys , as required by the time - reversal symmetry , so the lowest - energy electron and hole states have opposite spin projections in the opposite valleys , as shown schematically in fig . [ fig : tmdcbands ] . it is important that the two valleys can be addressed separately by optical means . namely , in each valley optical transitions with only one of the two in - plane circular polarizations are allowed  @xcite . this opens an exciting perspective of manipulating the spin / valley degree of freedom optically @xcite . ( color online . ) a schematic view of the conduction and valence band extrema of a tungsten - base tmdc monolayer . the solid arrows show the spin projections . in molybdenum - based compounds , the spin projections in the conduction band are the opposite . the dashed arrows represent circularly polarized optical transitions for @xmath4 excitons in each valley . , width=302 ]    the large spin splitting in the valence band ( a fraction of ev ) gives rise to two prominent features in the absorption spectrum , called @xmath4 and @xmath5 excitons , formed by a hole in the upper or lower spin - split valence band , respectively , and an electron in the conduction band . the spin splitting in the conduction band is much weaker ( on the mev scale ) ; moreover , its sign depends on the material : in molybdenum - based compounds , the top valence band and the lowest conduction band have the same spin projection in the same valley , while in tungsten - based ones , they are opposite ( fig .  [ fig : tmdcbands ] ) @xcite . this implies that the lowest - energy @xmath4 excitons in wx@xmath1 involve electronic states either from the same valley but with opposite spins , or with the same spin but from different valleys . as the photons , polarized in the plane , couple only to the orbital degrees of freedom and carry a very small momentum , they are decoupled from such excitons , which therefore have been named `` dark '' . this picture provided a natural explanation of the experimentally observed rise of the luminescence intensity in wse@xmath1 with increasing temperature in terms of the increasing thermal population of the higher - energy bright @xmath4  excitons @xcite . at sufficiently low temperatures , several peaks at energies below the neutral and charged bright @xmath4 excitons have been seen in the wse@xmath1 luminescence spectrum @xcite . these observations pose an important question about existence of possible mechanisms for radiative decay of the dark excitons . while the radiative decay of bright excitons in tmdcs fits in the standard picture for 2d excitons @xcite and has been described in several theoretical works @xcite , the dark exciton radiative decay requires unlocking the spin and the valley degrees of freedom . namely , for intervalley dark excitons , a large momentum should be provided to the electron or to the hole without flipping its spin , which requires a third body ( a phonon , an impurity , another electron or a hole ) . for intravalley dark exciton decay , a spin flip is required . investigation of this latter possibility is the subject of the present work .    in this paper , we give an estimate of the intravalley dark exciton radiative decay rate due to several spin - flip mechanisms . one such mechanism could be the bychkov - rashba effect  @xcite , as mentioned in ref . it requires breaking of the crystal symmetry with respect to reflection in the monolayer plane . this can be achieved by applying an external electric field , perpendicular to the plane ; the corresponding rashba coefficient has been calculated in ref . the reflection symmetry can also be broken if the dielectric environments on the two sides of the monolayer are different . having not found any information on the magnitude of this effect in the literature , we make our own crude estimate ; for a tmdc monolayer on a glass - like substrate with the empty space above , we obtain an equivalent electric field to be roughly of the order of 0.1  v /  . however , even for this relatively high field , the resulting radiative rate is about @xmath6 of the bright exciton decay rate  @xmath7 , which makes the rashba mechanism totally negligible with respect to other relaxation processes . the main reason for such smallness is that rashba spin - flip amplitude is proportional to the electron momentum , and radiative decay is possible only in a narrow radiative region of small momenta . another way to flip the electron spin is to apply an in - plane magnetic field and use the zeeman effect . it is momentum - independent , and for a quite high but still realistic field of 30  t we obtain a radiative rate exceeding @xmath8 . finally , we analyzed the intrinsic radiative decay mechanism due to the interband spin - flip dipole moment perpendicular to the monolayer plane , mentioned in refs . in fact , it is the presence of this dipole moment that gives rise to bychkov - rashba coupling in a perpendicular electric field  @xcite , so its magnitude can be deduced from the estimates of ref . @xcite for the rashba coefficient . this enabled us to estimate the associated radiative decay rate as @xmath9 . however , we also found that the same out - of - plane interband dipole gives rise to a coulomb local - field term which lifts the double degeneracy between the dark excitons made of electronic states from the two valleys . this local - field effect is analogous to the exchange energy shift of the @xmath10  excitons in semiconductor quantum wells  @xcite , and produces an energy splitting which we very roughly estimate as about 10  mev . its precise evaluation requires a microscopic treatment on the atomic scales , similar to that in ref . crucially , the whole oscillator strength of the interband spin - flip dipole goes into the higher - energy component . thus , we find that the dark intravalley @xmath4  exciton has , in fact , two components , one really dark and the other one about 1001000 times darker than the bright exciton , so it can be called `` dim '' . because of the splitting , at low temperatures the emission from the dim exciton should be suppressed by an activation factor , analogously to that from the bright one . note that the dark component can be made decay by applying an in - plane magnetic field , as mentioned above , and the corresponding rate can be made comparable to that of the dim component for a sufficiently high field . the paper is organized as follows . in sec . [ sec : model ] we describe the effective model for electrons in the tmdc monolayer , the excitonic states , and specify how different spin - flip mechanisms enter the model . in sec . [ sec : polarization ] , we discuss spin - mixing of excitonic states and calculate the `` mechanical '' susceptibilities of the tmdc monolayer , which determine the exciton coupling to the macroscopic electromagnetic field . in sec . [ sec : radiative ] we study the effect of the exciton coupling to the electromagnetic field and compute the radiative energy shifts and decay rates for the excitons . finally , in sec . [ sec : conclusions ] we summarize our results and discuss some of their implications and perspectives . some details of calculations are presented in several appendices . we set @xmath11 throughout the paper , and write the electronic hamiltonian of a tmdc monolayer as @xmath12 its three terms will be discussed in the following subsections . we assume the monolayer to be in the @xmath13 plane , the three - dimensional position vector @xmath14 represented @xmath15 , so that @xmath16 denotes the position in the monolayer plane . the first term in eq . ( [ h= ] ) , @xmath17 , is the usual effective two - band hamiltonian  @xcite @xmath18 written in terms of four - component column operators @xmath19^t$ ] , where @xmath20 labels the two valleys @xmath2 , and @xmath21 is the @xmath22 matrix , written in the block form , @xmath23.\ ] ] here @xmath24 is the third pauli matrix in the spin subspace , @xmath25 is the band gap , @xmath26  is the velocity matrix element between the band extrema bloch functions ( which can be made real by an appropriate choice of the relative phase between the conduction and valence band bloch functions ) . the coefficients @xmath27 are related to the electron and hole effective masses @xmath28 as @xmath29 typically , @xmath30 , @xmath31 , @xmath32 for mox@xmath1 and @xmath33 for wx@xmath1 ( @xmath34 being the free electron mass ) . @xmath35 in eq . ( [ htauk= ] ) is half of the spin splitting in the conduction / valence bands . we distinguish the @xmath2 valleys by assuming the valence band states to originate predominantly from the metallic orbitals with the @xmath36  projections of the angular momentum equal to @xmath37 at the @xmath2 point , respectively . this fixes @xmath38 , as well as the valley - dependent optical selection rules : the left / right circular polarisation can be absorbed in the @xmath2 valley , respectively , as the conduction band states originate mostly from zero - angular - momentum metallic orbitals and the angular momentum is conserved modulo  3 due to the three - fold crystal rotation symmetry . typically , @xmath39 in mox@xmath1 and @xmath40 in wx@xmath1 . @xmath41 is much weaker , usually a few tens of mev , with the exception of mos@xmath1 where it is extremely weak , @xmath42 . the sign of @xmath41 depends on the material : @xmath43 ( @xmath44 ) in molybdenum - based ( tungsten - based ) compounds . because of this , the lowest - energy interband transition in tungsten - based compounds involves either a spin flip or valley switching .      the second term in eq . ( [ h= ] ) , @xmath45 , represents the coulomb interaction between electrons , @xmath46 \hat\psi_\tau({\mathbf{r } } ) . \label{rhomatrix=}\end{aligned}\ ] ] here each element of the block @xmath47 matrix should be understood as the unit matrix in the spin space . we included only the intra - valley part of the electron density @xmath48 , as it is the long - range part of the coulomb interaction ( on the scale of lattice constant ) which is responsible for the exciton formation .    the contribution of the diagonal matrix elements in eq . ( [ rhomatrix= ] ) corresponds to the direct coulomb attraction between the electron and the hole , which gives rise to the excitonic bound state . the wave function of the electron - hole relative motion , @xmath49 , is obtained from the corresponding schrdinger equation , @xmath50\phi({\mathbf{r}})= e\,\phi({\mathbf{r}}),\ ] ] where @xmath51 is the reduced mass . the lowest eigenvalue @xmath52 of eq . ( [ excschrodinger= ] ) defines the exciton binding energy  @xmath53 . as we are working in the parabolic approximation for the electronic dispersion and assume the electron and the hole masses to be spin - independent , the wave function @xmath54 and the binding energy  @xmath53 are the same for all spin and valley configurations of the electron and the hole . we do not specify the explicit form of the pair interaction potential @xmath55 , as we will not be solving eq . ( [ excschrodinger= ] ) . it is known that because of strong dielectric confinement in the tmdc monolayer , the interaction potential is not @xmath56 , so the bound state wave functions do not have a hydrogenic form @xcite . instead , we will assume @xmath54 and @xmath53 to be known and treat them as input parameters . in fact , we will not need the whole wave function @xmath54 , but only its value @xmath57 at the coinciding electron and hole positions . by the order of magnitude , @xmath57 is the inverse radius of the excitonic bound state . estimating it as @xmath58 with the binding energy @xmath59 @xcite , gives @xmath60 , while from the measured diamagnetic shift of the exciton energy one infers values slightly below @xmath61 @xcite . the contribution from the off - diagonal matrix elements in eq . ( [ rhomatrix= ] ) corresponds to the exchange interaction between the electron and the hole . it will be taken into account in sec . [ sec : radiative ] in the framework of macroscopic electrodynamics .      the third term in eq . ( [ h= ] ) , @xmath62 , is the spin - flip , which arises when the reflection symmetry in the  @xmath36 direction is broken  @xcite . we write this term as  @xcite @xmath63,\end{aligned}\ ] ] here we assumed that the reflection symmetry is broken by an external perpendicular electric field @xmath64 , introduced explicitly . then , by definition , the operator multiplying @xmath64 is nothing but the @xmath36  component of the electric dipole moment operator . if there is no external electric field , but the reflection symmetry is broken by some other mechanism ( e.  g. , van der waals interaction with a dielectric substrate ) , the coupling hamiltonian still has the form  ( [ hrashba= ] ) which is fixed by the symmetry , but instead of @xmath65 its strength is determined by another parameter having the dimensionality of energy . for example , the parameter @xmath66 from ref . @xcite is related to @xmath65 as @xmath67 . note that the parameter @xmath68 can be made purely real or purely imaginary by an appropriate spin rotation around the @xmath36  axis . the peculiar valley structure of the matrix @xmath69 follows from the total ( orbital plus spin ) angular momentum conservation : in @xmath70 ( @xmath71 ) valley only spin - up ( spin - down ) states in the valence band can couple to only spin - down ( spin - up ) in the conduction band , the corresponding total angular momentum change being @xmath72 . the hamiltonian @xmath62 produces interband spin - flip transitions . combining it with the off - diagonal terms in eq . ( [ htauk= ] ) in the first - order perturbation theory , one can project the spin - flip terms on the conduction and valence bands ( appendix  [ app : diag_hamiltonian ] ) . upon projection , they acquire the familiar rashba - type form  @xcite , linear in momentum  @xmath73 : @xmath74_z{\mathop{\mathrm{im}}}{d}_z\right\ } \left[\begin{array}{cc } 1 & 0 \\ 0 & -1 \end{array}\right].\ ] ] this form matches the one studied in ref . @xcite where the rashba coupling strength was parametrized by @xmath75 , related to @xmath65 as @xmath76 . the parameter @xmath75 was estimated in ref . @xcite for mos@xmath1 , mose@xmath1 , ws@xmath1 , wse@xmath1 monolayers suspended in vacuum and subject to an external perpendicular electric field . this enables us to extract @xmath77 of about @xmath78 , @xmath79 , @xmath80 , and @xmath81 , for mos@xmath1 , mose@xmath1 , ws@xmath1 , and wse@xmath1 , respectively . we are not aware of any estimates in the literature for the rashba coupling strength induced by a dielectric substrate . therefore , we make our own crude estimate of the effective electric field in appendix  [ app : substrate ] , which gives @xmath82 for a wx@xmath1 monolayer lying on a glass - like substrate with a dielectric constant @xmath83 and the empty half - space above . finally , we note that in addition to rashba effect in a perpendicular electric field , the spin can be flipped by the zeeman effect if an in - plane magnetic field @xmath84 is applied . this effect is taken into account straightforwardly by adding @xmath85 to the diagonal matrix elements in eq . ( [ htauk= ] ) , where @xmath86 is the bohr magneton and @xmath87 is the in - plane gyromagnetic ratio for electrons in the conduction and valence band . the hamiltonian of interaction between the electrons and the transverse photons , which are described by the long - wavelength vector potential @xmath88 , is usually obtained from the requirement of gauge invariance by performing the peierls substitution in eq . ( [ htauk= ] ) : @xmath89 . however , besides coupling to photons via gauge potentials , electrons can couple directly to electric and magnetic fields . such terms are gauge invariant and can not be deduced from the bare electronic hamiltonian in the envelope function approximation , as they correspond to the effect of electromagnetic field on the microscopic bloch functions .    in the previous subsection , we have already presented such terms , corresponding to rashba and zeeman effects . it is crucial for our analysis that , once the dipole operator in eq . ( [ hrashba= ] ) is determined , it can also describe the electronic coupling to the photon field . namely , @xmath64 can be understood not just as a static external field , but as the electric field of photons as well . in this case , one should write @xmath90 and put it inside the space integral . the corresponding hamiltonian can then describe interband photon absorption / emission , accompanied by the spin flip ( the existence of such coupling was briefly mentioned in ref . the main quantity which we use to describe the interaction of excitons with light is the susceptibility @xmath91 , defined as the mechanical response of the excitonic polarization to the electric field at the frequency @xmath92 and the in - plane wave vector  @xmath93 . ( the term `` mechanical '' implies that the only the direct coulomb interaction is taken into account for the moment ; the exchange interaction will be included in the framework of macroscopic electrodynamics . ) we define the polarization @xmath94 as the dipole moment per unit area . it is a three - dimensional vector , whose @xmath36 component is defined by eq . ( [ hrashba= ] ) , while the in - plane components are most conveniently defined via the in - plane current @xmath95 . the current operator is obtained in the usual way from @xmath96 , where we retain only the leading terms at @xmath97 : @xmath98 \hat\psi_\tau({\mathbf{r}}),\ ] ] where @xmath99 are the unit vectors in the corresponding directions . the kubo formula gives    [ chidef=]@xmath100    where @xmath101 and @xmath102 are the ground and excited electronic states , @xmath103 is their energy difference ( that is , the energy of the excitation  @xmath104 ) , the indices @xmath105 , and we omitted the non - resonant terms . in the translationally invariant case , the susceptibilities depend only on the difference @xmath106 , so @xmath107 here the in - plane components @xmath108 are labeled by the greek indices @xmath109 , while the latin indices run over all three dimensions , @xmath110 .    in the following , we evaluate @xmath91 using the eigenstates @xmath102 which are the lowest - energy states of the intravalley excitons , whose spins are mixed by the rashba coupling . the effect of the zeeman coupling is treated analogously , the result is summarized in the end of sec . [ ssec : susceptibility ] . we start by constructing the zero - approximation states , i.  e. , those in the absence of the rashba coupling . first , each intravalley exciton can be characterized by the index  @xmath111 , indicating the valley in which the electronic transition takes place . it gives rise to two species of intravalley excitons , which can be distinguished by the valley index @xmath20 . further , in each valley the conduction and valence bands are spin split , so the excitonic states can be labeled by a pair of spin indices @xmath112 referring to the conduction and the valence band , respectively ( the hole spin is given by @xmath113 ) . finally , the excitonic state is characterized by a center - of - mass momentum @xmath93 . so , the zero - approximation excitonic states are written as @xmath114 here , @xmath115 is the sample area , @xmath116 is the center - of - mass coordinate , @xmath117 , and @xmath49 is the normalized wave function of the relative electron - hole motion , corresponding to the lowest bound state , which was discussed in sec . [ ssec : coulomb ] . states  ( [ state0= ] ) are normalized as @xmath118 and have the energies @xmath119 where we introduced the excitonic mass @xmath120 . the rashba coupling flips electron and hole spins and mixes the excitonic states with different @xmath121 . as @xmath122 , we neglect the spin - flip in the valence band , and consider it only in the conduction band . it can be described by an effective @xmath47 hamiltonian in the basis of @xmath123 and @xmath124 , obtained by the projection of the electronic rashba hamiltonian  ( [ rcvb= ] ) on the excitonic states : @xmath125,\ ] ] where the excitonic rashba coupling constant is given by @xmath126 description of spin flip in terms of the purely excitonic effective hamiltonian  ( [ rashbaex= ] ) assumes that the spin flip does not disturb the electron - hole relative motion , which is guaranteed by the condition @xmath127 @xcite . the @xmath47 hamiltonian ( [ rashbaex= ] ) can be diagonalized exactly ; however , we are interested only in small momenta  @xmath128 , so perturbative expressions will suffice for us . up to second order in @xmath129 , the mixed - spin eigenstates are given by    [ mixedspin= ] @xmath130 . to the same order , their energies are given by @xmath131 the states with @xmath132 from the @xmath70 valley and those with @xmath133 from the @xmath71 valley are called the bright @xmath4 excitons . those with the opposite @xmath121 have a higher energy ( due to the @xmath134 term ) and are called the bright @xmath5 excitons . their energies are @xmath135 the states obtained by flipping the spin in the conduction band can be called the dark @xmath4 and @xmath5 excitons , their energies are @xcite @xmath136      we calculate the susceptibility @xmath91 from eqs . ( [ chidef= ] ) using the matrix elements @xmath137 and @xmath138 , which , in turn , are obtained from expressions  ( [ mixedspin= ] ) for the mixed - spin eigenstates  @xmath102 and from the matrix elements between the zero - approximation states  ( [ state0= ] ) following from the definitions of the in - plane current and the @xmath36-dipole moment operators [ eqs . ( [ current= ] ) and  ( [ hrashba= ] ) ] . namely , for the in - plane current we have @xmath139 the nonzero matrix elements of the @xmath36-dipole moment operator between the ground state and the zero - approximation states are    @xmath140    summation over @xmath141 in combination with the independence of the energies @xmath142 on the valley index  @xmath111 restores the in - plane isotropy in the final expressions for the susceptibility , written to the second order in  @xmath68 :    [ chi=]@xmath143 ^ 2}{e_a^2 } \left[\frac{1-w_{\mathbf{q}}}{e_a^\mathrm{b}({\mathbf{q}})-\omega } + \frac{w_{\mathbf{q}}}{e_a^\mathrm{d}({\mathbf{q}})-\omega}\right ] \nonumber \\ & \hspace*{1.5cm}{}+2\delta_{\alpha\beta } \frac{[ev\phi(0)]^2}{e_b^2}\left [ \frac{1-w_{\mathbf{q}}}{e_b^\mathrm{b}({\mathbf{q}})-\omega } + \frac{w_{\mathbf{q}}}{e_b^\mathrm{d}({\mathbf{q}})-\omega}\right ] , \label{chinplane=}\\ & \chi_{{z}\alpha}({\mathbf{q}},\omega)=-\chi_{\alpha{z}}({\mathbf{q}},\omega)=\nonumber \\ & \quad=\frac{iq_\alpha\lambda_\mathrm{ex}^*}{\delta_c}\ , \frac{ev\phi^2(0)}{e_a}\,d_z \left[\frac{1}{e_a^\mathrm{b}({\mathbf{q}})-\omega } -\frac{1}{e_a^\mathrm{d}({\mathbf{q}})-\omega}\right],\\ & \chi_{zz}({\mathbf{q}},\omega)=2|d_z|^2\phi^2(0)\ , \frac{1}{e_a^\mathrm{d}({\mathbf{q}})-\omega}.\end{aligned}\ ] ]    here we omitted the infinitesimal imaginary part in the denominator to keep the formulas more compact and approximated @xmath144 in the non - resonant prefactor , neglecting the dispersion and the conduction band splitting . note also that @xmath145 as well as @xmath57 are real quantities .    from eqs . ( [ chi= ] ) , we see the crucial role played by the quantity @xmath146 , introduced in sec . [ ssec : excitons ] . it represents the spectral weight transferred from the bright to the dark excitons by the rashba coupling . let us estimate its order of magnitude . taking the numerical values typical of wse@xmath1 , @xmath147 , the experimentally determined splitting @xmath148  @xcite , and @xmath149 from ref . @xcite for a quite strong perpendicular electric field @xmath150 , we still obtain a very small value of @xmath151 . if instead of the perpendicular electric field  @xmath64 , an in - plane magnetic field  @xmath152 is applied , spin mixing due to the zeeman effect can be taken into account by full analogy with the rashba mixing . in fact , it is sufficient to replace @xmath153 , so eq .  ( [ chinplane= ] ) has the same form , but the transferred spectral weight @xmath154 . for a magnetic field @xmath155 , we obtain a numerical estimate @xmath156 . the in - plane direction of the off - diagonal component @xmath157 is determined not by @xmath93 , but by the magnetic field , @xmath158 . here we consider the interaction of excitons in the tmdc monolayer with the electromagnetic field . the monolayer is assumed to be sandwiched between two semi - infinite media with dielectric constants @xmath159 and @xmath160 occupying the half - spaces with @xmath161 and @xmath162 , respectively . ( note that the values of @xmath163 at optical frequencies @xmath164 should be taken . ) the free field in such a structure is fully characterized by the green s function @xmath165 , which represents the response of the electric field , @xmath166 , to an external oscillating polarization @xmath167 , located in the plane @xmath168 . in the quantum theory , this green s function represents the retarded propagator of the electric field ; at the same time , it can be found from the classical maxwell equations  @xcite . the radiative self - energy for the excitons at @xmath169 is proportional to @xmath170 , for which we introduce the short - hand notation @xmath171 [ more precisely , it represents the projection on the spatial profile of the excitonic polarization in the @xmath36-direction , here assumed to be just @xmath172 .    the long - range exchange part of the coulomb interaction shifts the exciton energies and lifts the valley degeneracy . exciton coupling to the photons also shifts the excitonic frequencies and leads to the radiative decay . all these effects can be described by studying the linear system @xmath173 for each  @xmath93 , it has non - trivial solutions for some complex values of  @xmath92 whose real parts give the shifted exciton energies , and the imaginary parts ( with the opposite sign and multiplied by  2 ) represent the radiative decay rates . this procedure is equivalent to finding the poles of the full layer susceptibility ( i.  e. , dressed by the exchange interaction and coupling to photons ) in the complex plane of  @xmath92 , or to finding the poles of the monolayer reflectivity . calculation of @xmath171 from the maxwell equations is quite standard and is given in appendix  [ app : maxwell ] . the result is    @xmath174    here @xmath175 and @xmath176 are the @xmath36  components of the three - dimensional wave vector in the corresponding media : @xmath177 where the infinitesimal imaginary part determines the behaviour in the complex plane of  @xmath92 , prescribing @xmath178 for real @xmath92 in the interval @xmath179 , which corresponds to evanescent waves . the parameter @xmath180 in eq . ( [ dzz= ] ) represents the electric field of a double layer , arising from the excitonic polarization in the @xmath36  direction ( appendix  [ app : maxwell ] ) , which is singular in the limit of an infinitely thin layer , and can not be determined in the macroscopic framework , used here . for excitons in semiconductor quantum wells of a sizable width , this local - field effect could be treated properly in the envelope - function approximation  @xcite . in the atomically - thin tmdc monolayer , it requires the full microscopic treatment of the short - range exchange interaction , such as that in ref . the consequences of this local - field effect will be discussed in the next subsection . the tensor structure of @xmath91 and @xmath181 , following from the in - plane isotropy of the problem , determines how the two - fold valley degeneracy of the mechanical excitons is lifted . the first family of solutions of eq . ( [ psystem= ] ) is characterized by @xmath94 lying in the @xmath13 plane , perpendicular to  @xmath93 . these transverse excitons emit @xmath182-polarized light ( transverse - electric , or te modes ) . the second family of solutions has @xmath94 in the plane formed by the vectors @xmath93 and @xmath183 ; its precise direction is determined by the relative magnitude of different components of @xmath91 . these are longitudinal excitons , which emit @xmath184-polarized light ( transverse - magnetic , or tm modes ) . each of the two linear polarizations represents a linear combination of the two circularly polarized excitons from each valley with equal probability weights .    in the following , we treat eq . ( [ psystem= ] ) perturbatively . namely , we assume @xmath92 to be close to one of the poles @xmath103 of @xmath91 . near the pole , it can be represented as @xmath185 with some vectors @xmath186 , @xmath187 , which describe the polarization of the transverse and longitudinal excitons . then , the corresponding solutions of eq . ( [ psystem= ] ) are @xmath188 and @xmath189 , and the shifted ( complex ) excitonic energies are approximately given by @xmath190 here , @xmath191 is nothing but the radiative self - energy of the exciton , in which we explicitly separate the radiative shift @xmath192 and the decay rate @xmath193 . this procedure is equivalent to the quantum - mechanical calculation of the energy shift by perturbation theory to the first order in the coulomb exchange and to the second order in the exciton - photon coupling , and to the calculation of the radiative decay rate by the fermi golden rule . if the spin mixing is due to the zeeman effect in an in - plane magnetic field  @xmath84 instead of the rashba effect , the @xmath182 and @xmath184 polarizations do not separate , as @xmath93 and @xmath84 define two different directions in the plane . we discuss this situation in appendix  [ app : zeeman ] .      for the transverse bright and dark @xmath4 excitons , the radiative shift and decay rate are straightforwardly evaluated as @xmath194 where the upper / lower line in the braces with @xmath146 refers to the bright / dark exciton , and the expression to the left of the braces is the decay rate of the transverse bright @xmath4  exciton in the absence of spin flip @xcite . in the square roots , @xmath195 , we neglected the small difference between @xmath196 and @xmath197 as well as their dependence on  @xmath93 , since we are interested only in a narrow region @xmath198 where at least one of @xmath175 , @xmath176 is real  @xcite .    for the longitudinal @xmath4 excitons we obtain more bulky expressions ,    @xmath199,\label{gammalab=}\\ \omega_{la}^\mathrm{d}({\mathbf{q } } ) -\frac{i}2\,\gamma_{la}^\mathrm{d}({\mathbf{q } } ) = { } & { } -\frac{8\pi{i}q^2\phi^2(0)}{q_{1z}^a/{\varepsilon}_1+q_{2z}^a/{\varepsilon}_2 } \left[\frac{e^2v^2}{e_a^2 } \frac{|\lambda_\mathrm{ex}|^2}{4\delta_\mathrm{c}^2}\ , \frac{q_{1z}^aq_{2z}^a}{{\varepsilon}_1{\varepsilon}_2}+|d_z|^2\right]+{}\nonumber\\ { } & { } + \frac{8\pi{}q^2\phi^2(0)}{q_{1z}^a/{\varepsilon}_1+q_{2z}^a/{\varepsilon}_2}\ , \frac{ev}{e_a}\frac{\lambda_\mathrm{ex}^*}{2\delta_\mathrm{c } } \left(\frac{q_{1z}^a}{{\varepsilon}_1}-\frac{q_{2z}^a}{{\varepsilon}_2}\right)d_z + 4\pi\kappa_0|d_z|^2\phi^2(0).\label{gammalad=}\end{aligned}\ ] ]    whose qualitative features are similar to the transverse case : the decay rates are nonzero only in the small - momentum region @xmath198 , the bright exciton radiative rate is dominated by the first line of eq . ( [ gammalab= ] ) to which the rashba term gives a small correction . however , an important difference from the transverse case is that @xmath200 does not vanish for @xmath201 . the longitudinal dark @xmath4 exciton can decay even in the absence of the rashba coupling , due to  @xmath68 . moreover , this latter mechanism is by far the dominant one , as the ratio between the two terms in the square brackets in the first line of eq . ( [ gammalad= ] ) is @xmath202 for a quite high electric field of @xmath150 .    the last term in eq .  ( [ gammalad= ] ) containing the unknown parameter @xmath180 is purely real and does not contribute to the radiative decay rate . however , it produces an energy shift , @xmath203 of the longitudinal excitons , which thus lifts the valley degeneracy even at @xmath204 and in the absence of rashba or zeeman effects @xcite . in the macroscopic framework , used here , @xmath205 represents the interaction energy of the exciton polarization in the @xmath36  direction with its own electric field , which is singular in the limit of an infinitely thin layer . by the order of magnitude , @xmath206 , where @xmath3  is the monolayer thickness and @xmath207 is the effective background dielectric constant in the perpendicular direction at optical frequencies ( see appendix  [ app : maxwell ] ) . taking @xmath208 , @xmath209 , @xmath210 , and @xmath211 , we obtain an estimate @xmath212 . this value agrees by the order of magnitude with the dark exciton energy shifts due to the short - range exchange interaction , which were calculated using a microscopic _ ab initio _ approach in ref . indeed , both are supposed to have the same origin , as the coulomb part of @xmath171 represents the exchange field . however , no lifting of valley degeneracy was mentioned in ref . note that the presence of the positive shift @xmath205 for the longitudinal excitons has an important consequence for the luminescence , as the lowest - energy states are the transverse excitons whose decay rate is only due to rashba or zeeman effect .    to illustrate the @xmath128 dependence of the radiative energy shifts and the decay rates , we plot them in figs . [ fig : ta ] and  [ fig : la ] for the transverse and longitudinal dark @xmath4 excitons , for @xmath213 and two values of the substrate dielectric constant , one typical for glass - like substrates , @xmath214 , the other one corresponding to a highly dielectric substrate , such as algasb with @xmath215 at optical frequencies  @xcite . it is convenient to normalize the energy shifts and decay rates by the bright exciton decay rate at @xmath204 , @xmath216 , for a tmdc monolayer suspended in vacuum , @xmath217 for @xmath218 , @xmath219 , @xmath220 , this estimate gives @xmath221 , not very much different from the results of more precise calculations involving the full microscopic treatment of the exciton wave function  @xcite , which give @xmath222 for several tmdc materials . in the calculation presented in figs . [ fig : ta],[fig : la ] , we used the values @xmath219 , the experimentally determined splitting @xmath148  @xcite , @xmath210 as extracted from ref . @xcite , and the rashba coupling constant @xmath223 per each v /  of the static electric field  @xmath64  @xcite . the latter was taken to be 0.1  v /  for the glass - like substrate , giving @xmath149 , and 0.2  ev /  for the highly dielectric substrate ( @xmath224 ) , as estimated in appendix  [ app : substrate ] ( note that changing the electric field amounts to a simple rescaling of the @xmath225  axis @xmath226 in fig . [ fig : ta ] , while fig . [ fig : la ] is insensitive to  @xmath64 , as discussed above ) . ( black curves , negative @xmath225  axis ) and decay rates @xmath227 ( red curves , positive @xmath225  axis ) of transverse excitons for @xmath228 , @xmath229 ( solid curves ) and @xmath230 ( dashed curves ) , as a function of the dimensionless exciton momentum @xmath231.,width=321 ]    ) @xmath232/\gamma_0 $ ] ( black curves , negative @xmath225  axis ) and decay rates @xmath233 ( red curves , positive @xmath225  axis ) of longitudinal excitons for @xmath228 , @xmath229 ( solid curves ) and @xmath230 ( dashed curves ) , as a function of the dimensionless exciton momentum @xmath231 . , width=321 ]      so far , we calculated the decay rates for excitons with a given momentum  @xmath93 . if we want to characterize the radiative decay of a thermal population of excitons , these rates should be averaged over their thermal distribution  @xcite . this assumes that the thermalization rate is larger than the radiative rate , otherwise the full kinetic problem should be analyzed . we assume the exciton distribution to be non - degenerate :    @xmath234    where @xmath235 is the combined density of the @xmath4 excitons , both bright and dark ones , @xmath236  is the temperature , and we introduced the shorthand notation for the exciton mass , @xmath237 . the distribution functions @xmath238 are spread over the thermal momenta @xmath239 . then , the thermal average can be approximated as @xmath240 where the summation is over the four species of @xmath4  excitons . taking the leading contributions to expressions ( [ gammata= ] ) , ( [ gammalab= ] ) , ( [ gammalad= ] ) , we perform the integration analytically ( see appendix  [ app : integral ] ) , and obtain @xmath241 the explicit form of the functions @xmath242 , @xmath243 , @xmath244 is given in appendix  [ app : integral ] . for @xmath213 , @xmath214 , they are equal to 1.5 , 1.6 , and 1.0 , respectively .    much more important are the factors @xmath245 , @xmath246 , which characterize the relative oscillator strength of the dark @xmath4 excitons due to the rashba - induced in - plane dipole moment and due to the perpendicular dipole moment . they are defined as @xmath247 and their values are estimated to be @xmath248 for the field @xmath150 , and @xmath249 . if instead of a perpendicular electric field , an in - plane magnetic field is applied , we use the results of appendix  [ app : zeeman ] . the result amounts to the replacement @xmath250 in the first and the second terms in the numerator of eq . ( [ avgamma= ] ) , so instead of the rashba factor @xmath245 we have the zeeman factor @xmath251 for a field @xmath155 and @xmath252 , it is about @xmath253 . in this work we have investigated the radiative decay of spin - forbidden dark intravalley @xmath4 excitons in tungsten dichalcogenide monolayers due to several spin - flip mechanisms . our main result is represented by eq . ( [ avgamma= ] ) for the thermally - averaged radiative decay rate and by expressions ( [ etar=])([etaz= ] ) for the average efficiencies of different spin - flip mechanisms . as the bright excitons in tungsten - based compounds have higher energy than the dark ones , the bright exciton contribution to the average rate is suppressed at low temperatures by the thermal activation exponential @xmath254 with the bright - dark splitting @xmath255 of several tens of mev , of the order of the room temperature . the intrinsic mechanism of the dark exciton radiative decay is due to the interband spin - flip dipole moment , perpendicular to the monolayer plane . it was mentioned in refs . @xcite , and its magnitude can be deduced from the estimates of ref . we found , however , that the very same mechanism lifts the valley degeneracy of the dark excitons by a coulomb local - field effect , producing a splitting @xmath205 which we could very roughly estimate as about 10  mev . crucially , the whole oscillator strength of such interband spin flip ( about @xmath256 of the bright exciton oscillator strength ) is taken by the higher - energy component . thus , the spin - forbidden intravalley exciton , usually referred to as dark , in fact , has two components , one dark and the other which can be called `` dim '' . the contribution of this dim exciton to the average radiative rate  ( [ avgamma= ] ) is suppressed at low temperatures by another exponential , @xmath257 .    for the lowest - energy dark component , which should dominate the exciton population at low temperatures , we have analyzed extrinsic spin - flip mechanisms , which transfer some oscillator strength from the bright exciton to the dark one . one such mechanism is due to the rashba spin - orbit coupling which arises whenever the reflection symmetry in the monolayer plane is broken , e.  g. , by an externally applied perpendicular static electric field , or by different dielectric environment above and below the monolayer . this mechanism was mentioned in ref . @xcite . using our own estimate of @xmath258 for the effective electric field produced by the dielectric mismatch and the calculated value of the rashba coupling constant from ref . @xcite , we find that the fraction of the oscillator strength transferred from the bright exciton to the dark one is extremely low , about @xmath6 , mainly due to small momenta of excitons subject to radiative decay . another possible mechanism is the spin flip by the zeeman effect from a magnetic field directed along the monolayer plane . then , for a strong but still realistic field of 30  t , more than @xmath259 of the bright exciton oscillator strength can be transferred to the dark one . this suggests a way to manipulate the radiative properties of dark excitons . the results of the present work help to identify several directions for improvement of our understanding of the excitonic radiative processes in tungsten dichalcogenides . first , the value of the interband spin - flip dipole moment is currently known from only one source , the estimate of ref . . it would be quite helpful if more information were available , either theoretically or experimentally . e.  g. , by studying the effect of the rashba spin - orbit coupling on the carrier transport , as the rashba coupling has the same origin . second , a reliable microscopic calculation of the dark - dim exciton splitting  @xmath205 would help to determine how dramatic is the low - temperature suppression of the dim exciton population . finally , the role of collisions with defects , charge carriers , or phonons in the radiative decay also needs clarification . we are grateful to m.  potemski for stimulating discussions . a. o. s. acknowledges financial support from the ec graphene flagship project ( no . the @xmath260 matrix @xmath261 , can be transformed to the block - diagonal intra - band form by applying a unitary transformation @xmath262 ( here @xmath64 is only the static electric field , while the same term with the optical field remains inter - band ) : @xmath263 e^{\mathcal{s}_\tau({\mathbf{k}})}\ ] ] if the matrix @xmath264 is chosen in the form    @xmath265,\\ & \mathcal{s}_{-1}({\mathbf{k}})=\frac{1}{e_\mathrm{g}}\left[\begin{array}{cccc } 0 & 0 & vk_+ & \mathcal{e}_zd_z^ * \\ 0 & 0 & 0 & vk_+ \\ -vk_- & 0 & 0 & 0 \\ -\mathcal{e}_zd_z & -vk_- & 0 & 0 \end{array}\right],\end{aligned}\ ] ]    where @xmath266 , then , to the second order in @xmath267 , @xmath268 , we obtain a block - diagonal hamiltonian . the first term represents the kinetic energy of the electrons and holes , @xmath269,\ ] ] while the second one corresponds to the rashba coupling , @xmath270.\ ] ] here we also assumed @xmath271 . tmdc materials can have comparable values of the spin splitting in the valence band @xmath134 and the gap @xmath25 ; then the above calculation can be repeated without assuming @xmath271 , which would produce different rashba coupling in the conduction and the valence band . for tungsten compounds @xmath272 , so we neglect this difference . we also neglect the effect of the unitary transformation @xmath273 on the coulomb interaction which forms the excitonic states . an external electric field @xmath64 and interaction with a substrate via van der waals forces have the same qualitative effect of breaking the reflection symmetry in the @xmath36  direction and deforming the electronic wave functions in the tmdc monolayer . still , there is no reason for the deformation to be quantitatively similar in the two cases . thus , characterization of the substrate effect by a single parameter , an effective  @xmath64 , is quite a rough approximation , which gives only an order - of - magnitude estimate of the effect . furthermore , we adopt a macroscopic description of the substrate , characterizing it by the dielectric constant . as the substrate is only a few angstroms away from the monolayer center , such macroscopic description is also valid only qualitatively . we model the tmdc monolayer as a slab of thickness @xmath3 and dielectric constant  @xmath274 , sandwiched between two media with dielectric constants @xmath159 at @xmath275 and @xmath160 at @xmath276 . let us write the energy as a functional of the microscopic three - dimensional electronic density @xmath277 , @xmath278 where the substrate effect is included as a hartree - like term : @xmath279={}&{}e_0[\rho({\mathbf{r}})]-{}\nonumber\\ { } & { } -\int\left[v({\mathbf{r}},{\mathbf{r}}')-v_1({\mathbf{r}},{\mathbf{r}}')\right ] \times\nonumber\\ { } & \qquad{}\times \rho({\mathbf{r}})\,\rho_\mathrm{i}({\mathbf{r}}')\,d^3{\mathbf{r}}\,d^3{\mathbf{r}}'+\nonumber\\ { } & { } + \frac{1}{2}\int \left[v({\mathbf{r}},{\mathbf{r}}')-v_1({\mathbf{r}},{\mathbf{r}}')\right ] \times\nonumber\\ { } & \qquad{}\times \rho({\mathbf{r}})\,\rho({\mathbf{r}}')\,d^3{\mathbf{r}}\,d^3{\mathbf{r } } ' . \label{hartree=}\end{aligned}\ ] ] here @xmath280 $ ] is the functional for a tmdc monolayer suspended in vacuum , @xmath281 is the ionic density ( a sum of @xmath282  functions at the ion positions ) , @xmath283 is the interaction potential , represented by the green s function of the poisson equation in the full dielectric structure multiplied by  @xmath284 , @xmath285 is the same for @xmath286 . the second and the third terms in eq . ( [ hartree= ] ) describe interaction of the tmdc electrons with the polarization charges in the surrounding media . the deformed electronic wave functions can be found by varying the functional  ( [ hartree= ] ) . the whole description is analogous to the macroscopic description of polarons in ionic crystals @xcite ; indeed , deformation of electronic wave functions in the tmdc by interaction with a dielectric substrate can be viewed as a polaronic effect .    in the planar geometry considered here , evaluation of the integrals in eq . ( [ hartree= ] ) reduces to a summation over image charges . when all three dielectric constants @xmath287 are different , each charge produces an infinite number of images :    @xmath288 ^ 2}},\\ & \zeta_0=1,\quad \zeta_{\pm{1}}=\frac{{\varepsilon}-{\varepsilon}_{1,2}}{{\varepsilon}+{\varepsilon}_{1,2}},\quad \zeta_{\pm{2}}=\zeta_{\pm{1}}\ , \frac{{\varepsilon}-{\varepsilon}_{2,1}}{{\varepsilon}+{\varepsilon}_{2,1}},\quad\ldots.\end{aligned}\ ] ]    for @xmath289 , the decay of @xmath290 with @xmath291 is quite slow ( which is a manifestation of the confinement of the electric field lines to the interior of the dielectric ) , so many images contribute to the hartree terms in eq . ( [ hartree= ] ) . however , if we start with @xmath277 , symmetric with respect to @xmath292 , the image charge distribution due to  @xmath293 is also symmetric and does not produce any net electric field in the @xmath36  direction . as concerns the @xmath294  term , if at least one of @xmath295 , @xmath296 is not small compared to  @xmath297 , which we assume to be the case , @xmath290  decays exponentially with a decrement @xmath298 . thus , for an order - of - magnitude estimate , we can restrict ourselves to  @xmath299 . the electronic states we are interested in ( the bottom of the conduction band and the top of the balence band at @xmath2 points ) are known to originate mainly from @xmath3  orbitals of the metal atoms . assuming these orbitals to be concentrated around @xmath169 , we can expand the potentials in @xmath300 as a formal small parameter . of course , in reality the orbital spatial extent is of the same order as  @xmath3 , so this multipole expansion is good only for an order - of - magnitude estimate . the overall charge neutrality and the vanishing dipole moment of the unperturbed charge distribution around each metal atom ( due to its @xmath301 symmetry ) make the first non - vanishing multipolar moment to be the quadrupole . the quadrupole field decays with distance as @xmath302 , so , even though each atom feels the field from images of all other atoms , the dominant contribution comes from the nearest two images which are due to the atom itself . the electric field at the point @xmath303 produced by the two quadrupole images at @xmath304 is given by @xmath305 where @xmath306 is the quadrupole moment of the electronic cloud around a single metal atom , @xmath307    although the quadrupole moment of a metal atom deformed by the crystal field differs from that of an isolated atom , we assume them to be of the same order and estimate the latter . the electronic configuration of tungsten is [ xe]@xmath308 , and four out of five outermost @xmath3 orbital states are filled , according to the hund s rule . since filled atomic shells have zero quadrupole moment , @xmath306 for tungsten is determined by the single empty @xmath309  orbital which we take to be the @xmath310 one , as it is this one that is known to give rise to the lowest conduction band of tungsten dichalcogenides . the wave function of this orbital can be approximated by a hydrogen - like one with @xmath311 , which moves in the coulomb potential with effective charge @xmath312 the average quadrupole moment of a hydrogenic @xmath313 state  @xcite , @xmath314}{2}\,\frac{2l(l+1)-6m^2}{4l(l+1)-3},\ ] ] then gives @xmath315 for tungsten ( @xmath316 is the bohr radius ) . for molybdenum , the electronic configuration in the gaseous phase , [ kr]@xmath317 , has a filled @xmath318 shell with zero quadrupole moment . however , given the fact that the lowest conduction band of molybdenum dichalcogenides is formed mostly by the @xmath318 @xmath310 orbital , just like for tungsten dichalcogenides , the electronic configuration is likely to be changed by the crystal field to [ kr]@xmath319 . then , repeating the same calculation with @xmath320 , @xmath321  @xcite , we obtain @xmath322 for molybdenum . if a tmdc monolayer suspended in vacuum is subjet to an _ external _ electric field @xmath64 , the field inside the monolayer is @xmath323 . it is this one that should be matched with @xmath324 from eq . ( [ ez0= ] ) to obtain the effective external field due to the substrate : @xmath325 taking the static dielectric constants @xmath326 ( see ref . @xcite and references therein ) , @xmath213 ( vacuum ) , @xmath327 ( silica ) , and @xmath328 , we obtain @xmath329 for tungsten dichalcogenides . taking a highly dielectric substrate , @xmath330 , one can increase the effective field by about a factor of  2 . here we solve the maxwell equations in the presence of the oscillating monolayer polarization @xmath331 sandwiched between two semi - infinite media with dielectric functions @xmath159 and @xmath160 occupying the half - spaces with @xmath161 and @xmath162 , respectively . due to the in - plane isotropy of the maxwell equations and of the susceptibility , @xmath332 , @xmath333 , we can assume the wave vector @xmath93 to be along the @xmath334 axis , without any loss of generality . for the three - dimensional waves propagating in the two media , the @xmath36 component of the wave vector is given by @xmath335 . for the @xmath182  polarization , the fields in the incident , reflected and transmitted waves are parametrized by their electric field amplitudes @xmath336 , @xmath337 and @xmath338 , respectively . the non - zero components of the fields are given by ( we omit the common factor @xmath339 )    @xmath340    the boundary conditions on the tangential components of the electric and magnetic fields correspond to the continuity of the electric field and a jump in the magnetic field due to the surface current :    @xmath341    when the incident wave is absent , but the layer polarization acts as a source producing the outgoing field , the amplitudes of the latter are given by @xmath342 for the @xmath184 polarization , we choose the magnetic field amplitudes @xmath343 , @xmath344 , @xmath345 , to parametrize the fields , which are sought in the form    [ tm=]@xmath346    the boundary conditions in this case are more subtle , because the @xmath36-polarization @xmath347 produces an electrical double layer . let us assume @xmath348 to be spread over a narrow but finite region @xmath349 ( @xmath3  being the monolayer thickness ) with a background dielectric constant  @xmath297 . in the first maxwell equation for @xmath350 , @xmath351,\ ] ] the first term is finite at @xmath352 . in the layer region it can be neglected , which gives the following result for the field in the layer region ( we omit the terms which vanish at @xmath352 ) : @xmath353 where the first term represents the field at @xmath354 , related to the value at @xmath355 from eq . ( [ tmez= ] ) by the continuity of the normal component of the electric displacement .    in the third maxwell equation for @xmath356 , the @xmath348 term appears both in the polarization and displacement currents , so it cancels out , and the only singularity in @xmath357 comes from the in - plane current , @xmath358 however , from the faraday s law , @xmath359 it follows that @xmath360 must have a jump of @xmath361 : @xmath362 this leads to the following boundary conditions for the amplitudes in eqs . ( [ tm= ] ) :    @xmath363    if we wish to determine the outgoing field produced by a source layer polarization without the incident field , we have no ambiguity ; the amplitudes are given by    @xmath364    however , we face a problem when we want to couple the field back to the polarization , as we need the value @xmath365 , which is undetermined due to the singularity . here we note that coupling to the layer polarization is in fact determined by @xmath366 , with the same @xmath348 as in the spatial profile of the polarization itself , which is nothing but the product of the microscopic wave functions of the electron and the hole at coinciding points ( see , e.  g. , a microscopic treatment for excitons in a semiconductor quantum well  @xcite ) . in other words , we define @xmath367 i.  e. , as the projection on the polarization spatial profile in the @xmath36  direction . then , the uncertainty due to the integral terms in eqs . ( [ ez = delta ] ) , ( [ ex = delta ] ) is resolved as @xmath368 by symmetry . on the contrary , the term @xmath369 can only be determined from the microscopic theory . the projected fields are given by    @xmath370    to arrive at the final expressions ( [ dab=])([dzz= ] ) , one should recall that the field @xmath64 which appears in the definition of the dipole moment  ( [ hrashba= ] ) , and which was used in the rashba coupling constant estimate of ref . @xcite , is not the local field , but the external field , applied to a monolayer suspended in vacuum . hence , a factor @xmath371 should be absorbed into  @xmath68 . as the @xmath182 and @xmath184 polarizations do not separate in a parallel magnetic field , we return to the valley basis . namely , instead of using eq . ( [ chipole= ] ) , we represent the susceptibility near each pole @xmath103 as @xmath372 where the vector @xmath373 , describes the polarization of the exciton @xmath104 in the valley  @xmath111 . then , the radiative self - energy can be approximated by @xmath374 for the dark @xmath4 excitons in the parallel magnetic field , these vectors are given by    @xmath375    where @xmath376 . as we have seen in sec . [ ssec : gammaq ] , the most important term that lifts the valley splitting is the last term in eq . ( [ dzz= ] ) . thus , we first consider the splitting at @xmath204 , @xmath377 , determined by the self - energy matrix @xmath378,\ ] ] where @xmath379 is the phase of  @xmath68 . its eigenvectors determine the @xmath36-dipole - active state @xmath380 whose energy is shifted up by @xmath205 , and the orthogonal one , @xmath381 , which is @xmath36-dipole - inactive . in the case of rashba - induced splitting at @xmath382 these states evolve into the @xmath184- and @xmath182-polarized ones , respectively . at finite @xmath93 and @xmath84 , we project the self - energy on the two eigenvectors @xmath383 , which gives the radiative shift and the decay rate of the two states    @xmath384\nonumber\\ & + 4\pi\phi^2(0)\,\frac{ev|d_z|}{e_a}\ , \frac{g_\|\mu_bb_\|}{4\delta_\mathrm{c}}\,\frac{q_{1z}^a/{\varepsilon}_1-q_{2z}^a/{\varepsilon}_2}{q_{1z}^a/{\varepsilon}_1+q_{2z}^a/{\varepsilon}_2}\,q\cos\tilde\phi -\frac{8\pi{i}q^2|d_z|^2\phi^2(0)}{q_{1z}^a/{\varepsilon}_1+q_{2z}^a/{\varepsilon}_2}+\xi_0,\\ \omega_\|({\mathbf{q}})-\frac{i}2\,\gamma_\|({\mathbf{q}})= { } & { } -8\pi{i}\,\phi^2(0)\,\frac{e^2v^2}{e_a^2 } \left(\frac{g_\|\mu_bb_\|}{4\delta_\mathrm{c}}\right)^2 \left[\frac{(e_a / c)^2}{q_{1z}^a+q_{2z}^a}\cos^2\tilde\phi + \frac{(q_{1z}^a/{\varepsilon}_1)(q_{2z}^a/{\varepsilon}_2)}{q_{1z}^a/{\varepsilon}_1+q_{2z}^a/{\varepsilon}_2}\sin^2\tilde\phi\right ] , \label{gammab=}\end{aligned}\ ] ]    where @xmath385 , and @xmath386 are the polar angles of @xmath93 and @xmath84 in the @xmath13 plane , respectively . it is convenient to introduce a dimensionless variable @xmath387 , then the integration over @xmath93 transforms as @xmath388 , and the radiative decay rates of the @xmath4  excitons , given by eqs . ( [ gammata= ] ) , ( [ gammalab= ] ) , ( [ gammalad= ] ) , ( [ gammab= ] ) , can be written as      where we keep only the dominant terms . it is convenient to get rid of the square roots in the denominators and take the real part of each term in the numerators separately . then the integration becomes straightforward and gives    @xmath390 , \label{fbta=}\\ \int\frac{d^2{\mathbf{q}}}{(2\pi)^2}\ , \frac{\gamma^\mathrm{d}_{la}({\mathbf{q}})}{\gamma_0}= { } & { } \eta_z\,\frac{(e_a / c)^2}{2\pi } \left[\frac{2{\varepsilon}_1 ^ 2{\varepsilon}_2 ^ 2}{({\varepsilon}_1+{\varepsilon}_2)^2}\ , \frac{\sqrt{{\varepsilon}_1{\varepsilon}_2}+({\varepsilon}_1+{\varepsilon}_2)/3}{\sqrt{{\varepsilon}_1}+\sqrt{{\varepsilon}_2 } } \right.+\nonumber\\ & \hspace*{3cm}{}+\left . \frac{2{\varepsilon}_1 ^ 3{\varepsilon}_2 ^ 3}{({\varepsilon}_2 ^ 2-{\varepsilon}_1 ^ 2)({\varepsilon}_1+{\varepsilon}_2)^{3/2 } } \ln\left(\sqrt{\frac{{\varepsilon}_2}{{\varepsilon}_1}}\ , \frac{\sqrt{{\varepsilon}_2+{\varepsilon}_1}+\sqrt{{\varepsilon}_2 } } { \sqrt{{\varepsilon}_2+{\varepsilon}_1}+\sqrt{{\varepsilon}_1}}\right)\right ] . \label{fdta=}\end{aligned}\ ] ]                k .- k . liu , w. zhang , y .- h . lee , y .- c . chang , c .- y . su , c .- s . chang , h. li , y. shi , h. zhang , c .- s . lai , and l .- j . li , _ growth of large - area and highly crystalline mos@xmath1 thin layers on insulating substrates _ , nano . lett . * 12 * , 1538 ( 2012 ) . y. zhang , t .- chang , b. zhou , y .- t . cui , h. yan , z. liu , f. schmitt , j. lee , r. moore , y. chen , h. lin , h .- t . jeng , s .- k . mo , z. hussain , a. bansil , and z .- x . shen , _ direct observation of the transition from indirect to direct bandgap in atomically thin epitaxial mose@xmath1 _ , nature nanotech . * 9 * , 111 ( 2014 ) . a. chernikov , t. c. berkelbach , h. m. hill , a. rigosi , y.  li , o. b. aslan , d. r. reichman , m. s. hybertsen , and t. f. heinz , _ exciton binding energy and nonhydrogenic rydberg series in monolayer ws@xmath1 _ , phys . lett . * 113 * , 076802 ( 2014 ) . g. sallen , l. bouet , x. marie , g. wang , c. r. zhu , w.  p.  han , y. lu , p. h. tan , t. amand , b. l. liu , and b. urbaszek , _ robust optical emission polarization in mos@xmath1 monolayers through selective valley excitation _ , phys . rev . b * 86 * , 081301(r ) ( 2012 ) . g. wang , l. bouet , d. lagarde , m. vidal , a. balocchi , t. amand , x. marie , and b. urbaszek , _ valley dynamics probed through charged and neutral exciton emission in monolayer wse@xmath1 _ , phys . b * 90 * , 075413 ( 2014 ) . k. hao , g. moody , f. wu , c. k. dass , l. xu , c .- h . chen , l. sun , m .- y . li , l .- j . li , a. h. macdonald , and x. li , _ direct measurement of exciton valley coherence in monolayer wse@xmath1 _ , nature phys . 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( [ lambdaex= ] ) by replacing @xmath398 in the denominator which represents the interband excitation energy ; indeed , the excion binding energy @xmath53 is known to be quite large in tmdcs , values up to @xmath399 having been reported @xcite . however , if one corrects the denominator in eq . ( [ lambdaex= ] ) , then , to be consistent , one would also have to correct that in eq . ( [ meh= ] ) , as well as to go beyond the parabolic approximation for the electron and hole dispersion . thus , to be systematic , we assume to be in the limit @xmath400 and neglect corrections of the order of @xmath401 and higher . beyond this limit ( the extreme case being that of frenkel excitons ) , one can still work with the effective excitonic hamiltonian  ( [ rashbaex= ] ) , however , its parameters have no simple relation with the single - electron band structure parameters . in fact , the energy difference between the dark and bright excitons does not necessarily coincide with the single - particle spin splitting in the conduction band , due to an extra contribution from the exchange interaction  @xcite . here , @xmath255 should be understood as the excitonic splitting . in fact , lifting of the valley degeneracy for the dark @xmath4 excitons follows from a very simple and general consideration . consider the states at @xmath204 in the absence of external fields . then , we have two degenerate excitations corresponding to the @xmath2 valleys , whose transition dipole moments are both directed along  @xmath36 . then , it is always possible to form two linear combinations of them , one of which will have zero dipole moment and zero decay rate , while for the other one they will be finite . if the decay rates for the two states are different , so must be the energy shifts , which means lifting of the degeneracy . note that for @xmath402 , the denominator in eq . ( [ gammata= ] ) can vanish ; in this case , the perturbative approach  ( [ goldenrule= ] ) is not valid , so a more precise analysis of the original dispersion equation ( [ psystem= ] ) is required  @xcite . here we assume @xmath295 and @xmath296 to differ significantly , so this problem does not arise .
we perform a theoretical study of radiative decay of dark intravalley excitons in transition metal dichalcogenide monolayers . 
 this decay necessarily involves an electronic spin flip . 
 the intrinsic decay mechanism due to interband spin - flip dipole moment perpendicular to the monolayer plane , gives a rate about 1001000 times smaller than that of bright excitons . 
 however , we find that this mechanism also introduces an energy splitting due to a local field effect , and the whole oscillator strength is contained in the higher - energy component , while the lowest - energy state remains dark and needs an extrinsic spin - flip mechanism for the decay . 
 rashba effect due to a perpendicular electric field or a dielectric substrate , gives a negligible radiative decay rate ( about @xmath0 times slower than that of bright excitons ) . 
 spin flip due to zeeman effect in a sufficiently strong in - plane magnetic field can give a decay rate comparable to that due to the intrinsic interband spin - flip dipole .