arXiv:1307.0422v1 [cond-mat.mes-hall] 1 Jul 2013 h ad u oalresi ri opig hsmate- This valleytronics coupling. for orbit ideal spin considered of large is polarization a rial spin to is due there bands and gap the infrared band the direct valence in a fall Dirac by which separated the of are points bands zone conduction -K Brillouin and arrangement and lattice trigonal K honeycomb the a the At in symmetry. sulfur inversion of without layers two between oern r igelyrgopV ihloeie such dichalcogenides the VI to group come prominentas layer recently A single relatively are systems. only forefront some has fi- in which found with example also fermions are Dirac by mass protected (TRI). nite cones invariance Dirac reversal of states number time surface odd topological an metallic of have consisting but bulk their in alyidxi aiuae ndrc nlg otespin spintronics. the for to freedom analogy of direct in degree manipulated is index valley l.A motn xml fanwcaso materials topo- of are class fermions materi- new Dirac insulators(TI). a other logical helical of massless many example support to which important extended in- An been to continued als. has has and fermions prop- the Dirac tensify of relativistic study the of lattice, erties honeycomb a on atoms bon hnnsrcuei iprincre n est fstates of density and curves dispersion in structure Phonon igelyrdmtra ssilicene is material layered single lofudi iae graphene. are bilayer fermions in Dirac Massive found also scales. energy different very hr sabcln.IsmdlHmloincnb con- be for can that Hamiltonian of model subcase a Its as i.e. buckling. sidered sublattice other a the is of plane sublattice there the one of with out lattice shifted honeycomb slightly a on atoms silicon ino antcdpns hswsdn yCe et.al. Chen by done was This in introduc- dopants. the magnetic when of through tion massive violated is become symmetry can reversal time insulators topological in seen hntplgclisltrta o n otmsurfaces bottom and top that sufficiently use insulator to is topological alternative An thin gap. Dirac surface the ( Bi ihteioaino graphene, of isolation the With Bi MoS 0 . 99 2 Se Mn 2 3 . 13 with 0 . 01 tcnit falyro oydnmatoms molybdenum of layer a of consists It nrdcsnwsrcue nterdseso uvsad i and, curves dispersion their dichalcoge topology in by VI structures protected group new are introduces in states such found where are insulators some as such membranes mensional ASnmes 32 r 13 n 78.67.-n Cn, 71.38 c Pr, dispersion 73.22 both numbers: qua PACS in The modification important background. co to related longer leads phonon no this incoherent density and focus smaller spectral we a carrier particular and charge In in the present states. region of changes this density these curv their how dispersion of show carrier surements We charge dressed the gap. of the spectroscopy modify and shift ) ia emoseiti aysldsaessesicuigg including systems state solid many in exist fermions Dirac 2 Mn Se .INTRODUCTION I. 1 3 2 eateto hsc,MMse nvriy aitn Ont Hamilton, University, McMaster Physics, of Department smgei oat.Acomposition A dopants. magnetic as a u h hmclptnilinside potential chemical the put can aainIsiuefrAvne eerh oot,Ontari Toronto, Research, Advanced for Institute Canadian 3–12 hs aeil r insulators are materials These 25,26 MoS 1,2 20–24 18,19 igelyro car- of layer single a h ia fermions Dirac The 2 novn however involving 14–17 hc smd of made is which lsl related closely A huLi Zhou nwihthe which in Dtd pi 3 2018) 13, April (Dated: 1 ∗ n .P Carbotte P. J. and 27 ia emo a aeb jnnadKitagawa by and discussed Ojanen by as made surfaces was fermion two Dirac the of et.al. Dirac Lu each the on gaps this fermions and tunneling through communicate hteiso pcrsoy(RE)suyo h TI, the angle-resolved of their study lBi T in (ARPES) that spectroscopy mention photoemission we coupled electromagnetic Finally orbit THz spin polarized waves. dimensional circular with two gas a electron of irradiation ing hc a emaue nsann unln microscopy tunneling scanning in measured be can which esDrcsaein state Dirac less asv tt in state massive n si ig mechanism. Higgs break- authors a symmetry the in known spontaneous as yet of ing not possibility is the acquired suggest is mass mechanism which the While by before state. to mass non-topological a the topological acquire reaching from fermions Dirac (QPT) the transition non-topological phase quantum a pcrsoy(RE)adi h est fstate of density the emission in photo and resolved (ARPES) angular spectroscopy disper- effect in and Dirac the measured complex gapped curves calculate the sion now we on is coupling paper electron-phonon this theory of be In BCS would by dependent. which in energy modified gap real directly the and is and addi- constant channel In term gap electron-phonon lifetime. quasi- or the a the pairing provides in the as shift tion well a as the gives energies in which particle renormalizations channel to poten- leads particle pair This single constant theory. a BCS explicitly of by are tial modeled pairs than Cooper rather responsi- theory, into included condensation interaction Eliashberg the electron-phonon In for the ble of theory. details BCS the of analogous generalization is a finding theory This Eliashberg to dependence. also energy but acquires re- energy dispersion self bare electron-phonon modifies the an of not through modification leads lations usual hamiltonian the Holstein to a only with modeled phonon opigo asv ia emost nEinstein an to fermions Dirac massive of Coupling sadsann unln irsoymea- microscopy tunneling scanning and es ( h aeo asv ia emos can fermions, Dirac massive of case the n S opigo hs emost phonons to fermions those of Coupling . ssso oiatqaiatcepeak quasiparticle dominant a of nsists 1 ntergo rudtebn a.In gap. band the around region the on 31,32 − 28 re n est fstates. of density and urves x Se ahn,slcn n te w di- two other and silicene raphene, eydffrn rpslt rdc massive produce to proposal different very A iatcepcuehsboe down broken has picture siparticle ie,a ela ntesraeof surface the on as well as nides, 1 h a tefwihbcmscmlxand complex becomes which itself gap the , x 2 † ) nua-eovdphotoemission angular-resolved 2 aoet.al Sato , ro aaaLS4M1 Canada,L8S ario, ,Cnd 5 1Z8 M5G Canada o, lBiS T lBiSe T fmsieDrcFermions Dirac massive of 33–35 2 stesse osthrough goes system the As . 2 fsprodciiywihis which superconductivity of 30 oaodnr non-topological ordinary a to ofo oooia mass- topological a from go 29 N ( us- ω ) 2

(STM). with an electric field Ez applied perpendicular to the two In section II we provide the necessary formalism. Our sublattice planes. To describe a topological insulator sur- numerical results for the case of Topological Insulators face the last term is to be dropped (λ = 0) and the σ’s are found in Section III. Results specific to the single layer are to be understood as real spins rather than the pseu- MoS2 membrane and silicene are presented in section IV dospin of graphene, MoS2 and silicene. Also the valley with a summary and conclusion in section V. index τ is to take on a single value and spin degeneracy no longer arises. A Holstein hamiltonian has been widely used to describe the coupling of electrons to an Einstein II. FORMALISM phonon of energy ωE and this will be sufficient here. It takes the form

We begin with a two by two matrix Hamiltonian for † † H − = gω c ck′ (b ′ + bk−k′ ) (2) the bare bands which is sufficiently general to describe e ph − E k,s ,s k −k k,k′,s topological insulators as well as single layer MoS2 and X silicene. This Hamiltonian is given by Eq. (1) † where bq create a phonon of energy ωE and momentum † ∆ σz 1 q and c an electron of momentum k and spin s. The H = at[τk σ + k σ ]+ σ λτ − s (1) k,s 0 x x y y 2 z − 2 z coupling strength is g. For this very simple model the in- teracting two by two matrix Matsubara Green’s function where τ = 1 is a valley index, k is momentum and ± for the charge carriers σx σy σz are Pauli matrices with ∆ the gap parame- ter. To describe single layer MoS2 the spin orbit cou- 1 G(k,iω )= (1 + sHk σ)G(k,s,iω ) (3) pling 2λ = 0.15eV , t is the nearest neighbor hopping of n 2 · n sX=± 1.1eV with a the lattice parameter 3.193A˚ and sz is the z-component of spin. The same form of this Hamilto- with nian applies to silicene dropping the 1 in the last term ′ which leads to equal spin splitting in valence and con- ∆ Z∗ (atτkx,atky, 2 +Σ (iωn)) Hk = (4) duction band in contrast to MoS2 where it is large in ′ 2 2 2 ∆ Z 2 the valence band and small in the conduction band. The a t k + +Σ (iωn) | 2 | energy scales are however very different with λ of order q meV as is also the gap ∆ which is now associated here and

1 G(k,s,iω )= (5) n ′ I ∆ z 2 2 2 2 iωn + µ λτsz /2 Σ (iωn) s 2 +Σ (iωn) + a t k − − − q| |

′ I ′ In these formulas ∆ = ∆ λτsz and Σ (iωn) and 2 2 2 ∆ 2 Z s a t k + ( 2 ) . Note that Σ (iωn) in Eq. (7) is di- ΣZ (iω ) are the quasiparticle− and gap self energy cor- q n rectly proportional to ∆′ which appears linearly on the rections at Matsubara frequency iω . These quantities n right hand side of the equation. at temperature T can be written as g2ω2 ΣI (iω )= E n 2 III. NUMERICAL RESULTS Xq,s f (ε )+ f (ω ) f (ω )+1 f (ε ) F q,s B E + B E − F q,s (6) As we saw in the previous section the electron-phonon iω + µ + ω ε iω + µ ω ε  n E − q,s n − E − q,s interaction given by Eq. (2) leads to two self energy renormalizations in the Green’s function of Eq. (3). The and I ′ quasiparticle self energy Σ (iωn) given by Eq. (6) renor- 2 2 ∆ Z g ωE s 2 malizes the bare energies in the denominator of Eq. (5) Σ (iωn)= 2 ∆′ and remains even when the gap is set to zero. How- q 2 2 2 2 × X,s a t q + ( 2 ) ever there is also a second self energy correction ΣZ (iω ) q n fF (εq,s)+ fB(ωE) fB(ωE)+1 fF (εq,s) of Eq. (7), not present for the case of massless Dirac + − (7) fermions, which directly modifies the gap. It has both iωn + µ + ωE εq,s iωn + µ ωE εq,s  − − − real and imaginary part and is frequency dependent. It is I Here fF and fB are fermion and boson distribution this frequency dependence in both Σ (iωn ω + iδ) and functions 1/[e(ω−µ)/kBT 1] respectively and ε is ΣZ (iω ω + iδ) which carries the information→ about ± k,s n → the bare band quasiparticle energy εk,s = λτsz/2 + phonon structure and on how this structure manifests 3 itself in the dynamics of the Dirac fermions. For sim- and is now a frequency dependent quantity in sharp con- plicity we start with the case of massive Dirac fermions trast to the′ bare band case for which it is a constant and ∆ but with λ=0 in the bare band Hamiltonian of Eq. (1). equal to 2 in magnitude. Also, in general the bare In the top two frames of Fig. 1 we show respectively the band density| | of state is independent of filling factor i.e. renormalized self energy ΣZ and ΣI as a function of ω. of the chemical potential µ. Introducing the electron- For illustrative purpose the chemical potential was set phonon interaction lifts this simplicity and the DOS is, at µ=0.03 eV and the Einstein phonon energy ωE=7.5 in principle, different for each value of µ. meV with a gap of 40 meV. Our choice of phonon fre- In the lower frame of Fig. 1 we present results for the quency is motivated by the experimental finding of La Forge et.al36 who found phonon absorption feature in density of states of gapped Dirac fermions N(ω) vs. ω given in Eq. (10). The parameters are, chemical potential their infrared optical work at 61cm−1 and 133cm−1. The theoretical work of Zhu et.al37 identifies a dispersive sur- µ=30 meV above the center of the gap with ∆/2 = 20 meV. The dashed curve includes renormalizations due to face phonon branch ending at 1.8THz which they asso- ciate with a strong Kohn anomaly indicative of a large the coupling to phonons while the solid curve, given for comparison, is for the bare band. Some care is required electron-phonon interaction. This observation is further supported by the angular-resolved photoemission study in making such a comparison. The chemical potential of the interacting system (µ) is not the same as for the of Bi2Se3 where the electron-phonon mass enhancement parameter λ is found38 to be 0.25 so that we can ex- bare band (µ0). The two are related by the equation ep µ = µ + ReΣI (ω = 0). From the lower frame of the top pect significant effects of the electron-phonon interaction 0 panel of Fig. 1 we find ReΣI (ω = 0) 5 meV. This in the properties of topological insulators. This provides ≈ − a motivation for the present work. The electron-phonon gives µ0=35 meV so the gap in the bare band case falls between 15 meV (bottom of conduction band) to 55 coupling constant is set to be about 0.3, well within the − − range of the reported value of 0.25 in the reference [38] meV (top of valence band). Including interactions fur- ther shifts the bottom of the conduction band to lower and 0.43 in the reference [39]. The fermi surface falls, by arrangement, at ω=0. We see prominent logarithmic energies and the top of the valence band to higher en- ergies. These shifts effectively reduce the band gap in structure in the real part (solid curve) at ω = ωE with corresponding small jumps in the imaginary part± (dashed the interacting case to about 25 meV as compared with a bare band value of 40 meV. A feature to be noted is curve). There are additional phonon induced signature at µ + ∆/2 ω and µ ∆/2 ω . We note from that the value of the dressed density of states at ω=0 − − E − − − E (vertical dashed line) remains unchanged from its bare the mathematical structure of Eq. (6) and Eq. (7) that 33–35 these boson structures fall at precisely the same energies band value as is known for conventional systems. Z I In addition we note sharp phonon structures originating in both Σ and Σ . For the imaginary part there is a I Z Dirac Delta function of the form δ(ω+µ ω ε ) which from both Σ (ω) and Σ (ω). These structures in the E q,s self energies ΣZ (ω) and ΣI (ω) of top and middle frames leads to jumps, as we have noted. By± Kramers-Kronig− relations these imply logarithmic type singularities in the are at µ ∆/2 ωE, µ + ∆/2 ωE, ωE and ωE as identified− − in the− figure.− We emphasized− − that the last real part. These phonon structure get directly mirrored in the Dirac spectral function A (k,ω) which is given by two structures, placed symmetrically around the Fermi s surface at ω = ω , are very familiar in metal physics. ± E 1 The other two at µ ∆/2 ωE and µ + ∆/2 ωE As(k,ω)= ImG(k,s,iωn ω + iδ) (8) − − − − − −π → are not. As noted, the top of the valence band has been and works out to be shifted to higher energy by correlation effects but the shape of its profile is not very different. By contrast, the 1 ImΣI (ω) onset of the conduction band has a very much altered As(k,ω)= (9) π [˜ω s√M]2 + [ImΣI (ω)]2 shape. In particular it shows sharp phonon structure at − ′ µ+∆/2 ωE. Further, there is a prominent dip around whereω ˜ = ω + µ λτsz ReΣI (ω) and M = [ ∆ + ω− = ω .− In the energy region between these two struc- − 2 − 2 E ReΣZ (ω)]2 + [ImΣZ(ω)]2 + a2t2k2. The density of states tures,− electrons and phonon are strongly mixed by the N(ω) follows as interactions. This can be seen clearly in Fig. 2 where we show a plot of the Dirac fermion spectral function N(ω)= As(k,ω). (10) A(k,ω) vs. ω of Eq. (9) for various values of momen- Xk,s tum k. This function is measured directly in angular- We note that it is only the imaginary part of the quasipar- resolved photoemission spectroscopy. Fourteen values of ticle self energy ImΣI (ω) which broadens the Lorentzian k are considered as indicated and each curve is restricted form of Eq. (9). However both real and imaginary part of to the region below 600 for clarity. The solid vertical the gap self energy ΣZ (ω) modify the gap which becomes pink line identifies the fermi energy placed at ω = 0. The vertical dotted blue lines are at ω = ω and the an effective gap ± E dotted red lines identify ω = µ ∆/2 ωE. The two ′ − ± − ∆ (ω) ∆ curves close to k = kF are shown as red. In both curves eff = [ + ReΣZ (ω)]2 + [ImΣZ (ω)]2 (11) 2 r 2 we see a well defined quasiparticle peak near ω = 0. For 4 the bare band we would have a Dirac delta function at persion curves due to correlations have been observed in 2 2 2 ∆ 2 graphene.40–42 When electron-electron interaction are ac- ω = s a t k + ( 2 ) . If some residual scattering rate is included,q the delta function broadens into a Lorentzian counted for in a random phase approximation to lowest form. When the electron-phonon interaction of Eq. (2) order perturbation theory, one finds that the Dirac point is included there are further shifts associated with the at the intersection of valence and conductance band splits real part of ΣI (ω) and the gap is modified by both real into two Dirac points with a plasmaron ring inserted in and imaginary part of ΣZ (ω)(see top and middle frame between. This represents resonant scattering between of Fig.1). In addition to the quasiparticle peak seen in Dirac quasiparticles and plasmons (collective modes of the red curves of Fig. 2 there are also small incoherent the charge fluid) which are called plasmarons. Here it is the electron-phonon interaction which is involved in- phonon assisted side bands with onset at ω = ωE. As k ± stead. We hope the region where quasiparticles cease is increased beyond k = kF the phonon side band on the right hand side of the main quasiparticle peak becomes to be well defined excitations can be observed in future ARPES experiments. more prominent and as k is decreased below kF it is the left side phonon assisted band which becomes stronger. As the quasiparticle peak falls at smaller energy with de- creasing k, the onset of the boson structure remains fixed IV. RESULTS SPECIFIC TO SINGLE LAYER in energy and eventually they meet. When this happens MoS2 AND SILICENE the phonon and the electron lose individual identity and merge into a composite whole. As k is reduced towards We turn next to the specific case of single layer 22,43–45 zero the quasiparticle peak crosses the boson structure MoS2. In this material both valence and conduc- and we see that it re-emerges on the opposite side. It tion bands are spin polarized. This is shown schemati- is clear from this discussion that the bottom of the con- cally in the inset of the top panel of Fig. 4 for the valence duction band falls precisely in the region where electron band as well as in the bottom panel where valence and quasiparticle and phonon are not separately well defined. conduction bands are both shown schematically in color, Instead the electron spectral density consists of a com- spin up valence band in blue and down in red. We note posite incoherent entity with no identifiable sharp quasi- that the spin splitting in the conduction band (in gold) is particle peak. This situation is very different when the small and does not appear in the schematic. In a related top of the valence band is considered. In this case, there material silicene23,24 the splitting is the same size in both appears a well defined quasiparticle peak near k = 0 and bands (see lower panel of Fig. 6) as we will discuss later. consequently this region looks very much quasiparticle It is the -1 in the last term of Eq. (1) which controls like. Of course as k is increased the spectral density of the amount of spin polarization seen in the conduction the valence band starts to show a sideband with onset band and this term is missing in silicene. Our numeri- at µ ∆/2 ω , as the quasiparticle peak approaches − − − E cal results for the energy dependence of the valence band more closely this energy. density of states (DOS) are shown in the top and bottom In frame (a) of Fig. 3 we show a color plot of the frame (top panel) for two values of chemical potential (see spectral density A(k,ω) (ω = E(k), the dressed dis- heavy black dashed vertical lines) respectively -0.845 and persion curves) for the bottom of the conduction band -0.995 mev. The electron-phonon coupling constant is set (top left) and for the top of the valence band (bottom so that the mass enhancement parameter λep=0.1. In the left). In the right frames we show results for the corre- first instance the chemical potential falls below the top sponding bare band case with residual smearing Γ = 0.1 of the spin up band but above the top of the spin down meV. For the bare band the chemical potential has been band while in the second instance the chemical potential shifted because renormalized (µ) and bare (µ0) chem- falls below the top of both up and down valence bands. I ical potential are related by µ0 = µ ReΣ (ω = 0). The density of states for the spin up band alone is shown In frame (b) we provide similar results− but have halved as the heavy dotted red curve and for the combined up the value of the electron-phonon coupling g in Eq. (2), and down band by the solid black curve. The phonon consequently the renormalization effects are smaller but structures are at ∆/2+ λ + ωE, µ + ωE, µ ωE and still quite significant. Returning to the top left frame ∆/2 λ ω in− the top frame and at ∆/2+− λ + ω , − − − E − E of panel Fig. 3(a) we noted the region of the conduction ∆/2 λ + ωE, µ + ωE and µ ωE in the bottom frame. band between ω = 7.5 meV (dotted blue in Fig. 2) and −They are− ordered according to− the larger energy first. In ω = 12.5 meV (dotted− red in Fig. 2) which showed addition to phonon kinks, the electron-phonon interac- complex− changes due to the phonon. Below this re- tion has also shifted and modified the top of each of the gion however we see a much more conventional type of two bands. As we have already seen there are two dis- gapped Dirac fermion dispersion curve which is recogniz- tinctly different behaviors which characterize the shape of able as a distortion of a bare dispersion curves (shown the top (bottom) of these renormalized bands. We refer on the right). This is also true for the valence band to the first as quasiparticle like. This designation applies dispersion in the bottom left frame. Here it is only be- to the spin up band in the top frame. The band edge rises low -52.5 meV that significant phonon distortions can be smoothly although rather sharply. The second behavior seen. Analogous modifications of the Dirac fermion dis- seen is referred to as correlation dominated. It applies 5 to the other three cases. Here a quasiparticle descrip- ωE of 7.5 meV and in the lower frame the Einstein energy tion ceases to be possible and a sharp phonon induced is increased to 16.5 meV. We keep the electron-phonon structure is associated with the ending of the band. coupling g in Eq. (2) fixed. Comparing upper and lower These issues are elaborated upon and better illustrated frame shows that this increase in ωE has lead to much in Fig. 5 where we show a blow up of the band edges. For more filling of the gap between valence and conduction both top and bottom frame the chemical potential has a band. In fact the band gap in the lower frame has almost value of -0.995 eV and thus falls below the top of both closed. For comparison with the dressed case the band spin up and spin down valence bands. The solid black edges in the bare bands is shown as the heavy red vertical curve is the bare band case and is shown for compari- lines. It should be emphasized that increasing the value son with the dressed case. The long dashed red curves of the Einstein frequency (ωE) effectively increases the I 2 include only the quasiparticle self energy Σ while the electron-phonon coupling because of the ωE factor which dotted blue curve includes in addition the gap renormal- appears in the numerator of Eq. (6) and Eq. (7) although ization ΣZ . In both top and bottom frames we note that there are certain amount of cancelations from the denom- when we include only the quasiparticle self energy, the inator containing ωE. The electron-phonon mass renor- edge shifts slightly to higher energy but does not change malization is λep=0.3 for ωE=7.5 meV and λep=0.45 for its shape which remains characteristic of the existence ωE=16.5 meV. This change in the electron-phonon cou- of good Dirac quasiparticles as is the case for the bare pling is largely responsible for the near closing of the gap bands. On the other hand, the dotted blue curve which noted above. includes gap renormalizations as well as quasiparticle self energy corrections has change radically as compared to the bare band. The shift of the edge to higher energies V. SUMMARY AND CONCLUSIONS is greater and its shape is also very different. There is a sharp rise at ∆/2+ λ + ω and at ∆/2 λ + ω for − E − − E Coupling of Dirac fermions to a phonon bath changes up and down bands respectively which is followed by a their dynamical properties. For momentum k near the second rise of magnitude and shape much more compa- Fermi momentum kF well defined quasiparticles exist in rable to that for the bare band. In the lower frame there the conduction band with energies shifted from bare to is also a second phonon kink at µ + ωE. dressed value controlled by the real part of this self en- In the right panel of Fig. 5 we show results for the ergy. The imaginary part of the quasiparticle self energy spectral density A(k,ω) at k = 0 for the valence bands gives them a finite lifetime. In addition there are phonon of the left hand frame. The solid black curve is for the sidebands to which part of the spectral weight under the bare band but includes a small constant scattering rate spectral density A(k,ω) has been transferred. The onsets Γ=1 meV so as to broaden the Dirac delta function of the of these sidebands are determined by the singularities in pure case. The long dashed red curve is for the electron- the self energy. For an Einstein optical phonon with en- phonon dressed band where we include only the quasipar- ergy ωE, these onsets are found to be at ωE,-ωE on either ticle self energy correction which renormalizes directly side of the chemical potential. As momentum k is moved the bare quasiparticle energies. When gap self energy away from kF , the energy of the renormalized quasiparti- renormalization is additionally included we get the dot- cle peak approaches more closely these onset energies and ted blue curve which has entirely lost its sharp quasi- the spectral weight transfer to the sidebands increases. particle peak. The spectral density also shows a phonon Eventually the quasiparticle picture itself breaks down peak at ∆/2+ λ + ωE µ and at ∆/2 λ + ωE µ entirely and the spectral density takes on the appear- − − − − − in top and bottom frame respectively. These kinks are ance of a broad incoherent background with no sharp directly mirrored in the DOS of the left hand frames. Of quasiparticle peaks. Electron and phonon are no longer course we expect that it is not just the k=0 value of the individually defined and a Green’s function formalism as spectral density that contributes to the density of state we have used here is needed for a proper description. around the band edge but the results given are enough Near k = 0, which corresponds to states near the top of to understand how the band edge becomes modified from the valence band and bottom of the conduction band its bare band behavior. with a gap in between for massive Fermions, we find In Fig. 6 (top panel) we present additional results for that the spectral function can remain largely coherent the dressed density of state when the -1 in the last term and quasiparticle-like, while in other instances it can be of our Hamiltonian Eq. (1) is left out and we use much quite incoherent. In the first case the density of states reduced energy scales with ∆/2= 20 meV and λ/2=10 (DOS) near the gap edge retains the general character- meV which is more representative of silicene (see illus- istic of bare bands, while in the second case, the shape trative figure in right hand frame). In both upper and of the band edge becomes much more complex reflecting lower frames the black solid line gives the contribution the incoherence due to correlations and the absence of to the total DOS of the spin down band and the dot- dominant quasiparticle peaks. ted of the spin up band. In this case the green and blue In addition to the familiar quasiparticle self energy, in color apply to the conduction and valence band respec- the case of gapped Dirac fermions, the electron-phonon tively. In the upper frame we employ a phonon energy interaction also renormalizes directly the gap through a 6 new self energy which is complex and energy dependent. (STM). This self energy shifts up the top of the valence band, and shifts down the bottom of the conduction band, thus re- duces the effective gap. It does not however introduce ACKNOWLEDGMENTS additional damping of the spectral density. We found that it is this self energy that plays the major role in re- This work was supported by the Natural Sciences and shaping the gap edges in the DOS making it go from co- Engineering Research Council of Canada (NSERC) and herent to incoherent behavior. The effects described here the Canadian Institute for Advanced Research (CIFAR). can be measured in angular-resolved photoemission spec- troscopy (ARPES) and in scanning tunneling microscopy REFERENCES

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0.01 Re Z 5*Im Z

0.00

Re

-0.01 5*Im

0.02 I I

0.00

Re Re

-0.02 (- 5*Im) -5*Im

=7.5m eV =0.03eV

E

0.15

E

Dressed band

Bare band

0.10 - - -

E DOS

Fermi surface - + -

0.05

E

-

E

0.00

-0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02

(eV)

FIG. 1. The real(solid) and imaginary(dashed) part of the self energy of a massive Dirac Fermion as a function of energy ω. The top frame of the top panel gives the mass renormal- ization ΣZ (ω) while the lower frame is for the quasiparticle renormalization ΣI (ω). The middle panel is the charge carrier density of states DOS as a function of ω. It compares the bare band case (solid line) with the dressed case (dashed line). The bare band chemical potential was set at µ = 30meV and the bare band gap extends from -55 meV to -15 meV (solid line). Coupling to a phonon at ωE = 7.5 meV introduces structure at −µ−∆/2−ωE, −µ+∆/2−ωE, −ωE and ωE . The gap for the interacting case is shifted with respect to its bare value and its magnitude is effectively reduced to about 25 meV. In the bottom panel we show a schematic of the band structure of ungapped (left) and gapped (right) Dirac fermions. The blue dot in the middle of the Brillouin zone is the Γ point. 8

-0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01

600

300 -52.5 meV

0

-12.5 meV

-7.5 meV

7.5 meV ka=0.025

/2= 20 meV ka=0.02

)

= 25 meV ka=0.015

A(

=7.5 meV

ka=0.01

E

ka=0.005

600

300 ka=0.0

0

-0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01

(eV)

FIG. 2. (Color online) The Dirac fermion spectral density A(k,ω) vs. ω in eV for various values of momentum as la- beled. Each curve is restricted to the region below 600 for ease of viewing. The chemical potential is 25 meV and both gap and quasiparticle self energies ΣZ (ω) and ΣI (ω) are in- cluded. The vertical dotted lines mark important energies 7.5, -7.5(blue), -12.5 and -52.5 meV(red). 9

FIG. 3. (Color online) Color plots of the dressed Dirac fermion dispersion curves ω = E(k) (left frame) as a function of momentum k compared with the bare case (right frame). In this last case a small constant residual scattering rate of Γ = 0.1meV was included. The chemical potential is set at 25 meV and the bare gap ∆ is 40 meV. For ease of comparison between left and right frames we have used for the bare case a value of chemical potential shifted by the value of ReΣI (ω) at ω = 0. Fig. 3b is same as Fig. 3a but the electron-phonon coupling has been halved to show the progression from bare to dressed case. 10

6

spin up + spin down

spin up

= -0.845 eV

4

- -

E DOS

E

2

E

+ +

E

0

= -0.995 eV

E

-0.755eV E

4

-0.905eV DOS

2

- + + +

E

E

0

-1.2 -1.0 -0.8 -0.6 -0.4

(eV)

FIG. 4. (Color online) The dressed Dirac fermion density of state as a function of ω for the case of a single layer MoS2 with spin polarized bands. In the top frame of the top panel the chemical potential has been chosen to be equal to -0.845 eV which falls below the top of the spin up and above the top of the spin down valence band. The chemical potential is shown by the vertical black dashed line. The phonon structures are at −∆/2 − λ − ωE , µ − ωE , µ + ωE and −∆/2+ λ + ωE . In the lower frame of the top panel µ = -0.995 eV and so falls below the top of both spin up and down bands as shown. The phonon structures are at µ − ωE, µ + ωE,−∆/2 − λ + ωE and −∆/2+ λ + ωE. The dotted red curves are for the spin up band and the solid black for sum of up and down. The bottom panel is a schematic of the band structure of MoS2. The spin splitting of the conduction band is small and not visible in the schematic. The electron-phonon mass renormalization was set at λep=0.1. 11

2.4 1000

spin up =-0.995eV

800

1.8

bare ) spin up

I ,

600 k=0

1.2

DOS +

+ + A (

E 400

+ + -

0.6 E

200

0.0 0

-0.80 -0.75 -0.70 -0.65 0.225 0.250 0.275 0.300

1000

spin down spin down 800

2 ) ,

600 k=0

DOS - + A( E - + - 400

E E 1

200

0 0

-0.95 -0.90 -0.85 -0.80 0.06 0.09 0.12 0.15

(eV)

(eV)

FIG. 5. (Color online) The dressed density of state for the massive Dirac fermions of single layer MoS2. The chemical potential is µ =-0.995 eV and falls below the top of both spin up (left top frame) and down (left bottom frame) valence bands as illustrated in the inset of Fig. 4. In each frame the solid curves are the bare band results shown for comparison with red dashed curve for which we have included only the quasiparticle self energy ΣI (ω), and the blue dotted curves which involves both ΣI (ω) and ΣZ (ω). The bare case has been shifted by the constant ReΣI (ω) at ω=0. The right frames show the corresponding spectral density A(k = 0,ω) vs. ω. 12

0.20

spin down

spin up

0.15

spin up

=7.5meV

E

0.10

=20meV

=10meV

Bare bands

0.05

DOS 0.00

- /2- + +

E E

=16.5meV

0.25 E

E

0.20

- +

E

0.15

=-20m eV

+ +

E

0.10

E

0.05

0.00

-0.04 -0.02 0.00 0.02 0.04

(eV)

FIG. 6. (Color online) The density of state vs. ω for massive Dirac fermions described by the Hamiltonian Eq. (1) (with the -1 in the last term left out) and coupling to phonons described by Eq. (2). The parameters used are much smaller than those for MoS2. They are illustrative of silicene with ∆/2= 20 meV, ωE= 7.5 meV and λ/2= 10 meV. Both conduction and valence bands are shown. Solid curves are for spin down and dotted for spin up (green is for the conduction band and blue for the valence band). The bare band edges are shown as heavy vertical red lines and an arrow indicates how they are shifted by interactions. The phonon structures are identified in terms of ∆, λ and ωE . In the lower frame of the top panel the phonon energy has been shifted to 16.5 meV. The bottom panel is a schematic of the bands in silicene. The spin splitting is the same size in both valence and conduction bands. The electron-phonon mass renormalization is λep=0.3 for ωE=7.5 meV and λep=0.45 for ωE =16.5 meV.