Dynamical properties of quasiparticles in a gapped graphene sheet
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Dynamical properties of quasiparticles in a gapped graphene sheet
A. Qaiumzadeh,
1, 2
F. K. Joibari, and Reza Asgari Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, 45195-1159, Iran School of Physics, Institute for research in fundamental sciences, IPM 19395-5531 Tehran, Iran
We present numerical calculations of the impact of charge carriers-carriers interactions on thedynamical properties of quasiparticles such as renormalized velocity and quasiparticle inelastic scat-tering lifetime in a gapped graphene sheet. Our formalism is based on the many-body G W -approximation for the self-energy. We present results for the many-body renormalized velocitysuppression and the renormalization constant over a broad range of energy gap values. We find thatthe renormalized velocity is almost independence of the carrier densities at large density regime. Wealso show that the quasiparticle inelastic scattering lifetime decreases by increasing the gap value.Finally, we present results for the mean free path of charge carriers suppression over the energy gapvalues. PACS numbers: 71.10.Ay, 81.05.Uw, 71.45.Gm
I. INTRODUCTION
The latest rival to succeed silicon’s status is graphene, a single atomic layer of graphite make a truly tiny transistorto decrease the size and to improve the operational speed of the electronic devices. Silicon lost it’s brilliant electronicproperties in pieces smaller than about 10nm and practically the smallest silicon chips which has been used in silicon-based electronics is 45nm. Furthermore, silicon has some limitations in speed of operations. These restrictions leadto serious challenges for the Moore’s law which states that the number of transistors can be placed inexpensively onan integrated circuit has increased exponentially, doubling approximately every two years. This growth cannot bemaintained forever and thus the search is on to find and use new materials which may be able to produce higherperformance and better functionality.The recent discovery of graphene in 2004, and its fabrication into a field-effect transistor , has opened up a newfield of physics and offers exciting prospects for new electronic devices and apparently possible to come over thoseaforementioned limitations. Graphene has instructive and unique physics with special intriguing electronic propertieswhich has attracted remarkable attentions. First, the electronic properties of graphene are improved in sizes lessthan 10nm . Second, the massless Dirac-like electrons move through graphene with almost near-ballistic transportbehavior with less resistance because back-scattering is suppressed. Third, graphene is itself a good thermal conductorsuch that graphene’s thermal conductivity is about ∼ . × W/mK at room temperature which is greater thanthe thermal conductivity of carbon nanotubes. Interestingly, the mobility of carriers in graphene is quite high andit is about 10 cm /Vs at room temperature. It is important to note that the highest electron mobility recordedon the semiconductor junction H-Si(111)-vacuum FET is 8 × cm /Vs at 4 . (100) MOSFET systems is 25 × cm /Vs at low temperature , make graphene promising for differentapplications in devices.Providing capability to control a type and density of charge carriers by gate voltage or by the chemical doping madegraphene instructive for novel nano-electronic devices. However, a gapped semiconducting behavior would be moresuitable for electronic applications. There have been some proposed in literature for a gap generation in graphene dueto breaking of the sublattice symmetry by some substrates (such as SiC , graphite and boron nitride ), to adsorbesome molecules (such as water, ammonia and CrO ), spin-orbit interaction and finite size effect. In case, weare interested to carry out the microscopic theory to calculate some physical quantities of gapped graphene.Theoretical calculations of quasiparticle properties of electron in conventional two-dimensional electron liquid areperformed within the framework of Landau’s Fermi liquid theory whose key ingredient is the quasiparticle conceptand its interactions. As applied to the electron liquid model this entails the calculation of effective quasiparticle-quasiparticle interactions which enter the many-body formalism allowing the calculation of various physical prop-erties. A number of calculations considered different variants of the G W -approximation for the self-energy intwo-dimensional electron gas from which density, spin-polarization, and temperature dependence ofquasiparticle properties are obtained.There is a mechanism for quasiparticle scattering against quasiparticles because they interact through the Coulombinteraction. This is an inelastic process and induced a finite lifetime of the quasiparticles. The carrier lifetime inan epitaxial graphene layers grown on SiC wafers has been recently measured. Since experiments carried out theirmeasurements on graphene placed on SiC, we expect that graphene was gapped. The experimental measurements arerelevant for understanding carrier intraband and interband scattering mechanisms in graphene and their impact onelectronic and optical devises.
In this paper we focus on the effect of energy gap on the renormalized velocity, the inelastic scattering lifetimeof quasiparticles and the inelastic mean free path in gapped graphene sheets over the broad range of energy gap.Our formalism is based on the Landau-Fermi liquid theory incorporating the G W -approximation for the self-energy.These quantities are related to some important physical properties of both theoretical and practical applications suchas the band structure of ARPES spectra , the energy dissipation rate of injected carriers and the width of thequasiparticle spectral function. The contents of the paper are described briefly as follows. In Section II we discuss about our theoretical modelwhich contains the effect of gap in the renormalized velocity of quasiparticles and the inelastic scattering lifetime τ in ,of gapped graphene due to electron-electron interactions by using G W -approximation. Our numerical results aregiven in Section III. Finally, Section V contains the summery and conclusions. II. THEORETICAL MODEL
Among the methods designed to deal with the intermediate correlation effects, of particular interest for its physicalappeal and elegance is Landau’s phenomenological theory dealing with low-lying excitations in a Fermi-liquid.Landau called such single-particle excitations quasiparticles and postulated a one-to-one correspondence betweenthem and the excited states of a non-interacting Fermi gas. He wrote the excitation energy of the Fermi-liquid interms of the energies of the quasiparticles and of their effective interaction. The quasiparticle-quasiparticle interactionfunction can in turn be used to obtain various physical properties of the system and can be parameterized in termsof experimentally measurable data. In this paper, we will compute the energy gap dependence of the renormalizedvelocity, renormalization constant and the inelastic scattering lifetime of quasiparticle in a gapped graphene sheet. A. Quasiparticle renormalized velocity
The dynamics of quasiparticles in a gapless graphene are described by two-dimensional (2D) massless Dirac Hamil-tonian ˆ H = ~ vσ · k , with eigenvalues ε s k = s ~ vk , where s = +( − ) representing right- and left-handed helicity orchirality for the electrons and holes, respectively. Note that chirality is the same as helicity for the massless particles. v = 10 m / s is the Fermi velocity. As it has been shown before , contrary to conventional 2D electron systems, theinteractions increase the velocity of quasiparticles in graphene because of interband exchange interactions and thedifference between positive and negative energy branches due to the chirality.The dynamics of quasiparticles in a gapped graphene are described by 2D massive Dirac Hamiltonian given byˆ H = ~ v F σ · k + mv σ with eigenvalues E s k = s p ( ~ vk ) + ∆ where ∆ = mv is the gap energy. Due to massive termin the Hamiltonian, the chirality differs from the helicity and also the helicity is conserved but is frame dependence.From the microscopic point of view, the quasiparticle energy can be calculated by solving the Dyson equation, δε QPs k = ξ s k + ℜ e [ δ Σ rets ( k , ω )] | ω = δε QPs k / ~ , (1)where ξ s k = E s k − E F is the energy of a quasiparticle relative to the Fermi energy. The Fermi wave vector ingraphene is given by k F = (4 πn/g s g v ) / where g s = g v = 2 are spin and valley degeneracy, respectively. TheFermi energy of gapless graphene is ε F = ~ vk F . The retarded self-energy of gapped graphene is Σ rets and we define δ Σ rets ( k , ω ) = Σ rets ( k , ω ) − Σ rets ( k F , , on the other hand, the above equation mustbe solved by setting ω = ξ s k / ~ .In the G W - approximation , the self-energy of gapped graphene at finite temperature ( β = 1 / ( k B T )) is given byΣ s ( k , iω n ) = − β X s ′ Z d q (2 π ) F ss ′ ( k , k + q ) (2) × + ∞ X m = −∞ W ( q , i Ω m ) G (0) s ′ ( k + q , iω n + i Ω m ) , where the dynamic screened effective interaction is W ( q , i Ω m ) = V q /ǫ ( q, i Ω m ) and ǫ ( q, i Ω m ) is the dynamical dielectricfunction and the bare Coulomb interaction is V q = 2 πe /κq where κ is the averaged background dielectric constant ofgraphene is placed on a substrate. G (0) s ( q, i Ω m ) = 1 / ( i Ω m − ξ s k / ~ ) is the standard noninteracting Green’s function.The overlap function for gapped graphene F ss ′ ( k , k + q ), is given by F ss ′ ( k , k + q ) = 12 (1 + ss ′ ~ v k · ( k + q ) + ∆ E k E k + q ) . (3)To evaluate of the zero temperature retarded self-energy, we decompose the self-energy into the line which is purely areal function and residue contributions, Σ rets ( k , ω ) = Σ lines ( k , ω ) + Σ ress ( k , ω ), where Σ line is obtained by performingthe analytic continuation before summing over the Matsubara frequencies, and Σ res is the correction which must betaken into account in the total self-energy,Σ lines ( k , ω ) = − X s ′ Z d q (2 π ) V q F ss ′ ( k , k + q ) (4) × Z ∞−∞ d Ω2 π ǫ ( q , i Ω) 1 ω + i Ω − ξ s ′ ( k + q ) / ~ , and Σ ress ( k , ω ) = X s ′ Z d q (2 π ) V q ǫ ( q , ω − ξ s ′ ( k + q ) / ~ ) F ss ′ ( k , k + q ) × [Θ( ω − ξ s ′ ( k + q ) / ~ ) − Θ( − ξ s ′ ( k + q ) / ~ )] , (5)where the dynamic dielectric function is given by ǫ ( q , ω ) = 1 − V q χ (0) ( q, ω ) in the random phase approximation(RPA) and χ (0) ( q, ω ) is the noninteracting polarization function for gapped graphene. The noninteracting polarizationfunction has been recently calculated on both along the imaginary and real frequency axis . The noninteractingpolarization function expressions along the real frequency axis are given in appendix A.Note that there are two independent parameters in the self-energy. One of them is the Fermi energy E F , and theother is the dimensionless coupling constant α gr = g s g v e /κ ~ v . The coupling constant in graphene depends onlyon the substrate dielectric constant while in the conventional 2D electron systems the coupling constant is densitydependent. For graphene placed on SiC or graphite substrates, the coupling constant is about α gr ≃ k for the isotropic systems. Expanding δε QP + k to first orderin k − k F , we obtain δε QP + k ≃ ~ v ∗ ( k − k F ) which effectively defines the renormalized velocity as ~ v ∗ = dδε QP + k /dk | k = k F .The renormalized velocity in the Dyson scheme is thus given by v ∗ v = (1 + ∆ ) − / + v − ∂ k ℜ e [ δ Σ ret + ( k , ω )] | ω =0 ,k = k F − ∂ ω ℜ e [ δ Σ ret + ( k , ω )] | ω =0 ,k = k F . (6)In the on-shell approximation, on the other hand, the renormalized velocity is given by v ∗ /v = (1 + ∆ ) − / + v − ∂ k ℜ e [ δ Σ ret + ( k , ω )] | ω =0 ,k = k F + (1 + ∆ ) − / ∂ ω ℜ e [ δ Σ ret + ( k , ω )] | ω =0 ,k = k F . The renormalized velocity in this approx-imation demonstrates qualitatively the same behavior obtained by the Dyson equation, Eq. (6) but its magnitudeis larger than the one calculated within the Dyson scheme. There is an ultraviolet divergence in the wave vectorintegrals of the line contribution in a continuum model formulated as discussed above. We introduce an ultravioletcutoff for the wave vector integrals, k c = Λ k F which is the order of the inverse lattice spacing and Λ is dimensionlessquantity. For definiteness we take Λ = k c /k F to be such that π (Λ k F ) = (2 π ) / A , where A = 3 √ a / a ≃ .
42 ˚A the carbon-carbon distance. With this choice,Λ ≃ ( gn − √ / . / × , where n is the electron density in units of 10 cm − .An important quantity in the Fermi-liquid theory is the renormalization constant Z , defined as the square of theoverlap between the state of the system after adding (or removing) of an electron with the Fermi wave vector andthe ground-state of the system. The non-zero renormalization constant value is always smaller than the one for thenormal Fermi-liquid systems and can be calculated explicitly as follow Z = 11 − ∂ ω ℜ e [ δ Σ ret + ( k , ω )] | ω =0 ,k = k F . (7)We will show that Z is a finite number for gapped graphene and it confirms as well that the system is a Fermi-Liquid. B. Inelastic scattering lifetime
In this subsection, we compute the inelastic scattering lifetime of quasiparticles due to carriers-carriers interactionsat zero temperature and disorder-free for gapped graphene sheets. This is obtained through the imaginary part ofthe self-energy when the frequency evaluated at the on-shell energy. τ − in ( k ) = Γ in ( k , ξ + k / ~ ) = − ~ ℑ m Σ ret + ( k , ξ + k / ~ ) , (8)where Γ in ( k , ξ s k / ~ ) is the quantum level broadening of the momentum with eigenstate | s k > . It is worthwhile tonote that the expression of τ − in is identical with a result obtained by the Fermi’s golden rule summing the scatteringrate of electron and hole contributions at wave vector k . Note again that the total contribution of the imaginarypart of the retarded self-energy comes from the residue term both intra- and interband contributions, ℑ m Σ ret + ( k , ω ) = ℑ m Σ resintra ( k , ω ) + ℑ m Σ resinter ( k , ω ). However, the total contribution of the imaginary part of the retarded self-energyevaluated at the on-shell energy comes only from intraband term, ℑ m Σ ret + ( k , ξ k / ~ ) = ℑ m Σ resintra ( k , ξ k / ~ ). We willdiscuss about that with more details in the appendix B and C.We turn our attention to investigate the imaginary part of the retarded self-energy with more details. By startingfrom Eq. (5), we end up to an expression for the imaginary part of self-energy which is given by, ℑ m Σ ret + ( k , ω ) = ℑ m Σ resintra ( k , ω ) + ℑ m Σ resinter ( k , ω )= Z d q (2 π ) V q ℑ m [ ǫ − ( q , ω − ξ + ( k + q ) / ~ )] F ++ ( k , k + q ) × [Θ( ω − ξ + ( k + q ) / ~ ) − Θ( − ξ + ( k + q ) / ~ )]+ Z d q (2 π ) V q ℑ m [ ǫ − ( q , ω − ξ − ( k + q ) / ~ )] F + − ( k , k + q ) × [Θ( ω − ξ − ( k + q ) / ~ ) − Θ( − ξ − ( k + q ) / ~ )] . (9)where the imaginary part of the inverse dielectric function in RPA level is obtained by ℑ m [ ǫ − ( q , ω )] = V q ℑ mχ (0) ( q , ω )[1 − V q ℜ eχ (0) ( q , ω )] + [ V q ℑ mχ (0) ( q , ω )] . (10)It is worth to note that the plasmon contributions in the imaginary part of self-energy comes from the zero-solutionsof denominator in Eq. (10). III. NUMERICAL RESULTS
We turn to a presentation of our main numerical results. We present some illustrative results for the quasipar-ticle dynamic properties such renormalized velocity, renormalization constant and inelastic scattering lifetime. Allnumerical data are calculated in the Dyson scheme at α = 1.The Fermi liquid phenomenology of Dirac electrons in gapless graphene and conventional 2D electron liquid have the same structure, since both systems are isotropic and have a single circular Fermi surface. The strength ofinteraction effects in a conventional 2D electron liquid increases with decreasing carrier density. At low densities, thequasiparticle renormalization constant Z is small, the renormalized velocity is suppressed , the charge compressibilitychanges sign from positive to negative, and the spin-susceptibility is strongly enhanced . These effects emerge froman interplay between exchange interactions and quantum fluctuations of charge and spin in the 2D electron liquid.In the 2D massless electron graphene, on the other hand, it has been shown that interaction effects also becomenoticeable with decreasing density, although more slowly, the quasiparticle renormalization constant, Z tends to largervalues, that the renormalized velocity is enhanced rather than suppressed, and that the influence of interactions on thecompressibility and the spin-susceptibility changes sign. These qualitative differences are due to exchange interactionsbetween electrons near the Fermi surface and electrons in the negative energy sea and to interband contributions toDirac electrons from charge and spin fluctuations.In this paper we have shown the results for gapped graphene which are determined values between the gaplessgraphene evaluated at △ = 0 and the conventional 2D electron liquid where △ → ∞ .In Fig. 1, we have plotted the renormalized velocity as a function of carrier density for the various energy gap. Asa result, we see that the impact of energy gap on quasiparticles velocity which is similar to the effect of impurityto that on graphene . The renormalized velocity is almost density independent in gapped graphene at large carrierdensities. The renormalized velocity reduces dramatically by increasing the energy gap especially in the low carrierdensities. Importantly, the renormalized velocity becomes less than the bare velocity at large energy gap and lowdensity values. It is physically accepted since the system tends to conventional 2D electron liquid by increasing theenergy gap values. Note that in the conventional 2D electron systems, the renormalized velocity is suppressed byincreasing the coupling constant or reducing the density.We have shown the renormalization constant Z , as a function of the energy gap in Fig. 2. The renormalizationconstant enormously reduces by increasing the energy gap in mild densities, however it decreases quite slowly in highdensities.Fig. 3(a) is shown the absolute value of ℑ m Σ ret+ ( k , ω ) as from Eq. (9), evaluated at ω = ξ k / ~ . By increasing the gapvalue, this function takes a finite jump at the wave number of the plasmon dip and at large △ values, a discontinuityappears. The discontinuity is peculiar to 2 D electron liquid. It is absent in gapless graphene and starts to arisefrom the fact that the oscillator strength of the plasmon pole is non-zero at special k value for gapped graphene.Fig. 3(b) is clearly shown the behavior of the energy gap dependence of the inverse inelastic scattering lifetime.As it is argued in the Appendix B, the imaginary part of self-energy evaluated at the on-shell energy start from △ − p ε + △ and in case the results are truncated below that. The quasiparticle lifetime decreases by increasingthe gap value and it is a clear difference between 2D massless Dirac electron and gapped graphene. Consequently,the inelastic scattering lifetime in graphene is always larger than the conventional 2D electron liquid. In the case ofgapless graphene, scattering rate is a smooth function because of the absence of both plasmon emission and interbandprocesses, nevertheless with generating a gap and increasing the amount of it, plasmon emission causes to arise adiscontinuity in the scattering time, similar to conventional 2D electron liquid. We have thus two mechanisms forscattering of the quasiparticles. The excitation of electron-hole pairs which is dominant process at low wave vectorsand the excitation of plasmon appears in a specific wave vector. We also see in Fig. 3(b) that the scattering rate isquite sensitive to the gap energy and the scattering rate increases by increasing the energy gap.In Fig. 4, we have depicted the inelastic mean free path l in ( k ) = v ∗ τ in ( k ), as a function of the on-shell energyfor various gap energies. To this purpose we multiplied the results of τ in ( k ) to a proper renormalized velocity. As aresult the mean free path of a gapped graphene is shorter than that obtained for gapless graphene. Furthermore, themassless graphene has larger l in and it decreases by increasing the energy gap values. Note that the typical value ofenergy gap due to breaking sublattice symmetry is △ = 10 −
100 meV corresponding the inelastic mean free path is l in = 20 −
50 nm which implies that the system remains in the semi-ballistic regime. . IV. SUMMERY AND CONCLUDING REMARKS
In summary, we have studied the problem of the microscopic calculation of the quasiparticle self-energy and many-body renormalized velocity suppression over the energy gap in a gapped graphene. We have carried out calculationsof both the real and the imaginary part of the quasiparticle self-energy within G W -approximation. We have alsopresented results for the renormalized velocity suppression and for the renormalization constant over a wide range ofenergy gap. We have shown that the renormalized velocity for a gapped graphene is almost independent of the carrierdensity at high density. We have finally presented results for the quasiparticle inelastic scattering lifetime suppressionover the energy gap and show that the mean free path of the charge carriers of a gapless graphene is larger than agapped graphene one. In case the mean free path of charge carriers decrease by increasing the energy gap.A possible role of correlations including the charge-density fluctuations beyond the Random Phase Approximation,remains to be examined. Acknowledgments
R. A. would like to thank the International Center for Theoretical Physics, Trieste for its hospitality during theperiod when part of this work was carried out. A. Q is supported by IPM grant.
APPENDIX A: THE DYNAMIC POLARIZATION FUNCTION FOR A GAPPED GRAPHENE
In this appendix we present the real and imaginary part of the noninteracting polarization function for a gappedgraphene, which is calculated recently by Pyatkovskiy. The dynamic polarization function for gapped graphenein the imaginary frequency axis is also calculated by us in Ref. [29]. Importantly, the noninteracting polarizationfunction along the imaginary frequency axis can be obtained by performing analytical continuation from real axis andthose results are the same. First, by introducing some following notations, f ( k, ω ) = g s g v k π p | ~ v k − ~ ω | ,g ± = 2 E F ± ~ ω ~ vk ,x = r ~ v k − ~ ω ,G < ( x ) = x q x − x − (2 − x ) cos − ( x/x ) ,G > ( x ) = x q x − x − (2 − x ) cosh − ( x/x ) ,G ( x ) = x q x − x − (2 − x ) sinh − ( x/ q − x ) , (A1)the real part of noninteracting polarization function is given by, ℜ eχ (0) ( k, ω ) = − g s g v E F πv F + f ( k, ω ) × , G < ( g − ) , G < ( g + ) + G < g − ) , G < ( g − ) − G < ( g + ) , G > ( g + ) − G > ( g − ) , G > ( g + ) , G > ( g + ) − G > ( − g − ) , G > ( − g − ) + G > ( g + ) , G ( g + ) − G ( g − ) ,
5B (A2)and the imaginary part of noninteracting polarization function is given by, ℑ mχ (0) ( k, ω ) = f ( k, ω ) × G > ( g + ) − G > ( g − ) , G > ( g + ) , , , , − G < ( g − ) , π (2 − x ) , π (2 − x ) , ,
5B (A3)with the followings regions in the ( k, ω ) space,1 A ~ ω < E F − p ~ v ( k − k F ) + ∆ , A | E F − p ~ v F ( k − k F ) + ∆ | < ~ ω < − E F + p ~ v ( k + k F ) + ∆ , A ~ ω < − E F + p ~ v ( k − k F ) + ∆ , A − E F + p ~ v ( k + k F ) + ∆ < ~ ω < ~ vk, B k < k F , √ ~ v k + 4∆ < ~ ω < E F + p ~ v ( k − k F ) + ∆ , B E F + p ~ v ( k − k F ) + ∆ < ~ ω < E F + p ~ v ( k + k F ) + ∆ , B ~ ω > E F + p ~ v ( k + k F ) + ∆ , B k > k F , √ ~ v k + 4∆ < ~ ω < E F + p ~ v ( k − k F ) + ∆ , B ~ vk < ~ ω < √ ~ v k + 4∆ , (A4) APPENDIX B: THE INTRABAND CONTRIBUTION OF SELF-ENERGY
Since we are interested in quasiparticle properties, we therefore need only s = + contribution. Let us focus on theintraband contribution of the retarded self-energy. The second argument of the dielectric function in Eq. (5) ( bysetting ~ = v = 1) is ω − ξ + ( k + q ) = ω + E F − p k + q + 2 kq cos φ + ∆ . (B1)In this case, we change the variable φ and integrate it over y = p k + q + 2 kq cos φ + ∆ . Using the new variable,the intraband contribution of self-energy changes toΣ resintra ( k , ω ) = e πκ √ k + ∆ Z + ∞ dq Z √ ( k + q ) +∆ √ ( k − q ) +∆ dy p k q − ( y − k − q − ∆ ) ( √ k + ∆ + y ) − q ǫ ( q , ω + E F − y ) × [Θ( ω + E F − y ) − Θ( E F − y )] . (B2)We can now simplify the Θ-functions further in Eq. (B2) by considering the positive and negative regions of ω asfollow 1) ω + E F − y > and E F − y < It implies that ω > , ω + E F − y < and E F − y > It implies that ω < . To consider the first case where ω >
0, the difference between the two Θ-functions in Eq. (B2) is equal to +1 if E F < y < ω + E F and p ( k − q ) + ∆ < y < p ( k + q ) + ∆ . (B3)Now we do need to find the overlap between these two intervals. We simply end up to inequivalent conditions whichare q > k − q ω + k + 2 ω p k + ∆ , q < k + q ω + k + 2 ω p k + ∆ and q > k F − k . Collecting everythingtogether and using the fact that q ≥
0, we finally findΣ resintra ( k , ω >
0) = e πκ √ k + ∆ Z k + q ω + k +2 ω √ k +∆ max (0 ,k F − k,k − q ω + k +2 ω √ k +∆ ) dq Z min ( ω + √ k +∆ , √ ( k + q ) +∆ ) max ( √ k +∆ , √ ( k − q ) +∆ ) dy × ( √ k + ∆ + y ) − q ǫ ( q , ω + E F − y ) p k q − ( y − k − q − ∆ ) (B4)By considering of the second case where ω <
0, the difference between the two Θ-functions in Eq. (5) is equal to − E F + ω < y < E F and p ( k − q ) + ∆ < y < p ( k + q ) + ∆ (B5)As what we did before, we calculate overlap between intervals and thus we find q > k − k F and q < k + k F , q > q ω + k + 2 ω p k + ∆ − k . Putting everything together and using the fact that q ≥ resintra ( k , ∆ − E F < ω <
0) = − e πκ √ k + ∆ Z k + k F max (0 ,k − k F , q ω + k +2 ω √ k +∆ − k ) dq × Z min ( √ k +∆ , √ ( k + q ) +∆ ) max (0 ,ω + √ k +∆ , √ ( k − q ) +∆ ) dy × ( √ k + ∆ + y ) − q ǫ ( q , ω + E F − y ) p k q − ( y − k − q − ∆ ) (B6)Σ resintra ( k , ω < − ∆ − E F ) = − e πκ √ k + ∆ Z k + k F max (0 ,k − k F ) dq Z min ( √ k +∆ , √ ( k + q ) +∆ ) max (0 ,ω + √ k +∆ , √ ( k − q ) +∆ ) dy × ( √ k + ∆ + y ) − q ǫ ( q , ω + E F − y ) p k q − ( y − k − q − ∆ ) (B7)The real and imaginary part of intraband contributions can be computed. APPENDIX C: THE INTERBAND CONTRIBUTION OF SELF-ENERGY
Now, we focus on the interband contribution of the retarded self-energy. The second argument of the dielectricfunction in Eq. (5) is ω − ξ − ( k + q ) = ω + E F + p k + q + 2 kq cos φ + ∆ . (C1)We change variable y = p k + q + 2 kq cos φ + ∆ , then we findΣ resinter ( k , ω ) = e πκ √ k + ∆ Z + ∞ dq Z √ ( k + q ) +∆ √ ( k − q ) +∆ dy p k q − ( y − k − q − ∆ ) q − ( y − √ k + ∆ ) ǫ ( q , ω + E F + y ) × [Θ( ω + E F + y ) − . (C2)Note that Σ resinter can be non-zero if ω + E F + y < and y >
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