Deformation Effects: Energy Gap in LDA and GW Approximation

Deformation Effects Energy Gap in LDA and GW ApproximationTheoretical calculations and experimental measurements indicating the importance of many-bodyeffects in slighten dimensional systems. We performed ab initio calculations based on density functional theory and many-body perturbation theory in the GW approximation. To illustrate ourresults, we consider a (8 0) whizz skirt one C nanotube and by solving the Bethe-Salpeter equalitycalculate the macroscopical dielectric function for both the undeformed and deformed nanotubes.The radial deformation is obtained by liquidity crisis the nanotube in the y direction and elongating inthe x direction. Results intend a decrease in knell violate and a red poke in exciton passageway energiesfor nanotubes of unsubdividedal cross-section. The deformation can be proposed as ideas for the achieveto less excitonic capacity. We implement the method in the ABINIT code for ground and crazy presents calculations.Single wall carbon nano tubes (SWCNTs) atomic number 18 cylindrical bodily structures that formed by rolling up a graphenesheet. SWNCTs geometric structures describe by chiralvector or positive integer pairs (n,m). Nanotubes with(n,0) chirality is said to be of zigzag carbon nanotubes,with (n,n) are armchair nanotubes and tubes with (n,m)are chiral nanotubes.In the zone folding approximation if the difference between these twain integers (n,m) is an integer multipleof 3, tubes are metal, otherwise, the tubes show semiconducting properties1.The observation and synthesis of single-walledcarbon nanotubes in recent years, making possiblethe experimental study of the opthalmic properties ofindividual SWCNTs. Beca mathematical function of the nature of quasione-dimensional carbon nanotubes many-body effectshave an important in uence on their optic propertiesand failure of single- subtracticle theories not unexpected.The rst optical data for carbon nanotubes wasobtained in 1999 by Kataura2, which reports the transitions energies (Eii) as a function of the tube diameterfor nanotube with different chirality (n,m).A few years later with the precise spectroscopy3 showedsome deviations from the summary of Kataura.In particular, the ratio (E22=E11), predicted to be equalto 2 in the approximation where portions are linear closeto Fermi energy4 was found to be smaller3, and thisproblem was not justi ed by single-particle theories, thisproblem so-called ratio problem in SWCNTs5.Recent predictions based on rst principles calculations and semi-empirical approaches show the existenceof exciton with high binding energy in the carbonnanotubes, so that the unknown effects observed inthe optical spectra of nanotubes can be attributed toexcitons and by considering the excitonic effects theratio probem would be solved.An usher for the excitons in carbon nanotubes isobtained in the theory6 and experiment3. A theoretical approach is the rst principles calculations of opticalspectra of carbon nanotubes, using the Bethe-Salpeter equivalency. These calculations show exciton with largebinding energy in semiconductor device nanotubes and even excitonic effects in metallic nanotubes6.In the present work, we obtain optical spectra with abinitio calculations for Bethe-Salpeter equation for nanotubes of simpleal cross-section.To illustrate our results, we consider a (8,0) single wallcarbon nanotube, past in our model squeezing the nanotube in the y direction and elongating in the x direction,we study quasiparticle band structures and kindling energies for nanotubes with elliptical cross-section. Withthis model, the deformation effects on the exciton energies is investigated.However, so uttermost the excitation energies are metricfor nanotubes with elliptical cross-section, but this calculation is done with single-particle approach that irrespective of the excitonic effects.Shan and Bao7 investigated the deformation effects onthe optical properties of carbon nanotubes based onthe tigh t-binding model and describe the deformationof SWCNT under stretching, compression, torsion, andbending, they were shown the shifting, merging, andsplitting of Van Hove Singularities in the DOS, and optical dousing properties variation with strains.We present a framework to predict the optical soaking up of deformed SWNTs using the Bethe-Salpeter equation with many-body approach, so far this work has notbeen done. The results can be employed to understandand guide experimental studies of electronic and optoelectronic devices based on the CNTs.With density functional theory can be calculatedground state energy and charge density for a many-bodyinteracting system. We obtain the DFT wave functionsand eigenvalues of (8,0) SWCNT by solving the Kohn-Sham equations8 within the local density approximation, with Teter Pade parametrization9 for the exchange correlational statistics functional implemented in the ABINIT computational package10. The code uses a plan-wave basis set and a perio dic supercell method. For all studied systems, we have used the ab initio normconserving Troullier-Martins pseudopotentials11 and (1140)Monkhorst-Pack k-grid sampling of the Brillouin zonewas taken, for the self-consistent calculations with an energy cutoff 60 Ry.In the end of LDA calculations, we compare our LDAcalculations with results obtained using the QUANTUMESPRESSO12 package with the Perdew-Burke-Ernzerhofapproximation and Ultrasoft pseudopotentials in a planewave basis. there is no difference between the dickens calculations for bandgap (8,0) carbon nanonotube13.Density functional theory is used to study the groundstate of the system and this theory cannot be used inthe prediction of excited states. In the investigation ofthe excited states, the amount of band gap is greaterthan that is observed with the LDA calculations. So beyond the DFT should use a theory that describes excitations correctly. Our approach is the many-body perturbation theory14 based on the concept of q uasi-particlesand Greens function. In this theory, the quasi-particleenergies obtain by solving the following equation thatso-colled Dyson equationWhere T is the kinetic energy, Vext is the external potential, and VHartree is the average Hartree potential.is the self energy of the electrons and the indices refer toBloch states n, k, thus problem of nding quasi-particleenergies decreases to the problem of nding self-energy.A good approach that has been used extensivelyfor nding of self-energy is the GW approximation ofHedin15. In the GW approximation, using the followingequation, self-energy (r rE) can be calculatedHere G is the Greens function of the electrons andW = ..1v is the screened Coulomb fundamental interaction determined by the inverse dielectric ground substance ..1(r rE) and+ is a positive in nitesimal time.Greens function is obtained with the Kohn-Sham wavefunctions and eigenvaluesSince the wave functions are obtained with the LDA areappropriate, a rst order approximati on is sufficient tocorrect the LDA energies, for this reason quasi-particleenergies derived from the rst-order perturbation theoryby the following equationWhere V LDAxc is the exchange-correlation potential andZnk is the renormalization factor of the orbital de nedas Znk = (1 emailprotected)1jE=ELDAnk.In the equation (2), sigma is a convolution of G and W,this part of the calculation is very complex, because thematrix 1GG (q ) (in reciprocal space) must calculate forall frequencies , in direction of the real and imaginaryaxis. Since only the value of the organic is important,with a simple and acceptable model can calculate theintegrals. In this model the frequency dependence thematrix 1GG (q ) calculate with a plasmon celestial pole model16In this equation, 2GG (q) and (q) are the parameters ofmodel, the nal values for the parameters in this modelis found in Ref. 16. Dielectric function in this model isapproximated as a single- anthesis structure, this peak placedin the plasmon f requency p. Plasmon pole model notonly reduces computation, but also makes an analyticcalculation of the relation (2).With the GW calculations, correction to the energygap of the carbon nanotube is obtained. The ABINITpackage has been used for the Hybertsen-Louie plasmonpole model calculations. For all the GW calculations,the energy cutoff is 36 Ry for the evaluation of the bareCoulomb exchange piece x, and 24 Ry for thecorrelation part c.With the many-body perturbation theory, can be calculate the excitation energies with obtaining self-energyusing the GW approximation. In fact, an optical absorption will build a pair of bound electron-hole or exciton.For the calculation of excitation energies, a good agreement between experiment and theory to be achievedwhen the interaction between the electron and hole arealso considered. BSE17 takes into account coupling between electron and hole and absorption spectra that obtained by solving this equation, is more consistent withthe experimen tal results. Bethe-Salpeter equation written for a bound two-body system, in condensed matterthis equation has the form of as followsWhere the quasiparticle energies EcEv enter on the diagonal, and the indices v, c refers to the occupied valenceand empty conduction band states,Wand V are screenedand bare Coulomb potentials, respectively.By solving the Bethe-Salpeter equation, exciton energies are calculated. In order to have an observation forthe excitonic energies, the macroscopic dielectric functionis calculated using the following equation18Where the Avcs is exciton amplitude and Es is excitonenergy. The relation between the imaginary part of Mto the frequency eliminates the absorption spectrum.In ABINIT, we use the option to evaluate the responsefunction recursively with Haydock algoritm19 and TammDancoff approximation14. Calculations of optical properties via BSE are more expensive computationally. Forboth undeformed and deformed SWNTs, the BSE kernel,in which the energy cuto ff is 16 Ry for V and W.Fig. 3 shows the band structure for undeformed (8,0)SWNT. According to the band structure, this SWNT isa semiconductor and amount of band gap is 057 eV. Werepeat the same calculation for elliptical tubes, with theprevious parameters (the same cutoff energy, numberof kpoint, and only the geometry of the tube willchange.Fig. 4 shows the band structure for deformed (8,0)SWNTs with different values of . In this calculation,the band gap decreases from Egap = 057 eV at = 10to the closing point, Egap = 00 eV at = 07. the energy gap is 049 eV , 026 eV for A, B elliptic nanotubes,respectively. For D, E and F, elliptic nanotubes no bandgap is found.In this calculation, A, B elliptic nanotubes remainedsemiconductor and the C elliptic nanotube representsthe boundary of the metal. In this approximation D, Eand F, elliptic nanotubes are metal.By this calculation we show that when the deformationis passing intense, the band gap decreases and oneinsulator-metal transitio n occurs.In the instant stage, we calculate correction bandgap energy and quasi-particle band structure with GWapproximation. Fig. 5 shows the quasiparticle bandstructure for undeformed (8,0) SWNT, in the GWapproximation band gap is 176 eV that is greater thanthe amount predicted in the LDA.Result for undeformed nanotube agrees well withab initio calculations presented in Ref. 6, that thecalculated value of the quasiparticle energy gap is given175 eV for undeformed (8,0) SWCNT. We performone-shot GW or G0W0 model where the convergencestudies have been carried out with respect to variousparameters (naumber of bands, cutoff energy, . . . ).In the previous stages, A, B elliptic nanotubesremained semiconductor and the C elliptic nanotuberepresents the boundary of the metal. We performedGW calculations only for semiconducting nanotubes.Fig. 6 shows the quasiparticle band structure and thecalculated value of the quasiparticle energy gaps thatthey are 165 eV , 134 eV for A, B elliptic nan otubes,respectively, and for the C elliptic nanotube no bandgap is found. For deformed nanotubes only the groundstate energy is calculated by many-body approach inRef. 20, so far no GW calculations have been done fordeformed nanotubes to compare our results with them.With the GW calculations, we conclude that when thedeformation is highly intense the band gap decreases,too. We show the evolution of the energy band gap(Egap) as a function of radial deformation in the Fig. 7,where the band gap in LDA and GW calculationsrepresents for nanotubes with different values of thedeformation.The values of the contributions of LDA exchangecorrelation potential Vxc, the exchange x and thecorrelation c part of the self energy are displayed inTable I-Table IV. Results are for plasmon pole models,in the Hybertsen-Louie approach presented in Ref.16. We calculate the screened interaction W( = 0)be expressed in terms of the inverse dielectric matrix..1(r rE), which describes screening in a solid whenl ocal elds due to density inhomogeneities and manybody effects are taken into account, to obtain self energyby (2).However, we found gap correction for undeformedand deformed nanotubes, but electron-hole interactiondecreased the excitation energy in these structures. Thecalculations include the electron-hole interaction (excitonic effects) are closer and split up values to experiment.In the third stage the macroscopic dielectric functionM() has been calculated by (7) including local eldeffects with solving the Bethe-Salpeter equation.In Fig. 8 A1 and B1 are peak for undeformed SWNT,A2, B2 and A3, B3 are for A, B deformed SWNT,respectively.The gure shows that with apply more deformationA,B peaks shift to lower energy, and red shift occurs inthe optical spectra of carbon nanotubes. Therefor thelow energy exciton can be occurred by deformation onthe nanotubes. Table V shows the values of lowest twooptical transition energies for the undeformed SWCNTin the present work and, ab initio ca lculations andexperiment. The value of ratio E11=E22 = 118 for the(8,0) tube is in agreement with the experiment findingsof Bachilo et al3. Bachilo and coworks in their workwith Spectrouorimetric measurements obtained rstand second transition energies for more than 30 semiconductor CNTs with different (n,m). their results showsratio equal to 1.17 for the (8, 0) nanotube and 1.85 fornanotube with a diameter larger, while a single-particlemodel, such as a tight bonding model is expected 2 value for this ratio. In considering excitonic effects theratio problem will be resolved and these calculationsgive us better results. We rst obtains values of therst and second excitation energy for the undeformedSWNT and were compared with computational andexprimental values, then we repeated calculations fordeformed nanotubes to get results. Table VI showslowest two optical transition energies for the undeformedand deformed SWNTs. The value of E11 and E22decreases with deformation.In conclusion, w e study the optical absorptionspectra of deformed and undeformed semiconducting small-diameter SWCNT and survey the agreement withavailable experimental data. We show by applyingdeformation on the nanotubes one insulator-metaltransition occurs, and peaks shift to lower energy,and red shift occurs in the optical spectra of carbonnanotubes. The deformation can be proposed as ideasfor achieving to less excitonic energy. The results canbe employed to understand and guide experimentalstudies of electronic and optoelectronic devices basedon the carbon nanotubes. So far GW calculations andabsorption spectra with excitonic effects for deformedtubes has not been obtained.We investigate deformation effects on the energy gapin LDA and GW approximation and optical spectra including excitonic effects. These calculations shows thatwith apply deformation on the SWNT structure, energygap decrease, and lowest two optical transition energiesfor the deformed SWNTs shift to lower energy. The deformatio n can be proposed as ideas for the achieve to lessexcitonic energy. The results can be employed to understand and guide experimental studies of electronic and optoelectronic devices based on the carbon nanotubes.We compare our results with experimental data andab initio calculations for undeformed nanotube, then repeat calculations for deformed nanotubes and investigatedeformation effects on the energy gap in LDA and GWapproximation and optical spectra. Investigation of excitonic effects so far has not been done with many-bodyapproach for Bethe-salpeter equation for deformed nanotubes. Results are agreed with sing-particle calculationsthat presented in Ref. 7.