Deformation Effects: Energy Gap in LDA and GW Approximation

Deformation Effects: Energy Gap in LDA and GW Approximation Theoretical calculations and experimental measurements indicating the importance of many-body  effects in reduced dimensional systems. We performed ab initio calculations based on density functional theory and many-body perturbation theory in the GW approximation. To illustrate our  results, we consider a (8; 0) single wall carbon nanotube and by solving the Bethe-Salpeter equation  calculate the macroscopic dielectric function à Ã‚ µ for both the undeformed and deformed nanotubes. The radial deformation is obtained by squeezing the nanotube in the y direction and elongating in  the x direction. Results show a decrease in band gap and a red shift in exciton transition energies  for nanotubes of elliptical cross-section. The deformation can be proposed as ideas for the achieve  to less excitonic energy. We implement the method in the ABINIT code for ground and excited  states calculations. Single wall carbon nanotubes (SWCNTs) are cylindrical structures that formed by rolling up a graphene  sheet. SWNCTs geometric structures describe by chiral  vector 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 two integers (n,m) is an integer multiple  of 3, tubes are metal, otherwise, the tubes show semiconducting properties1. The observation and synthesis of single-walled  carbon nanotubes in recent years, making possible  the experimental study of the optical properties of  individual SWCNTs. Because of the nature of quasione-dimensional carbon nanotubes many-body effects  have an important in uence on their optical properties  and failure of single-particle theories not unexpected. The rst optical data for carbon nanotubes was  obtained in 1999 by Kataura2, which reports the transitions energies (Eii) as a function of the tube diameter  for nanotube with different chirality (n,m). A few years later with the precise spectroscopy3 showed  some deviations from the analysis of Kataura. In particular, the ratio (E22=E11), predicted to be equal  to 2 in the approximation where bands are linear close  to Fermi energy4 was found to be smaller3, and this  problem was not justi ed by single-particle theories, this  problem so-called ratio problem in SWCNTs5. Recent predictions based on rst principles calculations and semi-empirical approaches show the existence  of exciton with high binding energy in the carbon  nanotubes, so that the unknown effects observed in  the optical spectra of nanotubes can be attributed to  excitons and by considering the excitonic effects the  ratio probem would be solved. An evidence for the excitons in carbon nanotubes is  obtained in the theory6 and experiment3. A theoretical approach is the rst principles calculations of optical  spectra of carbon nanotubes, using the Bethe-Salpeter  equation. These calculations show exciton with large  binding energy in semiconductor nanotubes and even excitonic effects in metallic nanotubes6. In the present work, we obtain optical spectra with ab  initio calculations for Bethe-Salpeter equation for nanotubes of elliptical cross-section. To illustrate our results, we consider a (8,0) single wall  carbon nanotube, then in our model squeezing the nanotube in the y direction and elongating in the x direction,  we study quasiparticle band structures and excitation energies for nanotubes with elliptical cross-section. With  this model, the deformation effects on the exciton energies is investigated. However, so far the excitation energies are calculated  for nanotubes with elliptical cross-section, but this calculation is done with single-particle approach that regardless of the excitonic effects. Shan and Bao7 investigated the deformation effects on  the optical properties of carbon nanotubes based on  the tight-binding model and describe the deformation  of SWCNT under stretching, compression, torsion, and  bending, they were shown the shifting, merging, and  splitting of Van Hove Singularities in the DOS, and optical absorption properties variation with strains. We present a framework to predict the optical absorption of deformed SWNTs using the Bethe-Salpeter equation with many-body approach, so far this work has not  been done. The results can be employed to understand  and guide experimental studies of electronic and optoelectronic devices based on the CNTs. With density functional theory can be calculated  ground state energy and charge density for a many-body  interacting system. We obtain the DFT wave functions  and eigenvalues of (8,0) SWCNT by solving the Kohn-Sham equations8 within the local density approximation, with Teter Pade parametrization9 for the exchange correlation functional implemented in the ABINIT computational package10. The code uses a plan-wave basis set and a periodic 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 zone  was taken, for the self-consistent calculations with an energy cutoff 60 Ry. In the end of LDA calculations, we compare our LDA  calculations with results obtained using the QUANTUM  ESPRESSO12 package with the Perdew-Burke-Ernzerhof  approximation and Ultrasoft pseudopotentials in a planewave basis. There is no difference between the two calculations for bandgap (8,0) carbon nanonotube13. Density functional theory is used to study the ground  state of the system and this theory cannot be used in  the prediction of excited states. In the investigation of  the excited states, the amount of band gap is greater  than 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 quasi-particles  and Greens function. In this theory, the quasi-particle  energies obtain by solving the following equation that  so-colled Dyson equation: Where 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 to  Bloch states n, k, thus problem of nding quasi-particle  energies decreases to the problem of nding self-energy. A good approach that has been used extensively  for nding of self-energy is the GW approximation of  Hedin15. In the GW approximation, using the following  equation, self-energy (r; r†²;E) can be calculated: Here G is the Greens function of the electrons and  W = ..1v is the screened Coulomb interaction determined by the inverse dielectric matrix ..1(r; r;E) and  + is a positive in nitesimal time. Greens function is obtained with the Kohn-Sham wave  functions and eigenvalues: Since the wave functions are obtained with the LDA are  appropriate, a rst order approximation is sufficient to  correct the LDA energies, for this reason quasi-particle  energies derived from the rst-order perturbation theory  by the following equation: Where V LDA xc is the exchange-correlation potential and  Zn;k is the renormalization factor of the orbital de ned  as Zn;k = (1 à ´Ã¢â€š ¬Ã¢â€š ¬Ã¢â€š ¬ @[emailprotected])à ´Ã¢â€š ¬Ã¢â€š ¬Ã¢â€š ¬1j E=ELDA n;k . In the equation (2), sigma is a convolution of G and W,  this part of the calculation is very complex, because the  matrix à ´Ã¢â€š ¬Ã¢â€š ¬Ã¢â€š ¬1 GG†² (q; !) (in reciprocal space) must calculate for  all frequencies !, in direction of the real and imaginary  axis. Since only the value of the integral is important,  with a simple and acceptable model can calculate the  integrals. In this model the frequency dependence the  matrix à ´Ã¢â€š ¬Ã¢â€š ¬Ã¢â€š ¬1 GG†² (q; !) calculate with a plasmon pole model16: In this equation, ÃŽ ©2 GG†² (q) and ~!(q) are the parameters of  model, the nal values for the parameters in this model  is found in Ref. 16. Dielectric function in this model is  approximated as a single-peak structure, this peak placed  in the plasmon frequency !p. Plasmon pole model not  only reduces computation, but also makes an analytic  calculation of the relation (2). With the GW calculations, correction to the energy  gap of the carbon nanotube is obtained. The ABINIT  package has been used for the Hybertsen-Louie plasmon  pole model calculations. For all the GW calculations,  the energy cutoff is 36 Ry for the evaluation of the bare  Coulomb exchange contribution x, and 24 Ry for the  correlation part c. With the many-body perturbation theory, can be calculate the excitation energies with obtaining self-energy  using 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 achieved  when the interaction between the electron and hole are  also considered. BSE17 takes into account coupling between electron and hole and absorption spectra that obtained by solving this equation, is more consistent with  the experimental results. Bethe-Salpeter equation written for a bound two-body system, in condensed matter  this equation has the form of as follows: Where the quasiparticle energies Ec;Ev enter on the diagonal, and the indices v, c refers to the occupied valence  and empty conduction band states,Wand V are screened  and bare Coulomb potentials, respectively. By solving the Bethe-Salpeter equation, exciton energies are calculated. In order to have an observation for  the excitonic energies, the macroscopic dielectric function  is calculated using the following equation18: Where the Avc s is exciton amplitude and Es is exciton  energy. The relation between the imaginary part of à Ã‚ µM  to the frequency ! gives the absorption spectrum. In ABINIT, we use the option to evaluate the response  function recursively with Haydock algoritm19 and TammDancoff approximation14. Calculations of optical properties via BSE are more expensive computationally. For  both undeformed and deformed SWNTs, the BSE kernel,  in which the energy cutoff 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 is  a semiconductor and amount of band gap is 0:57 eV. We  repeat the same calculation for elliptical tubes, with the  previous parameters (the same cutoff energy, number  of kpoint, : : 🙂 and only the geometry of the tube will  change. Fig. 4 shows the band structure for deformed (8,0)  SWNTs with different values of . In this calculation,  the band gap decreases from Egap = 0:57 eV at = 1:0  to the closing point, Egap = 0:0 eV at = 0:7. the energy gap is 0:49 eV , 0:26 eV for A, B elliptic nanotubes,  respectively. For D, E and F, elliptic nanotubes no band  gap is found. In this calculation, A, B elliptic nanotubes remained  semiconductor and the C elliptic nanotube represents  the boundary of the metal. In this approximation D, E  and F, elliptic nanotubes are metal. By this calculation we show that when the deformation  is highly intense, the band gap decreases and one  insulator-metal transition occurs. In the second stage, we calculate correction band  gap energy and quasi-particle band structure with GW  approximation. Fig. 5 shows the quasiparticle band  structure for undeformed (8,0) SWNT, in the GW  approximation band gap is 1:76 eV that is greater than  the amount predicted in the LDA. Result for undeformed nanotube agrees well with  ab initio calculations presented in Ref. 6, that the  calculated value of the quasiparticle energy gap is given: 1:75 eV for undeformed (8,0) SWCNT. We perform  one-shot GW or G0W0 model where the convergence  studies have been carried out with respect to various  parameters (naumber of bands, cutoff energy, . . . ). In the previous stages, A, B elliptic nanotubes  remained semiconductor and the C elliptic nanotube  represents the boundary of the metal. We performed  GW calculations only for semiconducting nanotubes.   Fig. 6 shows the quasiparticle band structure and the  calculated value of the quasiparticle energy gaps that  they are 1:65 eV , 1:34 eV for A, B elliptic nanotubes,  respectively, and for the C elliptic nanotube no band  gap is found. For deformed nanotubes only the ground  state energy is calculated by many-body approach in  Ref. 20, so far no GW calculations have been done for  deformed nanotubes to compare our results with them. With the GW calculations, we conclude that when the  deformation 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 calculations  represents for nanotubes with different values of the  deformation. The values of the contributions of LDA exchangecorrelation potential Vxc, the exchange x and the  correlation c part of the self energy are displayed in  Table 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; r;E), which describes screening in a solid when  local elds due to density inhomogeneities and manybody effects are taken into account, to obtain self energy  by (2). However, we found gap correction for undeformed  and deformed nanotubes, but electron-hole interaction  decreased the excitation energy in these structures. The  calculations include the electron-hole interaction (excitonic effects) are closer and better values to experiment. In the third stage the macroscopic dielectric function  Ãƒ Ã‚ µM(!) has been calculated by (7) including local eld  effects 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 deformation  A,B peaks shift to lower energy, and red shift occurs in  the optical spectra of carbon nanotubes. Therefor the  low energy exciton can be occurred by deformation on  the nanotubes. Table V shows the values of lowest two  optical transition energies for the undeformed SWCNT  in the present work and, ab initio calculations and  experiment. The value of ratio E11=E22 = 1:18 for the  (8,0) tube is in agreement with the experiment findings  of Bachilo et al3. Bachilo and coworks in their work  with Spectrouorimetric measurements obtained rst  and second transition energies for more than 30 semiconductor CNTs with different (n,m). their results shows  ratio equal to 1.17 for the (8, 0) nanotube and 1.85 for  nanotube with a diameter larger, while a single-particle  model, such as a tight bonding model is expected 2 value for this ratio. In considering excitonic effects the  ratio problem will be resolved and th ese calculations  give us better results. We rst obtains values of the  rst and second excitation energy for the undeformed  SWNT and were compared with computational and  exprimental values, then we repeated calculations for  deformed nanotubes to get results. Table VI shows  lowest two optical transition energies for the undeformed  and deformed SWNTs. The value of E11 and E22  decreases with deformation. In conclusion, we study the optical absorption  spectra of deformed and undeformed semiconducting   small-diameter SWCNT and survey the agreement with  available experimental data. We show by applying  deformation on the nanotubes one insulator-metal  transition occurs, and peaks shift to lower energy,  and red shift occurs in the optical spectra of carbon  nanotubes. The deformation can be proposed as ideas  for achieving to less excitonic energy. The results can  be employed to understand and guide experimental  studies of electronic and optoelectronic devices based  on the carbon nanotubes. So far GW calculations and  absorption spectra with excitonic effects for deformed  tubes has not been obtained. We investigate deformation effects on the energy gap  in LDA and GW approximation and optical spectra including excitonic effects. These calculations shows that  with apply deformation on the SWNT structure, energy  gap decrease, and lowest two optical transition energies  for the deformed SWNTs shift to lower energy. The deformation can be proposed as ideas for the achieve to less  excitonic 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 and  ab initio calculations for undeformed nanotube, then repeat calculations for deformed nanotubes and investigate  deformation effects on the energy gap in LDA and GW  approximation and optical spectra. Investigation of excitonic effects so far has not been done with many-body  approach for Bethe-salpeter equation for deformed nanotubes. Results are agreed with sing-particle calculations  that presented in Ref. 7.