Optical properties of the massive Vanadium dioxide

Abderrahim Ben Chaib1., Abdesselam Mdaa2 Izeddine Zorkani1., Anouar Jorio1. 1 Physics laboratory of solid FSDM Fès, university Sidi Mohamed Ben Abdellah, Morocco. 2 Laboratory of the thin layers and surface treatment by plasma ENS-Fès, Morocco. Abderrahim Ben Chaib abderrahim-196@hotmail.com Abdesselam Mdaaabdou261175@gmail.fr Izeddine Zorkani izorkani@hotmail.com Anouar Jorio a_jorio@hotmail.com ABSTRACT


INTRODUCTION
The vanadium dioxide VO₂ became a relevant material because of its transition from reversible phase [1,2,13,15] of the semiconductor state to the metal state at approximately 68 °C temperature in a few nanoseconds. Its optical and electronic properties change abruptly. Several studies were made on this material in a massive state and thin layer. At low temperature θ ˂ 68 °C, this material behaves like a semiconductor with a gap of 0.7 ev in a monoclinical structure (M) [5,14]. At high temperature θ ˃ 68 °C, this material becomes metal and is in a tetragonal rutile (R) structure. Mossaneck et al. [11,16] made optical studies of properties of massive VO₂. For incidental photons of energy there is an electronic transition from the energy band towards the band ; whereas, for incidental photons of energy we have electronic transitions from the band towards . For incidental photons of energy we have the excitons formation.
We will simulate by Maple the considered variation of R and T according to energy E (ev) of the exciton 1S. This variation occurs on the order of a few nanoseconds.

Methods
We will calculate the energy [6,7,8,10] of an exciton 1S of Hamiltonian: : the relative permittivity of material.
: the effective mass of the electron.
: the effective mass of the hole. : the ray of effective Bohr with 3D of the exciton 1S.
: the impulse of the electron. : the impulse of the hole. m : the total mass of the exciton.
: the reduced mass of the exciton.
: the wave vector of the exciton.
we will work in the center of the system of the exciton mass : We will solve the equation of Schrödinger : By a substitution in Schrödinger's equation [9], we have : We will do a calculation by variational method [3,4], where is the variational parameter. E: the energy of the exciton which we will minimize.
we pose :

Results
We integrate by part; we obtained the following results : We pose: where ; Solving the equation , we obtain : Thus, the expression of the energy E of an exciton 1S: : the gap energy of VO₂ in the semiconductor state : the reduced constant Planck : the elementary charge In the vicinity of the band gap (k→0) we have : That is to say, n is the index of refraction of VO₂ which we considered slightly absorbent : We use the approximation of the effective mass : We make a simulation by means of Maple software for ԑ, n, R and T according to the energy E of the exciton in the vicinity of the band gap (k→0) and for (k≠0 at the edge of the first zone of Brillouin): For k→0 we obtained the formula : and figures 1,2,3,4,5 and 6 by using the following data in the system (S.I) : Figure.1 The permittivity ԑ of VO₂ variation according to the energy E of the exciton 1S for k→0

Discussion :
The Hamiltonian is given in the system CGS; and for the energy of the exciton we  We see that the variations of R and T are simultaneously opposite. The transmittivity T decreases while the reflectivity R increases when energy E of the exciton approaches the conduction band. This result is logical because the Coulomb interaction of the exciton decreases and the electron becomes almost free.
These simulated results well prove that the creation of the exciton in the massive vanadium dioxide in a massive semiconductor state by incidental photons influences the reflectivity R and the transmittivity T of VO ₂ .
The interest of this optoquantic study concerning the excitons is to be able to control and exploit the variation of the optical constants of this material on a nanosecond scale for various nanotechnological applications at a very fast speed, like the optical detectors and the photovoltaic equipment. We think of making a similar study for the the vanadium dioxide's thin layers and of comparing the two results.