DIRECT INTERBAND LIGHT ABSORPTION IN A SPHERICAL QUANTUM DOT WITH THE MODIFIED PÖSCHEL-TELLER POTENTIAL

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1 Lase Physics Intenational Jounal of Moden Physics: Confeence Seies Vol. 5 () 4 Wold Scientific Publishing Company DOI:.4/S DIRECT INTERBAND LIGHT ABSORPTION IN A SPHERICAL QANTM DOT WITH THE MODIFIED PÖSCHEL-TELLER POTENTIAL Int. J. Mod. Phys. Conf. Se..5:4-. Downloaded fom by on 4//8. Fo pesonal use only. H.KH. TEVOSYAN, D.B. HAYRAPETYAN,, K.G. DVOYAN, E.M. KAZARYAN Dept. of Physics and Technology, Russian-Amenian (Slavic) nivesity, 3 Hovsep Emin St., 5 Yeevan, Amenia, hovhannes.tevosyan@gmail.com. Dept. of Physics, State Engineeing nivesity of Amenia, 5 Teian St., 9 Yeevan, Amenia. The enegy levels and diect inteband absoption of light in a spheical quantum dot with a modified Pöschel-Telle confining potential ae studied. Analytical expessions fo the paticle enegy spectum and absoption theshold fequencies in the egime of stong size quantization ae obtained. Selection ules fo quantum tansitions ae evealed. Red shift and blue shift of absoption theshold has been obseved depending on the values of half-width and depth of confining potential, coespondingly. Keywods: Light absoption; spheical quantum dot; modified Pöschel-Telle potential. Intoduction Inteest to the semiconducto quantum dots (QD) keeps gowing in ecent peiod of time which is attibuted to the new physical popeties of these zeo dimension objects which ae mainly a esult of size quantization (SQ) of chage caies in them [-3]. Duing the gowth pocess of QD in the esult of diffusion, fomation of confining potential is vey accuately appoximated though paabolic potential in most of the cases. But in eality paabolic potential is valid only fo low enegetic states. It is evident that confining potential is being sepaated fom paabolic paallel to the incement of quantum numbe. Fo bette and ealistic appoximation of foming confining potential application of modified Pöschel-Telle potential has been poposed in seveal woks [4,5]. In this pape the enegy levels and diect inteband light absoption in a spheical QD with the modified Pöschel-Telle potential have been consideed.. Theoy In the stong SQ egime, the enegy of Coulomb inteaction between electon and hole is much less than the enegy ceated by the confinement of the QD walls. In that egime, one can neglect the Coulomb inteaction. Thus, the enegy states of electon and hole should be detemined independently. Conside the motion of an electon in a spheical 4

2 Diect Inteband Light Absoption in a Spheical Quantum Dot 5 quantum dot with a modified Pöschel-Telle confining potential [6,7]. The potential enegy of an electon in spheical coodinates can be witten in the fom ɶ ( ρ) =, () ρ cosh ɶ Int. J. Mod. Phys. Conf. Se..5:4-. Downloaded fom by on 4//8. Fo pesonal use only. whee ɶ and ɶ ae espectively the depth and half-width of the potential well. View of the modified Pöschel-Telle potential is shown in Fig. a) and b) fo vaious paametes of the half-width and depth. =.5 =.5 = а) = 4 = 8 = Fig.. a) Modified Pöschel-Telle potential fo diffeent values of the half-width at = 4. b) Modified Pöschel-Telle potential fo diffeent values of the depth at =. The Hamiltonian of the system fo the zeo values of obital and magnetic quantum numbes ( l= m= ) in spheical coodinates can be witten as ˆ ħ Η= m ɶ, () cosh ɶ + e ρ ρ ρ ρ whee m e is the effective mass of electon. In the dimensionless quantities the Hamiltonian of the system takes the following fom: b) whee enegy, ˆ Hˆ Η ρ =, =, E a a B R e B ˆ H= +, (3) cosh a =, E ɶ = ɶ B, ER R = ħ is the effective Rydbeg m a e B κ = ħ is the effective Boh adius of electon, κ is the dielectic m e

3 6 H. Kh. Tevosyan et al. constant, e is the electon chage. The adial pat of the wave function of the electon is sought as R() = () χ. (4) Int. J. Mod. Phys. Conf. Se..5:4-. Downloaded fom by on 4//8. Fo pesonal use only. Afte substituting the wave function into the Schödinge equation and by intoducing new notations ξ= th( ) and s= 4 + +, we obtain the following equation: d dχ ( ξ ) + s( s+ ) χ=, (5) dξ dξ ξ whee = E / ER. This is the equation of the genealized Legende functions [8]. Pesent it to the hypegeometic type by substitution / ( ) ( ξ) χ= ξ Ω. (6) Substituting expession (6) in equation (5) and intoducing new notation u= ( ξ) we obtain the equation of Kumme [9]: u( u) Ω + ( + )( s) Ω ( s)( + s+ ) Ω=. (7) Solution of Kumme equation is given by the Gauss hypegeometic function: ξ Ω ( ξ) = F { s, + s+, +, }. (8) Finally, the adial wave function takes the following fom: / C th( ) R() = th F s, s,, + + +, (9) hee C is nomalizing constant. Fo the enegy of the system finally we get: = ( n) () whee n is the main quantum numbe.. Diect inteband light absoption We poceed to the consideation of the diect inteband absoption of light in a spheical QD with modified Pöschel-Telle potential in the egime of stong SQ, when the

4 Diect Inteband Light Absoption in a Spheical Quantum Dot 7 Coulomb inteaction between electon and hole can be neglected. We conside the case of * * heavy hole, when me mh, whee me and m h ae effective masses of electon and hole espectively. The absoption coefficient is defined by the expession K= A ΨΨ d ħ Ω E E E, () νν, ( g ν ) e h e h ν ν δ ν Int. J. Mod. Phys. Conf. Se..5:4-. Downloaded fom by on 4//8. Fo pesonal use only. whee ν and ν ' ae the sets of quantum numbes coesponding to an electon and heavy hole, E g is the fobidden band width of a bulk semiconducto, Ω is the fequency of the incident light, A is a quantity popotional to the squae of the matix element, taken ove the Bloch functions []. Finally, in the egime of stong SQ, fo the quantity K and absoption edge we obtain whee W Ω W = ħ and Eg n ( g e h) K= A δħ Ω E E E, () d a B 4 = + 4, (3) d= ħ. Expession (3) chaacteizes the dependence of µeg the effective fobidden band width on the paametes and. Conside now the selection ules fo quantum numbe n. Fo the quantum numbe n the tansitions between the levels with n= n' and n= n' ± ae allowed. Note that the analytic fom of expession () is pesented with allowance fo the mentioned selection ule of quantum numbe n. 3. Discussion As it is obvious fom the obtained esults, enegy of electon (see fomula ()) diffes fom chage caie s enegy expession in the case of paabolic confining potential. But thee is a mean to compae the obtained esults with the case of spheical QD, which confining potential has a fom of tuncated paabola []. It should be mentioned that in the case of small values of coodinate the modified Pöschel-Telle s potential is fits to the paabolic potential with a high accuacy, while in the case of high values of the same coodinate potential s appeaance is diffeent fom paabola. In othe wods, appoximation of confining potential though the modified Pöschel-Telle s potential gives an oppotunity to take in account the non paabolic shape of potential which is fomed in eality. It is evident that a few implementations ae possible in the case of compaison of potentials (see Fig. ). In the fist case the tuncated paabola is inscibed in the modified Pöschel-Telle s potential. Then gound state of electon in the case of modified Pöschel- Telle s implementation is located lowe (Fig. a)). The mentioned is a esult of

5 8 H. Kh. Tevosyan et al. Int. J. Mod. Phys. Conf. Se..5:4-. Downloaded fom by on 4//8. Fo pesonal use only. contibution of SQ of QD wells in the paticle s enegy. In the second case (Fig.. b)) modified Pöschel-Telle s potential is inscibed to the tuncated paabola. In this case the enegy coesponding to the QD with the modified Pöschel-Telle s potential is located uppe since the SQ is dominating upon the case of tuncated paabola. Finally in the thid case intemediate situation is implemented (Fig.. c)). As illustated in the figue, electon enegy of the gound state coesponding to the spheical QD with the modified Pöschel-Telle s potential is located uppe in the beginning, but with the incement of QD s adius situation is being changed. With the incement of adius of QD confining potential of QD fo the case of tuncated paabola is getting wide a) b) V() V() V() c) Fig.. Dependence of enegy of gound state in tuncated paabolic QD (dotted line) and in QD with modified Pöschel-Telle potential (solid line).

6 Diect Inteband Light Absoption in a Spheical Quantum Dot 9 Int. J. Mod. Phys. Conf. Se..5:4-. Downloaded fom by on 4//8. Fo pesonal use only. Numeical calculations fo the egime of stong SQ wee caied out fo a GaAs spheical QD with the following paametes: m = e.67m,. e me= mh, κ= 3.8, E R= 5.75 mev, a e= 4 Å and a h= 5 Å ae the effective Boh adii of an electon and hole, E g=.43 ev is the fobidden band width of a bulk semiconducto. Thee-dimensional Fig. 3 shows dependence of the fist thee levels of the electon enegy on the half-width and depth of the potential well. As can be seen fom the figue, with an incease of half-width, the electon enegy deceases, this is a consequence of educing the contibution of SQ. Fo the same eason with an incease of potential well depth the electon enegy deceases. Note that the enegy levels ae not equidistant. As it mentioned above, the modified Pöschel-Telle potential allows to descibe non-paabolic chaacte of the chage caie enegy, the fact clealy shown in Fig.3. Fig. 3. The dependence of the fist thee levels of the electon enegy on the half-width and depth of potential well. W = 4 = 6 = 8 a) W =.5 =.5 b) = а) Fig. 4. a) The dependence of the absoption edge on the half-width of the confining potential. b) The dependence of the absoption edge on the depth of the confining potential. Fig. 4 a) and b) shows the dependence of the absoption edges, espectively, fom the half-width and depth of the potential well unde the egime of stong SQ. As can be seen fom the figues with a incease in the half-width of the potential well the

7 H. Kh. Tevosyan et al. absoption theshold is educed (ed shift). And with inceasing depth of the potential well theshold inceases (blue shift), which is a consequence of the SQ incease (the effective fobidden band width inceases). 4. Conclusion Int. J. Mod. Phys. Conf. Se..5:4-. Downloaded fom by on 4//8. Fo pesonal use only. In the egime of stong size quantization analytical expessions fo the paticle enegy spectum and wave function ae obtained. It is shown that with inceasing of half-width of the potential well the paticle enegy deceases, and with deceasing of depth of potential well the paticle enegy inceases due to SQ contibution. Analytical expessions fo the absoption theshold fequencies ae obtained. Selection ules fo quantum tansitions ae evealed. Red shift of absoption theshold has been obseved depending on the values of half-width and blue shift of absoption theshold has been obseved depending on the values of depth. Refeences. P. Haison, Quantum Wells, Wies and Dots. Theoetical and Computational physics, (Wiley, New Yok, 5).. D. Bastad, Wave Mechanics Applied to Semiconducto Heteostuctues, (Les editions de physique, Pais, 989). 3. E.M. Kazayan, S.G. Petosyan, Physical Pinciples of Semiconducto Nanoelectonics, (RA, Yeevan, 5). 4. K.G. Dvoyan, D.B. Hayapetyan, E.M. Kazayan, A.A. Tshantshapanyan. Nanoscale Reseach Lettes, Vol. 4, Issue, pp.3-37 (9). 5. K.G. Dvoyan, D.B. Hayapetyan, E.M. Kazayan, A.A. Tshantshapanyan. Poceedings of the SPIE, Vol. 7998, pp. 7998F-7998F- (). 6. L.D. Landau, E.M. Lifshits. Quantum Mechanics. (Nauka, Moscow, 989). 7. S. Flügge. Pactical Quantum Mecannics, (Spinge-Welag, 97). 8. M. Abamowitz, I. Stegun, Handbook of mathematical functions, (Nauka, Moscow 979). 9. Gadshteyn and Ryzhik's Table of Integals, Seies, and Poducts, (Nauka, Moscow 96).. A.I. Anselm, Intoduction to semiconductos theoy, (Nauka, Moscow (978).. E.M. Kazayan, L.S. Petosyan, H.A. Sakisyan. Phys. of Pat. and Nucl., v. 34 p. 4 (3).