QUASI-STATIONARY ELECTRON STATES IN SPHERICAL ANTI-DOT WITH DONOR IMPURITY * 1. INTRODUCTION

Transcription

1 ATOMIC PHYSICS QUASI-STATIONARY ELECTRON STATES IN SPHERICAL ANTI-DOT ITH DONOR IMPURITY * V. HOLOVATSKY, O. MAKHANETS, I. FRANKIV Chenivtsi National Univesity, Chenivtsi, 581, Ukaine, ktf@chnu.edu.ua Received Septembe 5, 11 The electon enegy spectum in Al x Ga 1-x As/GaAs semiconducto quantum anti-dot with dono impuity, placed into the cente of a nanostuctue is studied. The enegies and semi-widths of the quasi-stationay states ae defined within the distibution of the pobability density of electon esidence in quantum anti-dot. Key wods: quantum anti-dot, quasi-stationay state, electon enegy spectum. 1. INTRODUCTION The investigation of quantum effects aising in atificial potential wells, ceated at the base of semiconductos attacts the attention of scientists studying the popeties of nanostuctues moe than twenty yeas. The majoity of investigations concen the so-called closed nano-systems with stationay enegy specta fo the quasi-paticles. Among them, the quantum dots, quantum wies and nano-films ae the most eseached [1-]. Recently, it is obseved the inceasing inteest to the open nano-systems o esonance tunnel semiconducto stuctues, whee the quasi-paticles can penetate the potential baie and move into infinity. The open nano-systems ae distinguished due to the spatial confinement and dimension. In the numbe of papes the open quantum dots, adial and axial open quantum wies and esonance tunnel plane nanostuctues ae studied. All these nano-systems ae chaacteized by quasi-stationay enegy specta of quasi-paticles and have the unique pespectives of utilization fo the ceation of field tansistos, diodes and quantum cascade lases [3-4]. In pape [5], the possibility of the ceation of open nano-system as quantum anti-dot (QAD) with dono impuity is discussed. Hee, the stationay states of electon bound by dono impuity, placed into the cente of ZnS/Cd x Zn 1-x S QAD * Pape pesented at the 1 th Intenational Balkan okshop on Applied Physics, July 6 8, 11, Constanta, Romania. Rom. Joun. Phys., Vol. 57, Nos. 9 1, P , Buchaest, 1

2 186 V. Holovatsky, O. Makhanets, I. Fankiv ae studied. The enegies of stationay states and distibution of pobability density of electon esidence in nanostuctue ae calculated. It is poven that depending on the potential baie height and enegy the electon can be localized in deep o shallow potential well, ceated by Coulomb and QAD potentials. The electon, having the bigge enegies, can tunnel though the potential baie and move into infinity. The esonance states, manifesting themselves in scatteing pocesses, ae obseved at the enegies highe than the potential baie. In this pape we study the quasi-stationay and esonance states of electon in semiconducto spheical nanostuctue (Al x Ga 1-x As/GaAs) with cental dono impuity. The investigation of electon enegy specta is pefomed using the method of limit tansition fom the open nanostuctue to the espective closed spheical coe-shell one with unpenetable oute inteface, inceasing GaAs-shell sizes till the macoscopic ones. The detailed appobation of the method fo open spheical systems is given in [4], whee it is shown that the basic popeties of an electon in a simple open spheical quantum dot can be epoduced to any specified accuacy in the model of a closed two-well spheical quantum dot with a sufficiently lage width of the oute well.. HAMILTONIAN OF ELECTRON AND SOLUTION OF SCHRODINGER EQUATION The nanostuctue: spheical semiconducto coe () with adius, embedded into the semiconducto shell (1) with adius (Fig.1) is unde study. The hydogen-like dono impuity, placed into the cente of nanostuctue, ceates the Coulomb potential fo the electon. The electon spectum is obtained within the effective masses appoximation with its m, <, m () = m1, 1. The Hamiltonian of the electon is witten as (1) Ze H = + V(), () m ( ) ε whee V, <, V () =, < 1,, = 1 and ε - dielectic constant of QAD. (3)

3 3 Quasi-stationay electon states 187 U() V "" "1" Fig. 1 Scheme of potential enegy fo electon in the nanostuctue with impuity. Solving the Schodinge equation in spheical coodinate system, it is clea that the adial ones have the fom ( + 1) e + R () + ( E V + ) R () = m ε ( + 1) e + R () ( ) () + E + R = m1 ε Using the convenient paametes < (4),, < < 1 (5) ξ = 8 ( - ) m V, ξ = 1 8m1 E, m,1e,1,1 η = ε ξ (6) and adial wave function witten as χ ( ξ ), <, R () = χ1( ξ1 ), < 1,, = 1 the diffeential equations ae obtained ( ) η 1/4 ( 1/) 1 χ ξ χ ( ξ ) = ξ 4 ξ ξ ( ) η 1/4 ( 1/) 1 χ1 ξ χ1( ξ 1) = ξ 4 1 ξ1 ξ1,, (7) < (8) >. (9)

4 188 V. Holovatsky, O. Makhanets, I. Fankiv 4 Thei geneal solution can be witten within hittake functions [6], χ ( ξ ) = A M( η, +, ξ ) + B ( η, +, ξ ), (1) 1 1 χ ( ξ ) = AM( η, +, ξ ) + B( η, +, ξ ). (11) As fa as (z) function is singula at z =, it is obtained B = fom the condition that the wave function must be finite. Fom the condition of the adial wave functions and thei densities of cuents continuity at the inteface () (1) n = n = = R () R (), (1) () (1) n n 1 R () R () = m m = = = 1, (13) R (1) n () =, (14) the discete enegy spectum ( E ) of electon in spheical nanostuctue with cental dono impuity is obtained. The nomality condition fo the wave function 1 R () d = 1. (15) fixes the nomality coefficient. Now the electon enegy spectum ( E ) and its adial wave functions ( R ( ) ) fo the closed spheical coe-shell nanostuctue ae completely defined. 3. RESULTS OF CALCULATIONS AND DISCUSSION 3.1. ELECTRON ENERGY SPECTRUM AND AVE FUNCTIONS FOR THE CLOSED NANOSTRUCTURE ITH IMPURITY Compute calculations of the electon enegies and wave functions wee pefomed fo Al x Ga 1-x As/GaAs nanostuctue with physical paametes: V =.57(1155 x+37 x ) mev the height of potential baie, m(x) = ( x) m e electon effective mass, m e pue electon mass, ε=11.71 dielectic constant of QAD = 5.65(Å) GaAs lattice constant. The dependences of electon enegy spectum on the shell adius ( ) at diffeent values of coe adius ( ) and Al concentation (x) ae pesented in Fig. fo the QAD with and hydogen-like dono impuity in the cente. It is clea that electon enegy

5 5 Quasi-stationay electon states 189 spectum consists of the enegy states whee the electon is localized in the coe () and in the shell (1). At the inceases of the shell adius ( ), the width of the potential well becomes bigge. It bings to the weake effect of size quantization and the enegy levels, coesponding to the states of electon localized in the shell (1) ae shifting into the ange of lowe enegies. The enegies of electon localized in the coe () do not depend on shell adius ( ). As a esult, the effect of anticossing of electon enegy levels is obseved in Fig.. 5 x=. = 4 5 x=. = , a 1 GaAs 5 x=.3 = 4 5 x=.3 =4 4 3 n=3 3 1 n= 1 n= x=.4 = 4 5 x=.4 = Fig. Evolution of electon enegy spectum as function of shell adius ( ) fo the nanostuctue with the cental dono impuity at coe adius: = a GaAs ; 4 a GaAs, and Al concentation: x =.,.3,.4.

6 19 V. Holovatsky, O. Makhanets, I. Fankiv 6 The distibution of pobability density of electon esidence in nanostuctue (x =.3, = a GaAs, = 4 a GaAs ) is pesented at Fig. 3 fo the fist thee states (n = 1,, 3). The calculations wee pefomed oy fo spheically symmetic states (l=) but analogous ones can be fulfilled fo l. Fig. 3 poves that in the states n = 1 and n = the electon is localized in oute well and in the state n = 3 in inne one. (R ),8,6,4, n=3 n= n=1 U [mev] 3 1, [a GaAs ] Fig. 3 Distibution of pobability density of electon location in nanostuctue with cental dono impuity At > 5 а GaAs the distance between the neighbou enegy levels is less than.1 mev, since, such spectum can be assumed as quasi-continuous. Thus, at the shell (1) becomes a macoscopic medium and the nanostuctue efoms into the open QAD with cental impuity. 3.. THE QUASI-STATIONARY ENERGY SPECTRUM OF ELECTRON IN QAD ITH CENTRAL IMPURITY The quasi-stationay electon spectum is calculated within the distibution of pobability density of its esidence in the space of QAD ( < < ) ( E ) = R ( ) d. (16) The set of enegies ( E ), at which the pobability of electon location inside of QAD each the maxima, defines the position of quasi-stationay enegy levels. All of them ae chaacteized by thei own adial and obital quantum numbes. The adial quantum numbe fo evey quasi-stationay level is fixed by its odinal

7 7 Quasi-stationay electon states 191 numbe in enegy scale and the obital one is fixed by that l value, at which the calculation of enegy specta ( E ) fo the espective closed nanostuctue was pefomed. The semi-width of evey quasi-stationay level is defined by the distance between the points located at the half of the height of ( E ) function peak. The quasi-stationay electon spectum in QAD with cental impuity is shown in Fig. 4 at diffeent magnitudes of Al concentation (х) and coe adius ( ).,1, Γ 1 x=. =4 a GaAs,16,1 x=. = a GaAs,1,8,4 Γ 1 Γ 1 3 E [mev] E [mev], x=.3 =4 a GaAs,3 x=.3 = a GaAs Γ 1,3,,,1,1 Γ 1,36,3,4, 3 4 E [mev] E [mev] Γ 1 x=.4 =4 a GaAs,1,5 Γ 1 x=.4 = a GaAs 3 4 E [mev] Fig. 4 Quasi-stationay enegy spectum of electon in QAD with dono impuity at x =.,.3,.4; = a GaAs, 4 a GaAs. E [mev] Numeical value of the enegies, semi-width and the life-times of the lowest quasi-stationay sates ae shown in Table 1. Table 1 Enegy E 1 and semi-width Γ 1 of the quasi-stationay states and life-time τ 1 Paametes E 1 [mev] Γ 1 [mev] τ 1 [ps] x=., =4 a GaAs x=.3, =4 a GaAs x=.4, =4 a GaAs x=., = a GaAs x=.3, = a GaAs x=.4, = a GaAs

8 19 V. Holovatsky, O. Makhanets, I. Fankiv 8 Fig. 4 poves that when Al concentation (x) becomes bigge, the height of the potential baie inceases and the quasi-stationay levels shift into the egion of highe enegies. The inceasing QAD adius causes the shift of quasi-stationay levels into the egion of lowe enegies. Heein, the pobability of electon tunneling though the baie deceases, thus, the semi-width becomes smalle too and the distance between quasi-stationay levels also deceases. The lowe is the quasi-stationay level in the enegy scale, the smalle is its width and, consequently, the bigge is the life time of electon in this state. It is explained by the inceasing width of the baie though which the electon tunnels. The esonance levels with the enegies bigge than the baie height ae chaacteized by the big semi-width and small life-time, espectively. 4. SUMMARY The electon enegy spectum in QAD with cental dono impuity is obtained within the limit tansition fom open nano-system to the closed one at 1. The exact solutions of the Schodinge equation fo spheical coe-shell nanostuctue with hydogen-like dono impuity placed into its cente ae obtained fo the electon. The electon enegies and the semi-widths of quasi-stationay states ae defined by the distibution of the pobability density of its esidence inside the QAD. ithin the used method of limit tansition, the dependences of quasi-stationay states paametes on the coe adius ae obtained. It is shown that at the inceasing QAD adius the enegies of quasi-stationay states of the electon bound by cental dono impuity ae shifting into the low-enegy ange of spectum. Heein, thei semi-widths ae deceasing. The quasi-stationay states can manifest themselves in the pocesses of electon scatteing though the aay of QADs with impuities inside. REFERENCES 1. U.oggon, Optical popeties of semiconducto Quantum Dots, Spinge, Belin, P.Haison, Quantum wells, wies, and dots: theoetical and computational physics of semiconducto nanostuctues. iley, Chicheste, M.Tkach, V.Holovatsky, O.Voitsekhivska, Electon and hole quasistationay states in opened cylindical quantum wie. Physica E: Low dimensional systems and Nanostuctues. 11, 17 6 (1). 4. M.Tkach, Yu. A. Sety, Popeties of the electon spectum of a closed two-well spheical quantum dot and evolution of the spectum with the oute-well width. Semiconductos. 4, (6), tanslated fom ussian Fizika i Tekhnika Polupovodnikov. 4, (6). 5. V.Holovatsky, O.Makhanets, O.Voitsekhivska, Oscillato stengths of electon quantum tansitions in spheical nanosystems with dono impuity in the cente, Physica E. 41, (9). 6. A. Abamowitz and I. Stegun, Handbook of Mathematical Function with Fomulae, Gaphs and Mathematical Tables, ashington, 1964.