The exciton binding energies in finite-potential quantum dot discs of GaAs are obtained and the eigenstates and the eigenvalues of the exciton are calculated. We present the exciton binding energy for different values of the disc radius (R) and the disc half-width (L/2). The exciton-state stability for large and small sizes of the dot is discussed. We compare our results with the existing theoretical and experimental results. Our results give good estimates for the optimal quantum dot disc geometry, and represent useful data in studies of the optical properties of quantum dots in nano-scale devices.Izračunali smo energiju vezanja, svojstvena stanja i svojstvene vrijednosti eksitona u kvantnoj točki s konačnim potencijalom. Opisujemo energiju vezanja eksitona za više vrijednosti polumjera (R) i poluširine (L/2) diska. Raspravljamo stabilnost eksitona za male i veće dimenzije diska. Uspoređujemo naše rezultate s poznatim drugim teorijskim i eksperimentalnim rezultatima. Naši rezultati daju dobre ocjene za povoljnu veličinu diska kvantne točke, i predstavljaju korisne podatke za proučavanje optičkih svojstava kvantnih točaka u napravama nano veličine

El-Meshad, N. A.

Hekmat, M. H.

Safwan, S. A.

English

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Printed ISSN 1330–0008Online ISSN 1333–9125CD ISSN 1333–8390CODEN FIZAE4EXCITON STATE IN A QUANTUM DOTS. A. SAFWAN, M. H. HEKMAT and N. A. EL-MESHADTheoretical Physics Department, National Research Center Cairo, EgyptReceived 21 December 2005; Accepted 21 November 2006Online 22 May 2007The exciton binding energies in finite-potential quantum dot discs of GaAs are ob-tained and the eigenstates and the eigenvalues of the exciton are calculated. Wepresent the exciton binding energy for different values of the disc radius (R) andthe disc half-width (L/2). The exciton-state stability for large and small sizes of thedot is discussed. We compare our results with the existing theoretical and experi-mental results. Our results give good estimates for the optimal quantum dot discgeometry, and represent useful data in studies of the optical properties of quantumdots in nano-scale devices.PACS numbers: 73.61.-r, 85.30.Vw UDC 538.915Keywords: quantum dot disc, GaAs, finite potential, exciton binding energy, eigenstates,eigenvalues1. IntroductionThe appearance of a peak in the absorption curve of direct-gap materials, suchas GaAs, is due to the formation of a complex known as the “exciton”. Excitons areelementary excitations in solids and arise due to the Coulomb interaction betweenelectrons and holes. When an excitation energy is transferred to a semiconductor byabsorption of a photon, an electron-hole pair is created. At low temperatures, theirbound states are formed because the Coulomb interaction between the electron andthe hole is prominent. In systems of small dimensions, especially in quantum dots(three-dimensional confinement), the picture is different [1]. Because of nanometerwidths and nanometer thickness, and of various shapes, the quantum confinementincreases considerably. Whereas the micro-crystallites are approximately spheri-cal, thin discs or cylinders describe better the quantum dots. In quantum dots,the “exciton” (i.e., the confined electron-hole state) remains present even at roomtemperature in both absorption and emission spectra. Therefore, many devicesusing exciton transitions have been proposed. These effects are expected to beparticularly important when the dimensions of the boxes become comparable toFIZIKA A (Zagreb) 16 (2007) 1, 1–10 1safwan et al.: exciton state in a quantum dotthe excition effective Bohr radius. Although many theoretical studies [10 – 12] havebeen devoted to the excitonic states in spherical micro-crystals, very few studiesconsider excitons in cylindrical quantum dots. Up to now, various shapes have beenconsidered, like square flat plates [13 – 14] and cylindrical [15] boxes. In the lattercases, infinite barriers have been used to confine the electron and the hole.In the present work, we study excitons in quantum discs using a variationalapproach and the effective-mass approximation with a finite confinement potential.There have been concerns as to whether the effective-mass approximation couldstill be valid in the quantum dot limit when the size of the exciton could be of theorder of the average lattice constant of the bulk semiconductor. Recently, Marin etal. [16] performed variational calculations of the exciton energies for spherical dotsof radius in the range of 1.5 − 4 nm to compare both with experimental and theirtheoretical data for CdS, CdSe, PbS and CdTe crystallites. They found that theeffective-mass approximation is still appropriate for these geometries. Moreover, inthe limit of a very small spatial confinement, when the exciton extends substan-tially into the barrier material, the effective-mass approximation could again be anappropriate approximation with the exciton described by the effective mass of theembedded barrier material (GaAlAs). The paper is organized as follows. In Sec. 2,we present the theoretical model used to describe the exciton in a cylindrical quan-tum disc. Section 3 presents the results obtained for the exciton binding energy inthe quantum dot with a finite barrier, and in Sec. 4 we give the conclusion.2. TheoryThe Hamiltonian [16] of an exciton confined in a QD disc, using the relativecoordinates r = |r̄e − r̄h|, can be written asH =−h̄22m∗e{∂2∂r2e+1re∂∂re+r2e − r2h + r2rer∂2∂re∂r}− h̄22m∗h{∂2∂r2h+1rh∂∂rh+r2h − r2e + r2rhr∂2∂rh∂r}− h̄22μ{∂2∂r2+1r∂∂r}− h̄22m∗e∂2∂z2e− h̄22m∗h∂2∂z2h+Ve(re, ze) + Vh(rh, zh) − e2ε√r2 + (ze − zh)2, (1)where μ is the reduced exciton mass μ = m∗em∗h/(m∗e + m∗h) and ε is the dielectricconstant. Assuming a finite potential well, the total confinement potential V forelectron or hole is written asV (ri) ={ 0, ri < R, |zi| < L/2 ,V0, otherwise ,(2)where i stands for e or h. The full three-dimensional Schrödinger equation for anexciton in a quantum dot isHΨ(re, rh, r, ze, zh) = EΨ(re, rh, r, ze, zh) . (3)2 FIZIKA A (Zagreb) 16 (2007) 1, 1–10safwan et al.: exciton state in a quantum dotThe Ritz variation principle is used to solve this equation numerically. We are ableto determine the ground state by choosing the following trial wave function whichtakes into account the electron-hole correlationΨ(re, rh, r, ze, zh) = f(re)f(rh)g(ze)g(zh) exp{−α√r2 + (ze − zh)2} . (4)The variational parameter α is determined by minimizing the value of the excitonenergy, and at the corresponding value of α, the exciton energy isEex(α) =〈Ψ|H |Ψ〉〈Ψ|Ψ〉 ,where f(ri) and g(zi) are the ground wave functions of both the electron and thehole (see our previous paper in Ref. [17]). Thus we have〈Ψ|H |Ψ〉 =∞∫−∞∞∫−∞∞∫0∞∫0∞∫0Ψ∗HΨ redre rhdrh rdr dze dzh. (5)Substituting the Hamiltonian (1) and the envelope wave function Ψ (4) into theintegral (5), and after some tedious algebra, we obtained the following five integralsfor the kinetic-energy terms of the Hamiltonian [we used the transformation ofvariables (re, rh, r, ze, zh) to (xe, xh, x, ye, yh)]k1 =∞∫−∞∞∫−∞∞∫0∞∫0∞∫0h̄22me1R2f2(xe)f2(xh)g2(ye)g2(yh)×e−2α√R2x2 + a2(ye − yh)2[(f ′(xe)f(xe))2+αR2(x2e − x2h + x2)xe√R2x2 + a2(ye − yh)2f ′(xe)f(xe)]R8a2 xedxe xhdxh xdxdye dyh,k2 =∞∫−∞∞∫−∞∞∫0∞∫0∞∫0h̄22mh1R2f2(xe)f2(xh)g2(ye)g2(yh)×e−2α√R2x2 + a2(ye − yh)2[(f ′(xh)f(xh))2+αR2(x2h − x2e + x2)xh√R2x2 + a2(ye − yh)2f ′(xh)f(xh)]R8a2 xedxe xhdxh xdxdye dyh,k3 =∞∫−∞∞∫−∞∞∫0∞∫0∞∫0h̄22μ(α2R2x2R2x2 + a2(ye − yh)2)f2(xe)f2(xh)g2(ye)g2(yh)FIZIKA A (Zagreb) 16 (2007) 1, 1–10 3safwan et al.: exciton state in a quantum dot×e−2α√R2x2 + a2(ye − yh)2R8a2 xedxe xhdxh xdxdye dyh,k4 =∞∫−∞∞∫−∞∞∫0∞∫0∞∫0h̄22me1a2f2(xe)f2(xh)g2 (ye)g2(yh)×e−2α√R2x2 + a2(ye − yh)2[(g′(ye)g(ye))2− 2αa2(ye − yh)√R2x2 + a2(ye − yh)2g′(ye)g(ye)+α2a4(ye − yh)2R2x2 + a2(ye − yh)2]R8a2 xedxe xhdxhxdxdye dyh,k5 =∞∫−∞∞∫−∞∞∫0∞∫0∞∫0h̄22mh1a2f2(xe)f2(xh)g2(ye)g2(yh)×e−2α√R2x2 + a2(ye − yh)2[(g′(yh)g(yh))2+2αa2(ye − yh)√R2x2 + a2(ye − yh)2g′(yh)g(yh)+α2a4(ye − yh)2R2x2 + a2(ye − yh)2]R8a2 xedxe xhdxh xdxdye dyh.The potential terms take the formp1 =∞∫−∞∞∫−∞∞∫0∞∫0∞∫0Ve(xe, ye)f2(xe)f2(xh)g2(ye)g2(yh)×e−2α√R2x2+a2(ye−yh)2R8a2 xedxe xhdxh xdxdye dyh,p2 =∞∫−∞∞∫−∞∞∫0∞∫0∞∫0Vh(xh, yh)f2(xe)f2(xh)g2(ye)g2(yh)×e−2α√R2x2+a2(ye−yh)2R8a2 xedxe xhdxh xdxdye dyh,p3 =∞∫−∞∞∫−∞∞∫0∞∫0∞∫0−e24πεε0√R2x2 + a2(ye − yh)2f2(xe)f2(xh)g2(ye)g2(yh)×e−2α√R2x2+a2(ye−yh)2R8a2 xedxe xhdxh xdxdye dyh.where a = L/2, xe = re/R, xh = rh/R, x = |xe − xh|, ye = 2z/L and yh = 2z/L.The numerical evaluation of the above integrals gives the expectation value ofthe Hamiltonian, 〈H〉, for a certain value of the variation parameter α, from whichwe obtain Eb = Ee + Eh − Eex.4 FIZIKA A (Zagreb) 16 (2007) 1, 1–10safwan et al.: exciton state in a quantum dot3. Results and discussionFor the GaAs material, we used the electron effective mass m∗e = 0.0667me,the hole effective mass m∗h = 0.34me, the dielectric constant ε equal to 13.18, theband-gap energy Eg = 1.247 xAl (xAl is the aluminum concentration) and the ratiobetween the conduction and the valence band offsets of 65−35%. The 3D Rydbergenergy is equal to Ry = 0.0044 eV and the 3D Bohr radius is aB = 12.5 nm.Figure 1 shows the calculated exciton ground-state binding energy as a functionof the radius of the quantum disc for three different values of the half-width L/2 = 4,7 and 10 nm. The calculated values show the presence of the well known peaks ofthe binding energy curves in nano-structures, which depend strongly on the QDdisc radius (R values), but its dependence on the QD disc width is not strong.These results are in a good agreement with the previous data obtained by Le Goffand Stebe [18].Fig. 1. The binding energy of the exciton as a function of R. The three curves areat different values of QD disc half-width L/2 (as indicated). (xAl = 0.4).Here, we would like to add that the peak positions of the binding energy as afunction of L/2 occur nearly at the same value of R = 3 nm. We notice the suddendecrease of the exciton binding energy when decreasing the value of radius. WhenR increases from 7 to 10 nm, the binding energy changes almost by 10 meV, andchanges by about half of this value if the disc half-width L/2 increases from 7 to10 nm.FIZIKA A (Zagreb) 16 (2007) 1, 1–10 5safwan et al.: exciton state in a quantum dotFigure 2 displays the variation of the exciton binding energy as a function ofR for two different values of Al content (xAl = 0.15, 0.4). Here a right shift ofthe peak position by almost 1 nm, and by 20 meV in the height, is observed whenthe quantity of Al increases by the factor of about 0.25. The height of the peak ishigher due to the increase of the barrier height (larger x) because of the strongerconfinement of particles.Fig. 2. The variation of the exciton binding energy with R at two different valuesof Al concentration, xAl = 0.15 and 0.4.The position of the exciton binding energy peaks can be estimated to occuraround L/2 ∼= 4 nm and R ∼= 3 nm or for diameter ∼= 6 nm for the quantum dotdisc. It has been shown in Ref. [16] that there is a scaling rule for circular and squarequantum wires of the form L/(2R) = 0.9136, such that a square wire of width Lis equivalent to a circular wire of diameter 2R, if the ratio of 0.9136 is achieved.Using this scaling rule, the critical confinement width for quantum square wire ofwidth L = 5.4 nm is equivalent to the present quantum disc with radius R ∼= 3 nm.From the behavior of the binding energy positions discussed above, we mayconclude that the bulk effect sets in along one spatial axis around L/2 ∼= 4.5 to 6nm, fairly independently of the confinement conditions. The present results shouldbe useful for designers of nano-scale devices.Figure 3 represents the exciton energy Eex as a function of R for L/2 = 4, 7 and10 nm. Again, as we see the exciton energy does not depend strongly on L/2, aswe mentioned before about the dependence of the binding energy on L/2 (Fig. 1).6 FIZIKA A (Zagreb) 16 (2007) 1, 1–10safwan et al.: exciton state in a quantum dotFig. 3. The exciton energy Eex as s function of R, at the shown values of L/2.Fig. 4. The exciton energy Eex and the exciton binding energy Eb versus the radiusR. a) L/2 = 4 nm, b) L/2 = 7 nm and L/2 = 10 nm.FIZIKA A (Zagreb) 16 (2007) 1, 1–10 7safwan et al.: exciton state in a quantum dotIn Fig. 4 we present the exciton binding energy (Eb) and the correspondingexciton energy Eex as a function of the disc radius R for three different values ofL/2 = 4, 7 and 10 nm. In Fig. 4a there is no intersection between the exciton energyand exciton binding energy, Eex > Eb, so a small amount of energy is sufficient tobreak the combination of the electron-hole pair.At the intersection points (Figs. 4b and 4c), the binding energy is equal tothe expectation value of the Hamiltonian, and is equal to half of the free particleenergy (Ee + Eh). At energies below the intersection point, where Eex > Eb, theexciton state annihilates faster than above the intersection point where Eex < Eb.The crossover of the binding energy curve with the Hamiltonian curve confirmsour above discussion. In order to obtain a large exciton binding energy, we shouldchoose quantum dots with a larger radius (between 3 nm and 10 nm). However, ifthe radii of quantum dots are beyond the scale of nano-structure, the principle ofquantum theory is no longer valid and optical properties of the dots belong to theregion of bulk materials.4. ConclusionWe developed the computer programs to calculate the binding energy of anexciton which is confined in a quantum dot disc of GaAs material. We found thatthe binding energy depends strongly on the size of the disc. Any change of the Alcontent of the barrier material affects the exciton confinement in the nano-material.We present theoretical values for several quantum disc dimensions, and we suggestthat R should be greater than 3 nm and less than 10 nm. The obtained resultsare very important in studies of optical properties of the quantum dots and in thenano-scale devices [19,20].References[1] L. Esaki, in Physics and Application of Quantum Wells and Superlattices, Vol. 170,NATO Advanced Study Institute, series B: Physics, Eds. E. E. Mendez and K. vonKlitzing, Plenum Press, New York (1987).[2] L. Brus, Appl. Phys. A 53 (1996) 465.[3] L. Brus, IEEE J. Quantum Electron. 22 (1989) 1909.[4] P. M. Petrof, A. C. Gossard, R. A. Logan and W. Wiegmann, Appl. Phys. Lett. 41(1982) 635.[5] O. Brand, L. Tapfer, K. Ploog, R. Bierwolf, M. Hohenstein, F. Phillipp, H. Lage andA. Heberte, Phys. Rev. B 44 (1991) 8043.[6] T. Tsujikawa, S. Mori, H. Watanabe and M. Yoshita, Physica E 7 (2000) 308.[7] P. Modak, M. K. 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Imamog Lu et al., Phys. Rev. Lett. 83 (1999) 4204.[20] A. Imamog Lu, Physica E 16 (2003) 47.FIZIKA A (Zagreb) 16 (2007) 1, 1–10 9safwan et al.: exciton state in a quantum dotEKSITONSKA STANJA U KVANTNOJ TOČKIIzračunali smo energiju vezanja, svojstvena stanja i svojstvene vrijednosti eksitonau kvantnoj točki s konačnim potencijalom. Opisujemo energiju vezanja eksitonaza vǐse vrijednosti polumjera (R) i poluširine (L/2) diska. Raspravljamo stabilnosteksitona za male i veće dimenzije diska. Uspored–ujemo naše rezultate s poznatimdrugim teorijskim i eksperimentalnim rezultatima. Naši rezultati daju dobre oc-jene za povoljnu veličinu diska kvantne točke, i predstavljaju korisne podatke zaproučavanje optičkih svojstava kvantnih točaka u napravama nano veličine.10 FIZIKA A (Zagreb) 16 (2007) 1, 1–10

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