optical absorption of one particle electron states in quasi zero dimensional nanogeterostructures theory

1 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 PSRA54_proof ■ 18 December 2016 ■ 1/5 Pacific Science Review A: Natural Science and Engineering xxx (2016) 1e5 Contents lists available at ScienceDirect H O S T E D BY Pacific Science Review A: Natural Science and Engineering j o u r n a l h o m e p a g e : w w w j o u r n a l s e l s e v i e r c o m / p a c i fi c - s c i e n c e review-a-natural-science-and-engineering/ Q20 Optical absorption of one-particle electron states in quasi-zerodimensional nanogeterostructures: Theory Q19 Sergey I Pokutnyi a, Yuriy N Kulchin b, Vladimir P Dzyuba b, * a Q1 b Chuіko Institute of Surface Chemistry, National Academy of Sciences, Ukraine Institute of Automation and Control Processes, FEB Russian Academy of Sciences, Russia a r t i c l e i n f o a b s t r a c t Article history: Available online xxx The paper is devoted to the theory for the interaction of an electromagnetic field with one-particle quantum-confined states of charge carriers in semiconductor quantum dots It is demonstrated that the oscillator strengths and dipole moments of the transitions for one-particle states in quantum dots are large parameters, exceeding the corresponding typical parameters for bulk semiconductor materials In the context of the dipole approximation, it is demonstrated that the large optical absorption cross sections in the quasi-zero-dimensional systems enable the use of such systems as efficient absorbing materials Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Keywords: One-particle quantum-confined states of charge carriers Quantum dots Absorption and scattering of light Oscillator strength Dipole approximation Introduction At present, the optical and electro-optical [1e6] properties of quasi-zero-dimensional structures have been extensively studied Such structures commonly consist of spherical semiconductor nanocrystals, generally referred to as quantum dots (QDs), with a radius a z ¡ 102 nm grown in semiconductor (or dielectric) matrices The studies in this field are motivated by the fact that such heterophase systems represent new promising materials for the development of new components of nonlinear optoelectronics to be used, specifically, for controlling optical signals in optical computers or for manufacturing active layers of injection semiconductor lasers [1e6] The optical and electro-optical properties of such quasi-zerodimensional structures depend on the energy spectrum of a spatially confined electronehole pair (EHP), i.e., an exciton [1e8] By the methods of optical spectroscopy, the effects of quantum confinement on the energy spectra of electrons and excitons [5e8] were revealed in these heterophase structures Previously [7], the conditions for the localization of charge carriers near the spherical interface between the two dielectric media were analysed In this case, the polarization interaction of a * Corresponding author E-mail addresses: Pokutnyi_Sergey@inbox.ru (S.I Pokutnyi), kulchin@iacp.dvo ru (Y.N Kulchin), vdzyuba@iacp.dvo.ru (V.P Dzyuba) Peer review under responsibility of Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University charge carrier with the surface charge induced at the spherical interface, U (r, a), depends on the relative permittivity ε ¼ ε1=ε In this equation, r is the spacing between the charge carrier and the centre of the dielectric particle, a is the radius of the particle, and ε1 and ε2 are the permittivities of the surrounding medium and of the dielectric particle embedded in the medium, respectively For the charge carriers in motion near the dielectric particle, there are two possibilities: (i) Due to the polarization interaction U (r, a), the carriers can be attracted to the particle surface (to the outer or inner surface at ε < or ε > 1, respectively), with the formation of outer [8,9] or inner surface states [10] (ii) Due to the polarization interaction U (r, a), the carriers can be, at ε < 1, repelled from the inner surface of the dielectric particle, with the formation of bulk local states inside the particle bulk [11,12]; in this case, the spectrum of the lowenergy bulk states is of an oscillatory shape It has been shown [7e12] that the formation of the abovementioned local states is of a threshold-type nature and is possible if the radius of the dielectric particle a is larger than a certain critical radius ac: a ! ac ¼ 6jbi j À1 abi ; (1) http://dx.doi.org/10.1016/j.psra.2016.11.004 2405-8823/Copyright © 2016, Far Eastern Federal University, Kangnam University, Dalian University of Technology, Kokushikan University Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article in press as: S.I Pokutnyi, et al., Optical absorption of one-particle electron states in quasi-zero-dimensional nanogeterostructures: Theory, Pacific Science Review A: Natural Science and Engineering (2016), http://dx.doi.org/10.1016/j.psra.2016.11.004 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 S.I Pokutnyi et al / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1e5 where b ẳ 11 ỵ2 and abi is the Bohr radius of a charge carrier in a medium with the permittivity εi (i ¼ 1, 2) In [10e15], the optical properties of an array of InAs and InSb QDs in the GaAs and GaSb matrices and the corresponding operational characteristics of injection lasers, with the active region on the basis of this array, were studied experimentally In these studies, a large short-wavelength shift of the laser emission line was observed for the array of QDs In such an array, the energy spectrum of charge carriers is completely discrete [10e12,15] if the QDs are smaller than 1e7 nm in size In the first-order approximation, the spectrum of such quantum-confined states can be described as a spectrum of a charge carrier in a spherically symmetric well with infinitely high walls To date, there have been no theoretical investigations on optical absorption and scattering at such discrete states in arrays of QDs To close the gap in this area, here we develop a theory of the interaction between an electromagnetic field with one-particle quantum-confined states of charge carriers that originate in the bulk of a semiconductor QD In conclusion, we briefly discuss possible physical situations in which the results obtained above can be used for interpreting the experimental data H¼À Z2 Z2 ! ! ! De À D þ Veh ð! r e ; r h Þ þ U r e ; r h ; aị ỵ Eg 2me 2mh h (4) where the first two terms in the sum define the kinetic energy of the electron and hole, Eg is the energy bandgap in the bulk (un! ! bounded) semiconductor with the permittivity ε2, Veh ð r e ; r h Þ is the energy of the electron Coulomb interaction: ! ! Veh ð r e ; r h ị ẳ 2ae2 1  22 a re2 2re rh cos q ỵ rh2 (5) ! ! ! ! with the angle q between the vector r e and r h , and Uð r e ; r h ; aÞ is the energy of interaction of the electron and hole with the polarization field induced by the electron and hole at the spherical interface between the two media For arbitrary values of ε2 and ε1, ! ! the interaction energy Uð r e ; r h ; aÞ can be represented analytically as [7e11]  a   Z∞ a2 r y q y À a2 r dy 2 h h e b e b ! ! Uð r e ; r h ; aị ẳ ! ! ! !1 ỵ ịa 2    j r e À yð r e = r h Þj !! !! 2 ε2 a r e r h a À r e r h a cos q ỵ = Q2 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 Q3 58 59 60 61 62 63 64 65 PSRA54_proof ■ 18 December 2016 ■ 2/5  a   Z∞ a2 re y q y À a2 re dy e2 b ! ! ! 21 ỵ ịa j r h À yð r h = r e Þj (6) Spectrum of charge carriers in a quantum dot where q(x) is the unit step function and We consider a simple model in which a quasi-zero-dimensional system is defined as a neutral spherical semiconductor QD of radius a and permittivity ε2, embedded in a surrounding with permittivity ε1 Let an electron (e) and hole (h), whose effective masses are, correspondingly, me and mh, be motion in this QD Let the spacing between the electron or hole and the QD centre be re or rh, respectively We assume that the bands for the electrons and hole are parabolic Along with the QD radius a, the characteristic lengths of the problem are ae, ah and aex, where a¼ ae ¼ ε2 Z2 m e2 ; ah ¼ ε2 Z2 m e2 ; aex ¼ ε2 Z2 m e2 e h (2) are the Bohr radii of the electron, hole, and exciton in the semiconductor with the permittivity ε2, respectively, and m ẳ me mh=m ỵ m ị is the exciton effective mass All of the charh e acteristic lengths of the problem are much larger than the interatomic spacing a0: 1 ỵ (7) In the bulk of a QD, the electron (hole) energy levels can originate Their energies are dened as [17] En;l aị ẳ Z2 42n;l where the subscripts (n, l) refer to the corresponding quantum Q4 size-confined states In this equation, n and l are the principal and azimuthal quantum numbers for the electron (hole), respectively, and 4n,l are the roots of the Bessel function For the quantumconfined levels to originate, it is necessary that in the Hamiltonian (4), the electron (hole) energy (8) be considerably larger than the energy of the interaction of the electron (hole) with the polarization field (6) generated at the spherical QD-dielectric (semiconductor) matrix interface: En;l ðaÞ > > UðaÞz a; ae ; ah ; aex [a0 (3) which enables us to treat the motion of the electron and hole in the QD with the effective-mass approximation In the context of the above-described model and approximations for a quasi-zerodimensional system, the Hamiltonian of the EHP is [7e12]: (8) 2me;h a2 be2 2aε2 (9) Condition (9) is satisfied for QDs of radii a < < ae;h s ¼ 42n;l b ae;h Q5 (10) Please cite this article in press as: S.I Pokutnyi, et al., Optical absorption of one-particle electron states in quasi-zero-dimensional nanogeterostructures: Theory, Pacific Science Review A: Natural Science and Engineering (2016), http://dx.doi.org/10.1016/j.psra.2016.11.004 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 S.I Pokutnyi et al / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1e5 At room temperature T0, the discrete levels of the electron (hole) En,l (8) in the QD are slightly broadened if the energy separation between the levels is [9,11] DEn;l ðaÞ < < kT0 (11) Taking into account (8), we can rewrite inequality (11) as Z2 42nỵ1;l 42n;l 2me;h a2 kT0 ẳ haị < < (12) Formula (8), describing the spectrum of charge carriers in a QD, is applicable to the lowest states (n, l) that satisfy the inequality DEn;l ðaÞ < < DV0 ðaÞ (13) where DV0 (a) is the depth of the potential well for electrons in the QD For example, for the CdS QDs whose sizes satisfy inequality (10), the value DV0 (a) is 2.3e2.5 eV [18] If condition (10) is satisfied, we can use, for the electron (hole) wave function in a QD, the wave function of an electron (hole) in a spherical quantum well with infinitely high walls [19]: Jn;l;m r; q; 4ị ẳ Yl;m q; 4ị r J1ỵ1=2 4n;l J1ỵ3=2 4n;l a r (14) where r is the distance of the electron or hole from the QD centre, q and are the azimuthal and polar angles that define the orientation of the radius vector of the electron (hole), respectively, Yl,m (q, 4) are the normalized spherical functions, (m is the magnetic quantum number of the electron or hole), and Jn (x) are the Bessel functions that can be expressed [19] as À J1ỵ3 4n;l ẳ J1ỵ1 4n;l ¼ = Q6 = 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 PSRA54_proof ■ 18 December 2016 ■ 3/5 rffiffiffi p À Jlỵ1 4n;l 4n;l (15) r Jl 4n;l 4n;l p Dipole moments of charge carriers transitions in a quantum dot In the frequency region corresponding to the above-considered states of charge carriers in QD bulk, the wavelength of light is much larger than the dimensions of these states In this case, the operator of the dipole moment of the electron (hole) located in the QD bulk is expressed as [20] Drị ẳ 31 er 21 ỵ ε2 (16) To estimate the value of the dipole moment, it is sufficient to consider the transition between the lowest discrete states (8), e.g., between the ground states j1s〉 ¼ n ẳ 1; l ẳ 0; m ẳ 0ị and j1p〉 ¼ ðn ¼ 1; l ¼ 1; m ¼ 0Þ To calculate the matrix element of the dipole moment of the charge carrier transition from the 1s state to the 1p state, D1,0 (a), we assume that the uniform field of the light wave is directed only along the axis Z In this case, we take the dipole moment D1,0 (a) (16) induced by the light wave as the perturbation responsible for such dipole transition The expression for the dipole moment of the transition follows from formula (16) and the expression for the dipole moment of the transition in free space is 66 67 68 Taking into account (14) and (15), we can write the wave func69 tions of the 〈1sj and 〈1pj states as 70 À À ÁÁ 71 j0 40;1 r=a 72 À ÁÁ (18) j1s ẳ J1;0;0 r; q; 4ị ẳ Y0;0 qị 3=2 À j1 40;1 r=a a 73 74 À À ÁÁ 75 j1 41;1 r=a 76 À ÁÁ (19) j1s〉 ¼ J1;0;0 ðr; q; 4Þ ¼ Y1;0 ðqÞ 3=2 À j2 41;1 r=a a 77 78 Substituting (18) and (19) into formula (17) and integrating, we 79 obtain the expression for the dipole moment of the transition in 80 free space as follows: 81 pffiffiffi 82 2p 83   D01;0 aị ẳ 341;1 j2 41;1 421;1 À p2 84   85 2 41;1 À p sin 41;1 86 5ea ¼ 0; 433ea  4cos 41;1 À À Á 87 41;1 j2 41;1 421;1 À p2 88 (20) 89 90 Next, according to (20) and (16), the dipole moment of the 91 transition in the QD with the permittivity ε2 in the surrounding 92 matrix with the permittivity ε1 is 93 94 D1;0 aị ẳ L0; 433a (21) 95 96 where 97 98 21 Lẳ (22) 99 21 ỵ 100 101 102 103 104 Absorption of light at electron states in quantum dots 105 106 Using the above results for the matrix element of the dipole 107 moment of the transition (formulas (21), (22)), we can elucidate the 108 behaviour of the semiconductor quasi-zero-dimensional systems 109 on absorbing the energy of the electromagnetic field in the fre110 quency region corresponding to the energies of the quantum111 confined states in the QD (8) The absorption cross section of a 112 spherical QD of radius a can be expressed in terms of the polariz113 ability of the QD, A (u, a) as [20] 114 u 115 sabc u; aị ẳ 4p Au; aÞ (23) 116 c 117 where u is the frequency of the external electromagnetic field 118 The polarizability can be easily determined if the QD is consid119 ered as a single giant ion Let the radius of the QD be (10) In such 120 QD, the quantum-confined states of charge carriers are formed At 121 room temperature, these states are slightly broadened, satisfying 122 inequality (12) In this case, the polarizability of the charged, can be Q7 123 expressed in terms of the matrix element of the dipole moment of 124 the transition D1,0 (a) (21) between the lowest 1s and 1p states [14]: 125 126 e2 f0;1 127 à (24) Au; aị ẳ 128 me;h u21 aị u2 iuG1 aị 129 130 where D1;0 aị ẳ 〈1sjD0 ðrÞj1p〉 (17) Please cite this article in press as: S.I Pokutnyi, et al., Optical absorption of one-particle electron states in quasi-zero-dimensional nanogeterostructures: Theory, Pacific Science Review A: Natural Science and Engineering (2016), http://dx.doi.org/10.1016/j.psra.2016.11.004 Q8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Q9 51 52 53 54 55 Q10 56 57 58 59 60 61 62 63 64 65 PSRA54_proof ■ 18 December 2016 ■ 4/5 S.I Pokutnyi et al / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1e5 f0;1 ¼ 2me;h Ze2  2 ẵu1 aị u0 aịD1;0 aị (25) is the oscillator strength of the transition of a charge carrier from the ground 1s state to the 1p state, Zu1 ¼ E1;1 and Zu1 ¼ E1;0 are, correspondingly, the energies of the discrete 1s and 1p levels by formula (8) and G1 (a) is the width of the 1p level [9,11] Taking into account formulas (8) and (21), we can express the oscillator strength (25) of the transition as f0;1  2 L2 D21;0  ¼ 421;1 À p2 ¼ ¼ 0; 0967L2 e2 a2 (26) We assume that the frequency u of the wave of light is far from the resonance frequency u1 of the discrete 1p state and, in addition, that the broadening of the 1p level is small, i.e., G1= < < [9,11] u1 Then, for the qualitative estimate of the QD polarizability (24), we obtain, with regard to (8), the following expression: Aaị ẳ 4f0;1 me;h 441;1 m0  a aB 4 a3B (27) where aB is the Bohr radius of an electron in free space Now we write the expression for the cross section of elastic scattering of the electromagnetic wave with frequency u by the QD [20] as ssc u; aị ẳ 27 jAðuÞj2 u4 c 33 (28) The theory developed here for QDs is applicable only to the e;h intraband transitions of electrons (holes) whose spectrum En;l ðaÞ is defined by formula (8) The optical attenuation coefficient, which involves both the absorption and scattering of light by one-particle quantum-confined states of charge carriers in the QD bulk of radius a, can be written for the concentration of QDs N, as [21] gu; aị ẳ Nẵsabc u; aị ỵ ssc u; aị (29) Formula (29) is applicable to an array of QDs that not interact with each other The condition such that the QDs of radius a and concentration N not interact with each other is reduced to the requirement that the spacing between the QDs is considerably larger than the dimensions of the above-considered one-particle states, i.e., ae;h N À1=3 < < (30) With ae,h ~5 nm, criterion (30) is satisfied for the concentrations of QDs N ~ 1015 cm¡3 achievable under the experimental conditions of Refs [1e6] for a number of IIIeV semiconductors Comparison of theory with experiments Similar to Ref [18], we can assume that under the experimental conditions of Refs [1e7], the annealing of the arrays of InAs and InSb QDs in the GaAs and GaSb matrices at the temperature T ¼ 273 K induces the thermal emission of a light electron such that a hole alone remains in the QD bulk In this case, the electron may be localized at a deep trap in the matrix If the distance d, from this trap to the QD centre is large compared to the QD radius, the Coulomb electronehole interaction Ve,h (5) in the Hamiltonian (4) can be disregarded As a result, the one-particle quantumconfined hole states (n, l) with the energy spectrum En,l (a) described by formula (8), appear in the QD bulk Now we roughly estimate the cross sections of optical absorption sabs ((25) and (29)) and ssc(28) at the quantum-confined hole state in the QDs for the selected (1se1p) transition under the experimental conditions of Refs [1e7] For the rough estimation of the cross sections of Q11 optical absorption and scattering, we use expressions (23), (27), and (28) on the assumptions that the frequency of the light wave u is far from the resonance frequency u1 of the discrete hole state in the QD and that the broadening G1 of the energy level is small [9,11] (G1/u1