ii Due to the polarization interaction U r, a, the carriers can be, atε < 1, repelled from the inner surface of the dielectric particle, with the formation of bulk local states inside th
Trang 1Optical absorption of one-particle electron states in
quasi-zero-dimensional nanogeterostructures: Theory
Q20
Q19 Sergey I Pokutnyia, Yuriy N Kulchinb, Vladimir P Dzyubab,*
a Chuіko Institute of Surface Chemistry, National Academy of Sciences, Ukraine
b Institute of Automation and Control Processes, FEB Russian Academy of Sciences, Russia
Q1
a r t i c l e i n f o
Article history:
Available online xxx
Keywords:
One-particle quantum-confined states of
charge carriers
Quantum dots
Absorption and scattering of light
Oscillator strength
Dipole approximation
a b s t r a c t
The paper is devoted to the theory for the interaction of an electromagneticfield 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 sec-tions 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/)
1 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 1 ¡ 102 nm grown in semiconductor (or dielectric)
matrices The studies in thisfield 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
com-puters or for manufacturing active layers of injection
semi-conductor lasers[1e6]
The optical and electro-optical properties of such
quasi-zero-dimensional 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
charge carrier with the surface charge induced at the spherical interface, U (r, a), depends on the relative permittivityε ¼ ε1=ε2 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, andε1 andε2are 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
atε < 1 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, atε < 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 low-energy bulk states is of an oscillatory shape
It has been shown [7e12] that the formation of the above-mentioned 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:
* 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.
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
-r e v i e w - a - n a t u -r a l - s c i e n c e - a n d - e n g i n e e -r i n g /
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/ ).
Pacific Science Review A: Natural Science and Engineering xxx (2016) 1e5
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Trang 2whereb¼ε 1 ε 2
ε 1 þε 2and abiis the Bohr radius of a charge carrier
medium with the permittivityεi(i¼ 1, 2)
In[10e15], the optical properties of an array of InAs and InSb
QDs in the GaAs and GaSb matrices and the corresponding
opera-tional 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
approxi-mation, the spectrum of such quantum-confined states can be
described as a spectrum of a charge carrier in a spherically
sym-metric 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
inter-action between an electromagneticfield with one-particle
quan-tum-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
2 Spectrum of charge carriers in a quantum dot
We consider a simple model in which a quasi-zero-dimensional
system is defined as a neutral spherical semiconductor QD of radius
a and permittivityε2, embedded in a surrounding with permittivity
ε1 Let an electron (e) and hole (h), whose effective masses are,
correspondingly, meand 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, ahand aex, where
ae¼ ε2Z2
mee2; ah¼ ε2Z2
mhe2; aex¼ ε2Z2
are the Bohr radii of the electron, hole, and exciton in the
semi-conductor with the permittivity ε2, respectively, and
m¼ memh=ðmhþ meÞ is the exciton effective mass All of the
char-acteristic lengths of the problem are much larger than the
inter-atomic spacing
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-zero-dimensional system, the Hamiltonian of the EHP is[7e12]:
H¼ 2mZ2
eDe2mhZ2 Dhþ Vehð r!e; r!hÞ þ Uð r!e; r!h; aÞ þ Eg
(4)
where thefirst two terms in the sum define the kinetic energy of the electron and hole, Egis the energy bandgap in the bulk (un-bounded) semiconductor with the permittivityε2, Vehð r!e; r!hÞ is the energy of the electron Coulomb interaction:
Vehð r!e; r!hÞ ¼ 2ae2
2ε2a
r2
e 2rerhcosqþ r2
h
with the angleqbetween the vector r!eand r!
h, and Uð r!e; r!h; aÞ is the energy of interaction of the electron and hole with the polari-zation field induced by the electron and hole at the spherical interface between the two media For arbitrary values ofε2andε1, the interaction energy Uð r!e; r!h; aÞ can be represented analytically
as[7e11]
whereq(x) is the unit step function and
a¼ ε1
In the bulk of a QD, the electron (hole) energy levels can origi-nate Their energies are defined as[17]
En;lðaÞ ¼ Z
242
;l
Q4
where the subscripts (n, l) refer to the corresponding quantum 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 quantum-confined levels to originate, it is necessary that in the Hamilto-nian(4), the electron (hole) energy(8)be considerably larger than the energy of the interaction of the electron (hole) with the po-larizationfield(6)generated at the spherical QD-dielectric (semi-conductor) matrix interface:
En;lðaÞ > > UðaÞzbe2
Condition(9)is satisfied for QDs of radii Q5
a< < ae;hs ¼4
2
;l
ε2a
r
!
e!r
h
a22
2!r
e!r
h
a2 cosqþ 1
12= e2b 2ðε1þ ε2Þa
Z∞ 0
a2
rhya
qy a2
rh dy
j r!e yð r!e= r!hÞj
e2b
2ðε1þ ε2Þa
Z∞ 0
a2
rey
a
qy a2
re
dy
j r!h yð r!h= r!eÞj
(6)
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Trang 3At 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]
Taking into account(8), we can rewrite inequality(11)as
Z242
þ1;l 42
;l
Q6 Formula(8), describing the spectrum of charge carriers in a QD,
is applicable to the lowest states (n, l) that satisfy the inequality
whereDV0(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 valueDV0(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;mðr;q; 4Þ ¼ Yl;mðq; 4Þ J1þ1=2
4n;l
J1þ3=2
4n;la
ffiffiffi 2 r
r
(14)
where r is the distance of the electron or hole from the QD centre,q
and4 are the azimuthal and polar angles that define the
orienta-tion 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
J1þ3 =
2
4n;l¼
ffiffiffi 2 p
r
Jlþ1
J1þ1 =
2
4n;l¼
ffiffiffi 2 p
r
Jl
4n;l4n;l
3 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]
DðrÞ ¼ 3ε1
2ε1þ ε2
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 ðn ¼ 1; l ¼ 0; m ¼ 0Þ 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 uniformfield 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
Taking into account(14)and(15), we can write the wave func-tions of the〈1sj and 〈1pj states as
j1s〉 ¼J1;0;0ðr;q; 4Þ ¼ Y0;0ðqÞ 2
a3=2
j0
40;1r=a
j1
40;1
j1s〉 ¼J1;0;0ðr;q; 4Þ ¼ Y1;0ðqÞ 2
a3=2
j1
41;1r=a
j2
41;1
Substituting(18)and(19)into formula(17)and integrating, we obtain the expression for the dipole moment of the transition in free space as follows:
D0;0ðaÞ ¼ 2ppffiffiffi6
341;1j2
41;142
;1p2
2 4cos 41;1 3
42
;1p2 sin41;1
41;1j2
41;1
42
;1p2
3 5ea ¼ 0; 433ea
(20)
Next, according to (20) and (16), the dipole moment of the transition in the QD with the permittivityε2in the surrounding matrix with the permittivityε1is
where
L ¼ 2ε1
4 Absorption of light at electron states in quantum dots Using the above results for the matrix element of the dipole moment of the transition (formulas(21),(22)), we can elucidate the behaviour of the semiconductor quasi-zero-dimensional systems
on absorbing the energy of the electromagnetic field in the fre-quency region corresponding to the energies of the quantum-confined states in the QD(8) The absorption cross section of a spherical QD of radius a can be expressed in terms of the polariz-ability of the QD, A (u, a) as[20]
sabcðu; aÞ ¼ 4p u
whereuis the frequency of the external electromagneticfield
The polarizability can be easily determined if the QD is consid-ered as a single giant ion Let the radius of the QD be(10) In such
QD, the quantum-confined states of charge carriers are formed At room temperature, these states are slightly broadened, satisfying inequality(12) In this case, the polarizability of the charged, can be Q7
expressed in terms of the matrix element of the dipole moment of the transition D1,0(a)(21)between the lowest 1s and 1p states[14]:
Aðu; aÞ ¼ e2f0;1
me;h
where
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Trang 4Ze2 ½u1ðaÞ u0ðaÞ D1;0ðaÞ 2 (25)
is the oscillator strength of the transition of a charge carrier from
the ground 1s state to the 1p state,Zu1¼ E1;1andZu1¼ E1;0are
correspondingly, the energies of the discrete 1s and 1p levels by
formula(8)andG1(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;1¼42
;1p2L2
D2
;0
e2a2
2
We assume that the frequencyuof the wave of light is far from
the resonance frequencyu1of the discrete 1p state and, in addition,
that the broadening of the 1p level is small, i.e.,G1=u1< < 1[9,11]
Then, for the qualitative estimate of the QD polarizability(24), we
obtain, with regard to(8), the following expression:
AðaÞ ¼4f0;1me;h
44
;1m0
a
aB
4
where aBis 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 frequencyuby the QD[20]as
sscðu; aÞ ¼ 27jAðuÞj2
33
u c
4
(28)
The theory developed here for QDs is applicable only to the
intraband transitions of electrons (holes) whose spectrum Ene;h;lðaÞ 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]
Formula(29)is applicable to an array of QDs that do not interact
with each other The condition such that the QDs of radius a and
concentration N do 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.,
With ae,h~5 nm, criterion(30)is satisfied for the concentrations
of QDs N ~ 1015cm¡3achievable under the experimental conditions
of Refs.[1e6]
Q9 for a number of IIIeV semiconductors
5 Comparison of theory with experiments
Similar to Ref.[18]
Q10 , 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
T¼ 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
quantum-confined 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)) andssc(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 ofQ11
optical absorption and scattering, we use expressions(23),(27), and(28)on the assumptions that the frequency of the light waveu
is far from the resonance frequencyu1of the discrete hole state in the QD and that the broadeningG1of the energy level is small[9,11]
(G1/u1<<1) In this case, the absorption cross section and the scattering cross section take the following forms:
sabsðu; aÞ ¼16pf0;1umh
44
;1cm0
a
aB
4
sscðu; aÞ ¼11p3f2 ;1
3342
;1
mh
m0
2u c
4 a
aB
8
The quasi-zero-dimensional systems considered here are essential nonlinear media with respect to infrared radiation In fact, the dipole moments of the transitions in the QDs of radii 2e5 nm are rather large in magnitude: D1,0z 10D, being many times larger than the value 0, 1D typical for the bulk IIIeV semiconductors
[20,21] In addition, according to the selection rules for QDs in the electromagneticfield, the dipole transitions between the nearest levels En,l (8) are allowed [19] Under these transitions, the azimuthal quantum number changes by unity
From the estimates presented, it follows that for QDs of radii
2e5 nm, the absorption cross section can be as large as
sabs~ 1016cm2 This value is eight orders of magnitude larger thanQ12
the typical absorption cross sections for atoms [22] Since the scattering cross sections(32)under the experimental conditions of Refs.[1e7]are negligible compared to the corresponding absorp-Q13
tion cross sectionssabs(31) Thus, the optical attenuation coeffi-Q14
cientg(31)for QDs is controlled mainly by the absorption process at the quantum-confined states of charge carriers(8) If the concen-tration of QDs of radii satisfyingcondition (38)is N ~ 1015cm¡3, theQ15,16
optical attenuation coefficient, on absorption at the QDs, takes the valueg≈ 0, 1 cm¡1 The large optical absorption cross section and attenuation coefficients in the quasi-zero-dimensional systems treated above allow for the use of such heterophase structures as new efficient absorbers of electromagnetic waves in a wide wave-length range variable over wide limits in accordance with the na-ture of materials in contact
[16]
References
[1] V.P Dzyuba, A.E Krasnok, J.N Kulchin, Nonlinear refractive index of dielectric nanocomposites in weak optical fields, Tech Phys Lett 36 (No 11) (2010) 973e977.
[2] V.P Dzyuba, A.E Krasnok, J.N Kulchin, I.V Dzyuba, A model of nonlinear optical transmittance for insulator nanocomposites, Semiconductor 45 (3) (2010) 295e301.
[3] V.P Dzyuba, Y.N Kulchin, S.I Pokutnyi, Optical nonlinearity of dielectric nanocomposite”, Optics 3 (No 1) (2014) 22e37 Special Issue: Optics and spectroscopy of the charge carriers and exciton states in quasi-zero-dimensional nanostructures.
[4] V.P Dzyuba, Yu N Kulchin, V.A Milichko, Effect of the shape of a nano-object
on quantum-size states, J Nanopart Res 14 (No 8) (2012) 1208e1212.
[5] A.I Ekimov, A.A Onushchenko, Size quantization of the energy spectrum of electrons in semiconductor microcrystals, JETP Lett 40 (No 8) (1984) 1136e1139.
[6] S.I Pokutnyi, Quantum size Shtark effect in quantum dots, J Appl Phys 96 (No 2) (2004) 1115e1121.
S.I Pokutnyi et al / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1e5 4
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2
3
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9
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12
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20
21
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24
25
26
27
28
29
30
31
32
33
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35
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37
38
39
40
41
42
43
44
45
46
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48
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50
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52
53
54
55
56
57
58
59
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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130
Trang 5[7] N.A Efremov, S.I Pokutnyi, Macroscopic local charge states in the ultrafine
media, Phys Solid State 27 (No 1) (1985) 27e55.
[8] N.A Efremov, S.I Pokutnyi, Spectrum of local states in the ultrafine
environ-ments, Phys Solid State 32 (No.10) (1990) 2921e2930.
[9] N.A Efremov, S.I Pokutnyi, The broadening of the quasi-stationary states in
the ultrafine environments, Phys Solid State 33 (No 10) (1991) 2845e2851.
[10] S.I Pokutnyi, Theory of macroscopic local states in quasi-zero-dimensional
structures: surface local states, Phys Status Solidi B 165 (No 1) (1991)
109e118.
[11] S.I Pokutnyi, Theory of macroscopic local states in quasi-zero-dimensional
structures: bulk local states, Phys Status Solidi B 172 (No 2) (1992) 573e582.
[12] S.I Pokutnyi, Size quantization states in quasi-zero-dimensional structures,
Phys Solid State 35 (No 2) (1993) 129e136.
[13] S.I Pokutnyi, Optical absorption of one-particle electron states in quasi e zero
dimensional nanosystems: I Dipole moments, Phys Solid State 39 (No.4)
(1997) 634e638.
[14] S.I Pokutnyi, Optical absorption of one-particle electron states in quasi e zero
dimensional nanosystems: II Absorption cross section and scattering of light,
Phys Solid State 39 (No.4) (1997) 528e534.
[15] S.I Pokutnyi, Bulk local states in quasi-zero-dimensional nanosystems, Semiconductors 31 (No 12) (1997) 1247e1254.
[16] V.M Agranovich, V.L Ginzburg, Crystal Optics with Spatial Dispersion and Excitons, 2nd edn, Nauka, Moscow, 1979 Springer, New York, (1984). Q17
[17] Al.L Efros, A.L Efros, Interband absorption of light in spherical nanocrystals, Sov Phys Semicond 16 (No 8) (1982) 772e784.
[18] V Grabovskis, Y Dzenis, A Ekimov, Photoionization nanocrystals, Sov Phys Solid State 31 (No 1) (1989) 1255e1258.
[19] A.S Davydov, Quantum Mechanics, 2nd edn, Nauka, Moskow, 1973 Perga-mon, Oxford, (1976).
[20] L.D Landau, E.M Lifshitz, Course of Theoretical Physics, Vol 8: Electrody-namics of Continuous Media, 2nd edn, Nauka, Moskow, 1973 Pergamon, New York (1984).
[21] Y.I Petrov, Physics of Small Particles [in Russian], Nauka, Moskow, 1982.
[22] N.W Ashcroft, N.D Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1979.
S.I Pokutnyi et al / Pacific Science Review A: Natural Science and Engineering xxx (2016) 1e5 5
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12
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15 16 17 18 19 20 21 22 23 24 25 26 27 28
...[14] S.I Pokutnyi, Optical absorption of one- particle electron states in quasi e zero< /small>
dimensional nanosystems: II Absorption cross section and scattering of light,... quantization states in quasi- zero- dimensional structures,
Phys Solid State 35 (No 2) (1993) 129e136.
[13] S.I Pokutnyi, Optical absorption of one- particle electron. .. 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