direct interband light absorption in conical quantum dot

Research ArticleDirect Interband Light Absorption in Conical Quantum Dot

D B Hayrapetyan12 A V Chalyan1 E M Kazaryan12 and H A Sarkisyan123

1Russian-Armenian (Slavonic) University H Emin 123 0051 Yerevan Armenia2Yerevan State University A Manoogian 1 0025 Yerevan Armenia3Peter The Great Saint-Petersburg Polytechnic University Polytechnicheskaya 29 St Petersburg 195251 Russia

Correspondence should be addressed to D B Hayrapetyan dhayrap82gmailcom

Received 31 August 2015 Revised 3 November 2015 Accepted 4 November 2015

Academic Editor Paulo Cesar Morais

Copyright copy 2015 D B Hayrapetyan et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In the framework of the adiabatic approximation the energy states of electron aswell as the direct light absorption are investigated inconical quantum dot Analytical expressions for particle energy spectrum are obtained The dependence of the absorption edge ongeometrical parameters of conical quantum dot is obtained Selection rules are revealed for transitions between levels with differentquantum numbers In particular it is shown that for the radial quantum number transitions are allowed between the levels withthe same quantum numbers and any transitions between different levels are allowed for the principal quantum number

1 Introduction

The conical quantum dots (CQD) are zero-dimensionalsystems with nontrivial geometry the analytic descriptionof which is an extremely difficult problem [1ndash5] From theother side availability of many geometrical parameters char-acterizing the CQD allows realizing flexible manipulation ofthe energy levels of the charge carriers in these systems Theradius of the base and the height (or equivalently the coneangle) are from above-mentioned geometric parameters ofCQD

It is clear that changing these parameters we canmanipulate the energy levels of the electrons and holes inthese systems which lead to the change of the intrabandand interband quantum transitions behavior Theoreticalstudy of physical processes in the CQD was held in [5ndash9] It should be noted that for the theoretical descriptionthe form of confining potential of observed nanostructurehas fundamental importance In such systems an importantrole may play in particular effects of mechanical stress Thisquestion has been studied in [10 11] Thus the authors basedon the continuum elastic theory present a finite elementanalysis to investigate the influences of the elastic anisotropyand thickness of spacing layer on the strain field distribution

and band edges (both conduction band and valence band) ofthe InAsGaAs conical shaped quantum dots

In another work [12] authors studied intersubband linearand third-order nonlinear optical properties of CQD withinfinite barrier potential The electron structure of CQDsthrough effective mass approximation is determined analyt-ically Linear nonlinear and total absorption coefficients aswell as the refractive indices of GaAs CQDs are calculatedThe effect of the dots size and the effect of the incidentelectromagnetic field are investigated Results show that thetotal absorption coefficient and the refractive index of thedots largely depend on the size of the dots and on the intensityand polarization of the incident electromagnetic field

As follows from the results of these studies strain poten-tials are rather complicated and do not allow conduct ananalytical description of the physical process at the CQD Onthe other hand the authors of papers [5 8 9 12] consideredthe profile of the confining potential of CQD in the frameof the model of a rectangular infinitely deep well It isnoteworthy that the infinitely deep confining potential allowsthe partial separation of variables and this greatly facilitatesthe description of CQDs Along with the above descriptionof the mechanisms of the CQD in the case of a small coneangle effective analytical method description of such systems

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2015 Article ID 915742 6 pageshttpdxdoiorg1011552015915742

2 Journal of Nanomaterials

z

H

r

R

120579

0

Figure 1 Schematic figure of conical quantum dot

is the adiabatic approximation [13] Under this approach aquantum system can be represented as being linked to twosubsystems fast and slow

The Hamiltonian of the fast subsystem includes variablesas parameters of the slow subsystem and an effective poten-tial energy plays a slow subsystem (parametrically dependingon coordinates of slow system) energy fast [14ndash16] Thispeculiar method of separation of variables allows us to givean analytical description of single-particle states

In this paper the energy states of electron and directinterband light absorption inCQDare investigatedwithin theframework of adiabatic approximation

2 Theory

Let us consider an impenetrable conical QD (see Figure 1)where 119867 is height of the cone and 119877 is radius of the base ofthe cone Here the angle of the top of the cone is denoted by120579 Note that we consider the case of small angles of the top ofcone 120579 ≪ 1 It means that 119877 ≪ 119867 and hence

119905119892120579 =119877

119867≪ 1 (1)

Such a case is possible when during the experimentconical quantum dots were grown with high height [1ndash3]Condition (1) gives us the opportunity to apply geometricaladiabatic approximation for solving the problem [13] Thepotential energy of a charged particle (electron or hole) insuch structure has the following form

119880conf (120588 120593 119911) =

0 particle isin Δ

infin particle notin Δ(2)

In the regime of strong size quantization the energy ofthe Coulomb interaction between the electron and hole ismuch less than the energy caused by the size quantizationIn this approximation the Coulomb interaction betweenparticles can be neglected Then the problem is reduced tothe determination of separate energy states of the electronand hole It follows from the geometrical form of conicalQD that the particle motion along the radial direction occurs

more rapidly than along the 119911-direction This allows one touse the adiabatic approximation [13] The Hamiltonian of thesystem in the cylindrical coordinates has the form

(120588 120593 119911) = minusℏ2

2119898lowast119890

(1

120588

120597

120597120588(120588

120597

120597120588) +

1

1205882

1205972

1205971205932+1205972

1205971199112)

+ 119880conf (120588 120593 119911)

(3)

where 119898lowast119890is effective mass of charge carrier The frequency

of the radial ldquomotionrdquo is much greater than the frequencycharacterizing the ldquomotionrdquo along the axis of the cone Con-sequently the Hamiltonian of the system can be representedas a sum of the Hamiltonians for the ldquofastrdquo (

119891) and ldquoslowrdquo

(119904) subsystems

(120588 120593 119911) = 119891(120588 120593 119911) +

119904(120588 120593 119911) (4)

where

119891(120588 120593 119911) = minus

ℏ2

2119898lowast119890

(1

120588

120597

120597120588(120588

120597

120597120588) +

1

1205882

1205972

1205971205932)

+ 119880conf (120588 120593 119911)

119904(119911) = minus

ℏ2

2119898lowast119890

1205972

1205971199112

(5)

Axial variable 119911 in 119891(120588 120593 119911) plays the role of a constant

parameter According to geometrical adiabatic approxima-tion thewave function of the system is represented as follows

Ψ (120588 120593 119911) =119890119894119898120593

radic2120587

119891 (120588 119911) 120594 (119911) (6)

By substituting wave function in the Schrodinger equa-tion for the ldquofastrdquo subsystem for the radial wave function119891(120588 119911) we get

ℏ2

2119898lowast119890

(11989110158401015840(120588 119911) +

1

1205881198911015840(120588 119911) minus

1198982

1205882119891 (120588 119911))

+ 119864rad (119911) 119891 (120588 119911) = 0

(7)

where 119864rad(119911) is the radial part of energy The solution of thisequation is given through the Bessel functions of the first kind119869119898(120575120588) [17] where 120575 = radic2119898lowast

119890119864rad(119911)ℏ

2 The radial wavefunction should satisfy the following boundary conditions

119869119898((119867 minus 119911) 119905119892120579) = 0 (8)

From this boundary conditions finally we obtain the expres-sion for energy of ldquofastrdquo subsystem

119864rad119899120588 |119898|

(119911) =

ℏ21205822

119899120588 |119898|

2119898lowast119890(119867 minus 119911)

21199051198922120579

(9)

where 120582119899120588|119898|

are zeros of the Bessel functions of the first kind(119899120588= 0 1 2 ) For the lower levels of the spectrum the

Journal of Nanomaterials 3

Table 1 The comparison of the energy of Coulomb interaction (119864119890ℎ) between the electron and hole and size quantization energy (119864SQ)

119898 = 0 119899120588= 0 119899 = 0 119898 = 1 119899

120588= 0 119899 = 0 119898 = 0 119899

120588= 1 119899 = 0 119898 = 0 119899

120588= 0 119899 = 1

119864SQ119864119877 119864119890ℎ119864119877

119864SQ119864119877 119864119890ℎ119864119877

119864SQ119864119877 119864119890ℎ119864119877

119864SQ119864119877 119864119890ℎ119864119877

119877 = 05119886119861 29624 minus1121 70809 minus0877 141542 minus113 34483 minus095

119867 = 10119886119861


119867 = 10119886119861


119867 = 10119886119861

particle is mainly localized in the region |119911| ≪ 119867 Based onthis we expand 119864rad

119899120588 |119898|(119911) into a series

119864rad119899120588 |119898|

(119911) = 119864(0)

rad (1 minus119911

119867)

minus2

asymp 119864(0)

rad (1 + 2119911

119867) (10)

where 119864(0)

rad = ℏ21205822

119899120588|119898|2119898lowast

1198901198772 Relation (10) represents

an effective potential which is incorporated in Schrodingerequation of the ldquoslowrdquo subsystem

minusℏ2

2119898lowast119890

1205972

1205971199112120594 (119911) + 119864

(0)

rad (1 + 2119911

119867)120594 (119911) = 119864120594 (119911) (11)

This equation after simple transformations reduced to theform

minusℏ2

2119898lowast119890

1198892120594 (119911)

1198891199112+ 119865119911120594 (119911) = (119864 minus 119864

(0)

rad) 120594 (119911) (12)

where 119864ax = 119864 minus 119864(0)

rad is the axial part of the total energy and119865 = 2119864

(0)

rad119867 Introducing new notations 1199110= (ℏ22119898lowast

119890119865)13

and 119911 = (11199110)(119911 minus 119864119865) for the Schrodinger equation of

ldquoslowrdquo subsystem we get

1198892120594 (119911)

1198891199112

minus 119911120594 (119911) = 0 (13)

Solution of this equation is given by the Airy function of thefirst kind Ai(119911) [17]

120594 (119911) = 119888Ai (119911) = 119888 1

radic120587int

infin

0

cos(119906119911 minus 1

31199063)119889119906 (14)

From the boundary conditions we get expression for the axialenergy

119864ax = minus(ℏ21198652

2119898lowast119890

)

13

120572119899+1 (15)

where 120572119899+1 and (119899 = 0 1 2 ) are zeros of the Airy

function (1205721asymp minus2338 120572

2asymp minus4087 etc) Finally for the

total energy of the system we can write

119864 = 119864rad + 119864ax

=

ℏ21205822

119899120588 |119898|

2119898lowast1198901198772

minusℏ2

2119898lowast119890

(

21205822

119899120588 |119898|

1198671198772)

23

120572119899+1

(16)

As we mentioned above in the regime of strong size quan-tization the energy of the Coulomb interaction betweenthe electron and hole is much less than the energy causedby the size quantization and it can be neglected in firstapproximation Table 1 presents the comparison of the energyof Coulomb interaction between the electron and hole andsize quantization energy for different values of geometricalparameters of conical quantum dot The correction energy iscalculated by the help of perturbation theory According toperturbation theory the energy of the first correction is

1198641

] = ⟨]1003816100381610038161003816119881 (120588 119911)

1003816100381610038161003816 ]⟩ (17)

Here119881(120588 119911) = minus1198902120581radic1205882 + 1199112 where 120581 is the static dielectricconstant It can be seen from Table 1 the correction energyalways is negative and with the increase of radius 119877 therelative contribution of Coulomb energy of exciton becomessignificant

3 Direct Interband Light Absorption

Consider the direct interband absorption in conical quantumin the strong size quantization regime when the Coulombinteraction between electron and hole can be neglectedFurthermore consider the case of a heavy hole with 119898lowast

119890≪

119898lowast

ℎ where119898lowast

119890and119898lowast

ℎare effective masses of the electron and

hole respectivelyThen the absorption coefficient is given by[18]

119870 (Ω) = 119860sum

]]1015840

1003816100381610038161003816100381610038161003816intΨ119890

]Ψℎ

]1015840119889 119903

1003816100381610038161003816100381610038161003816

2

120575 (ℏΩ minus 119864119892minus 119864119890

] minus 119864ℎ

]1015840) (18)

where Ψ119890(ℎ)](]1015840) is given by expression (6) ] and ]1015840 are setsof quantum numbers corresponding to the electron andthe heavy hole respectively 119864

119892is the band gap of massive

semiconductorΩ is the frequency of the incident light and119860is a quantity proportional to the square of the matrix elementtaken by Bloch functions [19] In the regime of strong sizequantization for the absorption edge we finally get

ℏΩ000

= 119864119892+ℏ21205822

00

21205831198772+ℏ2

2120583(21205822

00

1198671198772)

23

1205721 (19)

where 120583 = 119898lowast119890119898lowast

ℎ(119898lowast

119890+119898lowast

ℎ) is the reduced electron-hole pair

effective mass

4 Journal of NanomaterialsEE

R

RaB

n = 2n = 1

n = 0

n120588 = 0

n120588 = 1 H = 10aBm = 0

180

150

120

90

60

30

006 08 10 12 14

Figure 2 The dependence of the energy levels of the electron fromthe base radius of CQD for the fixed value of the height

4 Result and Discussion

Let us proceed to the discussion of the results Note that thenumerical calculations are made for the conical QD fromGaAs with the following parameters 119898lowast

119890= 0067119898

119890 120581 =

1318 119864119877= 5275meV and 119886

119861= 104 A Figure 2 shows the

dependence of the energy levels of the charge carrier from thebase radius of CQD for the fixed value of the CQDrsquos height

Note that each level of the ldquofastrdquo subsystem has a familyof ldquoslowrdquo subsystem levels positioned thereupon One cansee from Figure 2 with increasing base radius energy of theparticle reduces since the contribution to the energy ofthe size quantization decreases The difference between theenergy levels of the same energy levelrsquos family is increasedwith increase in the axial quantum number For exampleΔ11986410

= 112119864119877 when 119877 = 15119886

119861and 119867 = 10119886

119861(119899120588= 0

119898 = 0) and Δ11986410= 34119864

119877 when 119877 = 15119886

119861and 119867 = 10119886

119861

(119899120588= 1 119898 = 0) Note that the transition frequency between

these energy levels is ΔΩ10

(119899120588= 0 119898 = 0) = 143 sdot 1012119888minus1

and ΔΩ10(119899120588= 1119898 = 0) = 43 sdot 1012119888minus1 which falls into the

IR part of spectrumFigure 3 shows the dependence of the energy levels of the

electron from the height of CQD for the fixed value of theCQDrsquos base radius The dependence of the energy levels onthe height of CQD has the same behaviour as the dependenceof radius with the increase of the height energy levels arereduced

Note that the total energy of the system ismore ldquosensitiverdquoto changes of the 119877 parameter which is a consequence ofthe higher contribution of size quantization into the electronenergy in radial directionThe same increase in the differencebetween the energy levels of the same energy levelrsquos familyoccurs for this dependence For example Δ119864

10= 058119864

119877

when 119877 = 2119886119861and 119867 = 15119886

119861(119899120588= 0 119898 = 0) and

Δ11986410= 176119864

119877 when119877 = 2119886

119861and119867 = 15119886

119861(119899120588= 1119898 = 0)

EE

R

60

50

40

30

20

10

010 12 14 16 18 20

HaB

n120588 = 0

n120588 = 1

n = 0n = 1

n = 2

R = 1aBm = 0

Figure 3 The dependence of the energy levels of the electron fromthe height of CQD for the fixed value of the base radius

300

295

290

285

280

275

27006 08 10 12 14

H = 5aBH = 10aB

H = 15aB

ℏΩ

000E

R

RaB

Figure 4 The dependence of absorption edge on the base radius ofCQD for the fixed value of height

Figure 4 shows the dependence of absorption edge onthe base radius of CQD for the fixed value of height in theregime of strong size quantization As can be seen from thefigure with decreasing base radius of CQD the absorptionedge increases It is the consequence of the following with thedecrease of parameter 119877 the effective width of the bandgapincreases by reducing the influence of the CQDrsquos walls Theenergy levels corresponding to high values of height arelocated above Note that the interband transition frequencybetween energy levels is Ω

000= 507 sdot 10

14119888minus1 for 119877 = 02119886

119861

and119867 = 15119886119861which falls into the visible part of spectrum

Figure 5 shows the dependence of absorption edge onthe height of CQD for the fixed value of base radius inthe regime of strong size quantization For the same reason


310

300

290

280

270

HaB

10 12 14 16 18 20

R = 05aBR = 1aB

R = 15aB

ℏΩ

000E

R

Figure 5The dependence of absorption edge on the height of CQDfor the fixed value of base radius

as we mentioned for Figure 4 with increasing height ofCQD the absorption edge increases Here the energy levelscorresponding to small values of base radius are locatedabove

Consider selection rules for transitions between levelswith different quantum numbers For the magnetic quantumnumber transitions between the levels with 119898 = minus119898

1015840areallowed and for the radial quantum number 119899

120588transitions

between the levels with 119899120588= 1198991015840

120588are allowed Consequently

there is no selection rule for the principal quantum numberand any transitions between different levels are allowed 119899 rarr

forall1198991015840

5 Conclusion

Summarizing the electronic states and optical properties ofCQD made of GaAs are studied The dependence of energylevels on the geometrical parameters of CQD is obtainedanalytically with the help of adiabatic approximation Eachlevel of the ldquofastrdquo subsystem has a family of ldquoslowrdquo subsys-tem levels positioned thereupon Note that the intrabandtransition frequency between energy levels falls into the IRpart of spectrum while the interband transition frequencyfalls into the visible part of spectrum It is shown that forradial quantum numbers transitions are allowed between thelevels with the same quantum numbers and for the principalquantum number any transitions between different levels areallowed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] S Pickering A Kshirsagar J Ruzyllo and J Xu ldquoPatternedmistdeposition of tri-colour CdSeZnS quantum dot films towardRGB LED devicesrdquo Opto-Electronics Review vol 20 no 2 pp148ndash152 2012

[2] J-HHuhCHermannstadter KAkahane et al ldquoFabrication ofmetal embedded nano-cones for single quantum dot emissionrdquoJapanese Journal of Applied Physics vol 50 no 6 Article ID06GG02 2011

[3] A Lenz R Timm H Eisele et al ldquoReversed truncated conecomposition distribution of In

08Ga02As quantum dots over-

grown by an In01Ga09As layer in a GaAs matrixrdquo Applied

Physics Letters vol 81 no 27 pp 5150ndash5152 2002[4] Y Liu W Lu Z Yu et al ldquoThe strain field distribution of

quantum dot array with conical shaperdquo Journal of NonlinearOptical Physics amp Materials vol 18 no 4 pp 561ndash571 2009

[5] R Khordad and H Bahramiyan ldquoOptical properties of a GaAscone-like quantum dot second and third-harmonic genera-tionrdquo Optics and Spectroscopy vol 117 no 3 pp 447ndash452 2014

[6] B Bochorishvili and H M Beka ldquoEnergy spectrum andoscillator strengths for spherical conical and cylindrical CdSequantum dotsrdquo IOP Conference Series Materials Science andEngineering vol 6 no 1 Article ID 012026 2009

[7] E M Kazaryan L S Petrosyan V A Shahnazaryan et alldquoQuasi-conical quantum dot electron states and quantum tran-sitionsrdquo Communications in Theoretical Physics vol 63 no 2pp 255ndash260 2015

[8] R Khordad and H Bahramiyan ldquoStudy of impurity positioneffect in pyramid and cone like quantum dotsrdquo The EuropeanPhysical Journal Applied Physics vol 67 no 2 pp 20402ndash204092014

[9] V Lozovskiy and V Pyatnytsya ldquoThe analytical study ofelectronic and optical properties of pyramid -like and cone-like quantum dotsrdquo Journal of Computational and TheoreticalNanoscience vol 8 pp 1ndash9 2011

[10] L Yu-Min Y Zhong-Yuan and R Xiao-Min ldquoThe influencesof thickness of spacing layer and the elastic anisotropy onthe strain fields and band edges of InAsGaAs conical shapedquantum dotsrdquo Chinese Physics B vol 18 no 1 pp 16ndash22 2009

[11] T O Cheche and Y-C Chang ldquoAnalytical approach for strainand piezoelectric potential in conical self-assembled quantumdotsrdquo Journal of Applied Physics vol 104 no 8 Article ID083524 2008

[12] M Dezhkam and A Zakery ldquoExact investigation of the elec-tronic structure and the linear and nonlinear optical propertiesof conical quantum dotsrdquo Chinese Optics Letters vol 10 no 12Article ID 121901 2012

[13] V Galitski B Karnakov and V Kogan Exploring QuantumMechanics A Collection of 700+ Solved Problems for StudentsLecturers and Researchers Oxford University Press New YorkNY USA 2013

[14] D B Hayrapetyan and E M Kazaryan ldquoAdiabatic descriptionof impenetrable particles in an infinitely deep potential wellrdquoJournal of Contemporary Physics vol 47 no 5 pp 230ndash2352012

[15] A A Gusev O Chuluunbaatar S I Vinitsky E M Kazaryanand H A Sarkisyan ldquoThe application of adiabatic method forthe description of impurity states in quantum nanostructuresrdquoJournal of Physics Conference Series vol 248 no 1 Article ID012047 2010


[16] D B Hayrapetyan ldquoDirect interband light absorption in astrongly prolated ellipsoidal quantum dotrdquo Journal of Contem-porary Physics vol 42 no 6 pp 292ndash297 2007

[17] M Abramowitz and I Stegun Handbook of MathematicalFunctions Applied Mathematics Series 1966

[18] Al L Efros and A L Efros ldquoInterband absorption of light in asemiconductor sphererdquo Semiconductors vol 16 no 7 pp 772ndash775 1982

[19] A Anselm Introduction to SemiconductorTheory Mir Publish-ers Moscow Russia 1982

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MetallurgyJournal of


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MaterialsJournal of


Nano

materials


Journal ofNanomaterials


z

H

r

R

120579

0

Figure 1 Schematic figure of conical quantum dot

is the adiabatic approximation [13] Under this approach aquantum system can be represented as being linked to twosubsystems fast and slow

The Hamiltonian of the fast subsystem includes variablesas parameters of the slow subsystem and an effective poten-tial energy plays a slow subsystem (parametrically dependingon coordinates of slow system) energy fast [14ndash16] Thispeculiar method of separation of variables allows us to givean analytical description of single-particle states

In this paper the energy states of electron and directinterband light absorption inCQDare investigatedwithin theframework of adiabatic approximation

2 Theory

Let us consider an impenetrable conical QD (see Figure 1)where 119867 is height of the cone and 119877 is radius of the base ofthe cone Here the angle of the top of the cone is denoted by120579 Note that we consider the case of small angles of the top ofcone 120579 ≪ 1 It means that 119877 ≪ 119867 and hence

119905119892120579 =119877

119867≪ 1 (1)

Such a case is possible when during the experimentconical quantum dots were grown with high height [1ndash3]Condition (1) gives us the opportunity to apply geometricaladiabatic approximation for solving the problem [13] Thepotential energy of a charged particle (electron or hole) insuch structure has the following form

119880conf (120588 120593 119911) =

0 particle isin Δ

infin particle notin Δ(2)

In the regime of strong size quantization the energy ofthe Coulomb interaction between the electron and hole ismuch less than the energy caused by the size quantizationIn this approximation the Coulomb interaction betweenparticles can be neglected Then the problem is reduced tothe determination of separate energy states of the electronand hole It follows from the geometrical form of conicalQD that the particle motion along the radial direction occurs

more rapidly than along the 119911-direction This allows one touse the adiabatic approximation [13] The Hamiltonian of thesystem in the cylindrical coordinates has the form

(120588 120593 119911) = minusℏ2

2119898lowast119890

(1

120588

120597

120597120588(120588

120597

120597120588) +

1

1205882

1205972

1205971205932+1205972

1205971199112)

+ 119880conf (120588 120593 119911)

(3)

where 119898lowast119890is effective mass of charge carrier The frequency

of the radial ldquomotionrdquo is much greater than the frequencycharacterizing the ldquomotionrdquo along the axis of the cone Con-sequently the Hamiltonian of the system can be representedas a sum of the Hamiltonians for the ldquofastrdquo (

119891) and ldquoslowrdquo

(119904) subsystems

(120588 120593 119911) = 119891(120588 120593 119911) +

119904(120588 120593 119911) (4)

where

119891(120588 120593 119911) = minus

ℏ2

2119898lowast119890

(1

120588

120597

120597120588(120588

120597

120597120588) +

1

1205882

1205972

1205971205932)

+ 119880conf (120588 120593 119911)

119904(119911) = minus

ℏ2

2119898lowast119890

1205972

1205971199112

(5)

Axial variable 119911 in 119891(120588 120593 119911) plays the role of a constant

parameter According to geometrical adiabatic approxima-tion thewave function of the system is represented as follows

Ψ (120588 120593 119911) =119890119894119898120593

radic2120587

119891 (120588 119911) 120594 (119911) (6)

By substituting wave function in the Schrodinger equa-tion for the ldquofastrdquo subsystem for the radial wave function119891(120588 119911) we get

ℏ2

2119898lowast119890

(11989110158401015840(120588 119911) +

1

1205881198911015840(120588 119911) minus

1198982

1205882119891 (120588 119911))

+ 119864rad (119911) 119891 (120588 119911) = 0

(7)

where 119864rad(119911) is the radial part of energy The solution of thisequation is given through the Bessel functions of the first kind119869119898(120575120588) [17] where 120575 = radic2119898lowast

119890119864rad(119911)ℏ

2 The radial wavefunction should satisfy the following boundary conditions

119869119898((119867 minus 119911) 119905119892120579) = 0 (8)

From this boundary conditions finally we obtain the expres-sion for energy of ldquofastrdquo subsystem

119864rad119899120588 |119898|

(119911) =

ℏ21205822

119899120588 |119898|

2119898lowast119890(119867 minus 119911)

21199051198922120579

(9)

where 120582119899120588|119898|

are zeros of the Bessel functions of the first kind(119899120588= 0 1 2 ) For the lower levels of the spectrum the



119898 = 0 119899120588= 0 119899 = 0 119898 = 1 119899

120588= 0 119899 = 0 119898 = 0 119899

120588= 1 119899 = 0 119898 = 0 119899

120588= 0 119899 = 1

119864SQ119864119877 119864119890ℎ119864119877

119864SQ119864119877 119864119890ℎ119864119877

119864SQ119864119877 119864119890ℎ119864119877

119864SQ119864119877 119864119890ℎ119864119877


119867 = 10119886119861


119867 = 10119886119861


119867 = 10119886119861


119899120588 |119898|(119911) into a series

119864rad119899120588 |119898|

(119911) = 119864(0)

rad (1 minus119911

119867)

minus2

asymp 119864(0)

rad (1 + 2119911

119867) (10)

where 119864(0)

rad = ℏ21205822

119899120588|119898|2119898lowast



minusℏ2

2119898lowast119890

1205972

1205971199112120594 (119911) + 119864

(0)

rad (1 + 2119911

119867)120594 (119911) = 119864120594 (119911) (11)


minusℏ2

2119898lowast119890

1198892120594 (119911)

1198891199112+ 119865119911120594 (119911) = (119864 minus 119864

(0)

rad) 120594 (119911) (12)

where 119864ax = 119864 minus 119864(0)


(0)


119890119865)13



1198892120594 (119911)

1198891199112

minus 119911120594 (119911) = 0 (13)


120594 (119911) = 119888Ai (119911) = 119888 1

radic120587int

infin

0

cos(119906119911 minus 1

31199063)119889119906 (14)


119864ax = minus(ℏ21198652

2119898lowast119890

)

13

120572119899+1 (15)





119864 = 119864rad + 119864ax

=

ℏ21205822

119899120588 |119898|

2119898lowast1198901198772

minusℏ2

2119898lowast119890

(

21205822

119899120588 |119898|

1198671198772)

23

120572119899+1

(16)


1198641

] = ⟨]1003816100381610038161003816119881 (120588 119911)

1003816100381610038161003816 ]⟩ (17)




119890≪

119898lowast





119870 (Ω) = 119860sum

]]1015840

1003816100381610038161003816100381610038161003816intΨ119890

]Ψℎ

]1015840119889 119903

1003816100381610038161003816100381610038161003816

2

120575 (ℏΩ minus 119864119892minus 119864119890

] minus 119864ℎ

]1015840) (18)




ℏΩ000

= 119864119892+ℏ21205822

00

21205831198772+ℏ2

2120583(21205822

00

1198671198772)

23

1205721 (19)


ℎ(119898lowast

119890+119898lowast


effective mass


R

RaB

n = 2n = 1

n = 0

n120588 = 0

n120588 = 1 H = 10aBm = 0

180

150

120

90

60

30

006 08 10 12 14




119890= 0067119898

119890 120581 =

1318 119864119877= 5275meV and 119886




= 112119864119877 when 119877 = 15119886

119861and 119867 = 10119886

119861(119899120588= 0

119898 = 0) and Δ11986410= 34119864

119877 when 119877 = 15119886

119861and 119867 = 10119886

119861



(119899120588= 0 119898 = 0) = 143 sdot 1012119888minus1





10= 058119864

119877

when 119877 = 2119886119861and 119867 = 15119886

119861(119899120588= 0 119898 = 0) and

Δ11986410= 176119864

119877 when119877 = 2119886

119861and119867 = 15119886

119861(119899120588= 1119898 = 0)

EE

R

60

50

40

30

20

10

010 12 14 16 18 20

HaB

n120588 = 0

n120588 = 1

n = 0n = 1

n = 2

R = 1aBm = 0


300

295

290

285

280

275

27006 08 10 12 14

H = 5aBH = 10aB

H = 15aB

ℏΩ

000E

R

RaB



000= 507 sdot 10

14119888minus1 for 119877 = 02119886

119861




310

300

290

280

270

HaB

10 12 14 16 18 20

R = 05aBR = 1aB

R = 15aB

ℏΩ

000E

R





120588transitions




forall1198991015840

5 Conclusion




References































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials





119898 = 0 119899120588= 0 119899 = 0 119898 = 1 119899

120588= 0 119899 = 0 119898 = 0 119899

120588= 1 119899 = 0 119898 = 0 119899

120588= 0 119899 = 1

119864SQ119864119877 119864119890ℎ119864119877

119864SQ119864119877 119864119890ℎ119864119877

119864SQ119864119877 119864119890ℎ119864119877

119864SQ119864119877 119864119890ℎ119864119877


119867 = 10119886119861


119867 = 10119886119861


119867 = 10119886119861


119899120588 |119898|(119911) into a series

119864rad119899120588 |119898|

(119911) = 119864(0)

rad (1 minus119911

119867)

minus2

asymp 119864(0)

rad (1 + 2119911

119867) (10)

where 119864(0)

rad = ℏ21205822

119899120588|119898|2119898lowast



minusℏ2

2119898lowast119890

1205972

1205971199112120594 (119911) + 119864

(0)

rad (1 + 2119911

119867)120594 (119911) = 119864120594 (119911) (11)


minusℏ2

2119898lowast119890

1198892120594 (119911)

1198891199112+ 119865119911120594 (119911) = (119864 minus 119864

(0)

rad) 120594 (119911) (12)

where 119864ax = 119864 minus 119864(0)


(0)


119890119865)13



1198892120594 (119911)

1198891199112

minus 119911120594 (119911) = 0 (13)


120594 (119911) = 119888Ai (119911) = 119888 1

radic120587int

infin

0

cos(119906119911 minus 1

31199063)119889119906 (14)


119864ax = minus(ℏ21198652

2119898lowast119890

)

13

120572119899+1 (15)





119864 = 119864rad + 119864ax

=

ℏ21205822

119899120588 |119898|

2119898lowast1198901198772

minusℏ2

2119898lowast119890

(

21205822

119899120588 |119898|

1198671198772)

23

120572119899+1

(16)


1198641

] = ⟨]1003816100381610038161003816119881 (120588 119911)

1003816100381610038161003816 ]⟩ (17)




119890≪

119898lowast





119870 (Ω) = 119860sum

]]1015840

1003816100381610038161003816100381610038161003816intΨ119890

]Ψℎ

]1015840119889 119903

1003816100381610038161003816100381610038161003816

2

120575 (ℏΩ minus 119864119892minus 119864119890

] minus 119864ℎ

]1015840) (18)




ℏΩ000

= 119864119892+ℏ21205822

00

21205831198772+ℏ2

2120583(21205822

00

1198671198772)

23

1205721 (19)


ℎ(119898lowast

119890+119898lowast


effective mass


R

RaB

n = 2n = 1

n = 0

n120588 = 0

n120588 = 1 H = 10aBm = 0

180

150

120

90

60

30

006 08 10 12 14




119890= 0067119898

119890 120581 =

1318 119864119877= 5275meV and 119886




= 112119864119877 when 119877 = 15119886

119861and 119867 = 10119886

119861(119899120588= 0

119898 = 0) and Δ11986410= 34119864

119877 when 119877 = 15119886

119861and 119867 = 10119886

119861



(119899120588= 0 119898 = 0) = 143 sdot 1012119888minus1





10= 058119864

119877

when 119877 = 2119886119861and 119867 = 15119886

119861(119899120588= 0 119898 = 0) and

Δ11986410= 176119864

119877 when119877 = 2119886

119861and119867 = 15119886

119861(119899120588= 1119898 = 0)

EE

R

60

50

40

30

20

10

010 12 14 16 18 20

HaB

n120588 = 0

n120588 = 1

n = 0n = 1

n = 2

R = 1aBm = 0


300

295

290

285

280

275

27006 08 10 12 14

H = 5aBH = 10aB

H = 15aB

ℏΩ

000E

R

RaB



000= 507 sdot 10

14119888minus1 for 119877 = 02119886

119861




310

300

290

280

270

HaB

10 12 14 16 18 20

R = 05aBR = 1aB

R = 15aB

ℏΩ

000E

R





120588transitions




forall1198991015840

5 Conclusion




References































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




R

RaB

n = 2n = 1

n = 0

n120588 = 0

n120588 = 1 H = 10aBm = 0

180

150

120

90

60

30

006 08 10 12 14




119890= 0067119898

119890 120581 =

1318 119864119877= 5275meV and 119886




= 112119864119877 when 119877 = 15119886

119861and 119867 = 10119886

119861(119899120588= 0

119898 = 0) and Δ11986410= 34119864

119877 when 119877 = 15119886

119861and 119867 = 10119886

119861



(119899120588= 0 119898 = 0) = 143 sdot 1012119888minus1





10= 058119864

119877

when 119877 = 2119886119861and 119867 = 15119886

119861(119899120588= 0 119898 = 0) and

Δ11986410= 176119864

119877 when119877 = 2119886

119861and119867 = 15119886

119861(119899120588= 1119898 = 0)

EE

R

60

50

40

30

20

10

010 12 14 16 18 20

HaB

n120588 = 0

n120588 = 1

n = 0n = 1

n = 2

R = 1aBm = 0


300

295

290

285

280

275

27006 08 10 12 14

H = 5aBH = 10aB

H = 15aB

ℏΩ

000E

R

RaB



000= 507 sdot 10

14119888minus1 for 119877 = 02119886

119861




310

300

290

280

270

HaB

10 12 14 16 18 20

R = 05aBR = 1aB

R = 15aB

ℏΩ

000E

R





120588transitions




forall1198991015840

5 Conclusion




References































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




310

300

290

280

270

HaB

10 12 14 16 18 20

R = 05aBR = 1aB

R = 15aB

ℏΩ

000E

R





120588transitions




forall1198991015840

5 Conclusion




References































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials















CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



direct interband light absorption in conical quantum dot

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