recent interest to semiconductor quantum dots ( qds ) is conditioned by new physical properties of these zero - dimensional objects , which are conditioned by size - quantization ( sq ) effect of the charge carriers ( ccs ) [ 1 - 3 ] .
such new structures were obtained by means of interrupted growth of qds within semiconductor media .
development of new growth technologies , in particular such as stranski - krastanov epitaxial method , made possible development of qds having various shapes and dimensions .
it is known that energy spectrum of ccs of a qd is fully quantized and similar to the energy spectra of atoms , in view of that the qds are often called  artificial atoms  .
most of the research in this area is focused on studies of the spherical qds .
however during last years the ellipsoidal , pyramidal , cylindrical and lens - shaped qds are also undergoing theoretical and experimental investigations [ 5 - 10 ] . due to presence of strong sq effect in the mentioned objects the physical characteristics of ccs in such systems strongly depend on the geometrical shapes of qds .
even slight variation of the shapes significantly affects the cc spectrum . in other words ,
qd geometric shapes and dimensions may serve as useful tools for cc energy spectrum and other characteristic parameters variation inside a qd for various practical applications in systems comprised of qd ensembles . monitored 
shaping during the process of growth makes possible simulation and development of samples with desired physical parameters .
another powerful factor affecting the cc energy spectrum shaping inside qd is the confinement potential of the qd 
the semiconductor nanostructures based on gaas / ga1xalxas - type systems are objects of intensive recent investigations due to wide band gap and availability of well elaborated growth techniques of various systems incorporating such materials . as a result of natural diffusion process during the growth of qds ,
the correspondently forming confinement potential is such that can be easily approximated in most cases by a parabolic potential .
also note , that for this approximation the kohn theorem is well generalized , this proves that such approximation is correct , the experimental verification is provided in ref . .
however , the effective parabolic potential may origin due to peculiarity of the qds shape .
, the rotational ellipsoids , or spheroids , in contrary to spheres , are known to be described by two parameters ( short and large half - axes instead of radius ) .
in addition to that the external electric and magnetic fields causing quantization are alternative tools of control of the energy spectrum of qds cc .
the strong external fields , at certain values of their intensities , may have the same , or even stronger sq effect on the energy spectrum than the quantum dot s shape variation .
note , that the magnetic field affects the cc motion only in transversal direction , in difference to the electrical field .
therefore two fields directed in parallel open possibility for a broad manipulation of the cc characteristics inside semiconductor sq systems . in particular in paper   the quantum effect of the magnetic field inside the of the strongly prolate qd is investigated .
the effect of electrical field on the cc energy spectrum inside the mentioned system has been considered in paper .
however , the combined effect of unidirectional electric and magnetic fields is not considered yet .
analysis of the optical absorption spectra of various semiconductor structures represents a powerful tool for obtaining numerous characteristics of these structures , namely : forbidden gap widths , effective masses of electrons and holes , their mobility , dielectric features , etc .
many papers study these spectra by experiments and analysis , both in massive and sq semiconductor structures ( see e.g. [ 16 - 18 ] ) .
indeed , presence of new sq energy levels makes possible to realize new inter - band transitions widening the scope of applications of devices based on such systems .
meanwhile existence of the external quantizing fields often results in restructuring of the energy levels , as well as creation of new selection rules during the process of the light absorption
. therefore electronic states and direct inter - band light absorption are considered below for strongly oblate ellipsoidal quantum dots ( soeqd ) and strongly prolate ellipsoidal quantum dots ( speqd ) at presence of unidirectional electric and magnetic fields ; the problem is considered for strong sq regime .
let us to consider an impenetrable soeqd located in unidirectional electric and magnetic fields ( see fig .
the potential energy of a charged particle ( electron , or hole ) in such structure has the following form:(1 ) where a1 and c1 are the short and long semiaxes of speqd , respectively .
( b ) strongly prolate ellipsoidal quantum dot as is known , in the strong sq regime , the energy of coulomb interaction between electron and hole can be considered much smaller than the energy created by the soeqd walls . in the framework of such approximation
thus , the problem is reduced to analytical determination of the energy separate expressions for electron and hole ( as for non - interacting particles ) .
the quantum dot shape indicates that particle motion along the z - axis takes place faster than in the normal direction , this also allows to utilize adiabatic approximation .
the system hamiltonian under these conditions has the following form:(2 ) in which   is the particle momentum operator ,   is the vector potential of the magnetic field ,   is the electrical field intensity ,   is the radius - vector , s is the light velocity in vacuum , and e is the magnitude of electron charge . assuming the calibration of vector potential in cylindrical coordinates to have a form
, one can express the system hamiltonian as(3 ) which may be represented as a sum of two hamiltonians of  fasth1 and  slowh2 subsystems in dimensionless variables:(4 ) with assumed notations :   where  is the effective mass of electron ,   is effective rydberg energy ,   is the effective bohr radius of electron , and   is the dielectric permeability .
the homogenous electrical and magnetic fields are given by relations   and   here we assume the wave function ( wf ) to have the following form:(7 ) following the above - mentioned adiabatic approximation , when the coordinaterof the  slow  subsystem is fixed , the particle motion is localized in one - dimensional potential well having effective width , wherea = a1/abandc = c1/ab . the schrdinger equation for
the  fast  subsystem will acquire the following form:(8 ) after a number of simple transformations and numerical simulation , for the close - to - bottom energy levels of the spectrum ( when the particle is predominantly localized in the regionr  a ) one can obtain the following power series expression with high degree of accuracy:(9 ) where n and n parameters depend on the value of the electrical field . the relation ( 9 )
represents an effective potential which is incorporated in the  slow  system schrdinger equation(10 ) thus by solving the eq .
10 ) , we shall obtain the ultimate energy relation for the charged particle ( electron , hole)(11 ) where n = 2nr + |m|,nr and m are the radial and magnetic quantum numbers , respectively .
now let us to consider an impenetrable speqd located in the unidirectional electric and magnetic fields ( see fig .
for this case the potential energy of the charged particle ( electron , or hole ) will have the form ( 1 ) under the conditiona1c1 , wherea1andc1are the short and long semiaxes of speqd , respectively .
as in the soeqd case , in the strong sq regime we neglect the coulomb interaction between the electron and hole .
1b makes possible the particle motion in the radial plane to be faster than along the z - axis , which also allows to use adiabatic approximation just like in the soeqd case . here
5and6 , for the speqd case we have(12)(13 ) the homogenous electrical and magnetic fields are represented as   and   just as is in the previous case , while the vector potential is . for this case
we assume the wf to have a structure(14 ) following the above - mentioned adiabatic approximation , when thezcoordinate of the  slow  subsystem is fixed , the particle motion is localized in two - dimensional potential well having effective width   the schrdinger equation for the  fast  subsystem has the form(15 ) after number of transformations we shall obtain the following energy expression 1(z ) for the  fast  sub - system(16 ) while n1 and n2 are some numbers depending on the magnetic field intensity . the expression ( 16 )
18gives the final energy expression for the charged particle ( electron , hole)(19 ) another notable issue of the problem is the direct inter - band absorption of the light by soeqd in the strong sq regime . in other words ,
the relationc  { ab , ab } holds , whereabis the effective bohr radius of electron ( or hole ) .
hole , when e  h , eand hare the effective masses of electron and hole , respectively .
the absorption coefficient is given by the expression ( 20 ) where v and v are sets of quantum numbers ( qn ) corresponding to the electron and heavy hole , eg is the forbidden gap width in the bulk semiconductor ,  is the incident light frequency and a is a quantity proportional to the square of matrix element in decomposition over bloch functions .
numerical calculations were made for the qd fromgaas , with parameters : e = 0.067me , e = 0,12h , er = 5.275 mev , ab = 104  , eg = 1.43 ev .
finally , for the quantitykand for the absorption edge ( ae ) we have obtained:(21)(22 ) where , while   and   are certain integral quantities .
let us now to consider the selection rules in this case . in the absence of the fields ,
the transitions between the levels with qnn = n,n = n andm = m are allowed .
when the external fields are present the transitions are possible in all cases of magnetic quantization in the radial direction , between the levels with magnetic qnsm = m. under weak magnetic quantization , whenah  a1 , where   is magnetic length , transitions are allowed between the levels with radial qnsnr = nr , and thus between the oscillatory qnsn = n. in the case whenaha1 , the selection rules among the oscillatory qns are cancelled , which is a result of competition between size and magnetic quantization . a magnetic field of extremely high - intensity , for whichah  a1 , restores the selection rules between the oscillatory qns .
it should be also noted that the selection rules in the direction of electrical field are becoming obsolete .
consider now a direct inter - band light absorption inside speqds in the strong sq regime .
in this particular case for thekand ae we obtained the following expressions(23)(24 ) respectively ; where , while , where   and   are certain integral quantities .
at absence of the fields at the speqd case , the transitions between the qnsn = n,nr = nr , andm = m are allowed .
the application of the external fields to the speqd the following alternations from the previous case can be indicated .
when the magnetic quantization is either weak or moderate , i.e. whenah  a1oraha1(herea1is the short half - axis ) , the selection rules in the radial direction are becoming completely obsolete . as a result of this ,
meanwhile , transitions between the magnetic qns remain unchanged : m = m. note that in the hypothetical case of extremely strong magnetic field ( whenah  a1 ) the selection rules for transitions between the oscillatory qns are completely restoring .
let us to consider an impenetrable soeqd located in unidirectional electric and magnetic fields ( see fig .
the potential energy of a charged particle ( electron , or hole ) in such structure has the following form:(1 ) where a1 and c1 are the short and long semiaxes of speqd , respectively .
( b ) strongly prolate ellipsoidal quantum dot as is known , in the strong sq regime , the energy of coulomb interaction between electron and hole can be considered much smaller than the energy created by the soeqd walls . in the framework of such approximation
thus , the problem is reduced to analytical determination of the energy separate expressions for electron and hole ( as for non - interacting particles ) .
the quantum dot shape indicates that particle motion along the z - axis takes place faster than in the normal direction , this also allows to utilize adiabatic approximation .
the system hamiltonian under these conditions has the following form:(2 ) in which   is the particle momentum operator ,   is the vector potential of the magnetic field ,   is the electrical field intensity ,   is the radius - vector , s is the light velocity in vacuum , and e is the magnitude of electron charge . assuming the calibration of vector potential in cylindrical coordinates to have a form
, one can express the system hamiltonian as(3 ) which may be represented as a sum of two hamiltonians of  fasth1 and  slowh2 subsystems in dimensionless variables:(4 ) with assumed notations :   where  is the effective mass of electron ,   is effective rydberg energy ,   is the effective bohr radius of electron , and   is the dielectric permeability .
the homogenous electrical and magnetic fields are given by relations   and   here we assume the wave function ( wf ) to have the following form:(7 ) following the above - mentioned adiabatic approximation , when the coordinaterof the  slow  subsystem is fixed , the particle motion is localized in one - dimensional potential well having effective width , wherea = a1/abandc = c1/ab . the schrdinger equation for
the  fast  subsystem will acquire the following form:(8 ) after a number of simple transformations and numerical simulation , for the close - to - bottom energy levels of the spectrum ( when the particle is predominantly localized in the regionr  a ) one can obtain the following power series expression with high degree of accuracy:(9 ) where n and n parameters depend on the value of the electrical field . the relation ( 9 )
represents an effective potential which is incorporated in the  slow  system schrdinger equation(10 ) thus by solving the eq .
10 ) , we shall obtain the ultimate energy relation for the charged particle ( electron , hole)(11 ) where n = 2nr + |m|,nr and m are the radial and magnetic quantum numbers , respectively .
now let us to consider an impenetrable speqd located in the unidirectional electric and magnetic fields ( see fig .
for this case the potential energy of the charged particle ( electron , or hole ) will have the form ( 1 ) under the conditiona1c1 , wherea1andc1are the short and long semiaxes of speqd , respectively .
as in the soeqd case , in the strong sq regime we neglect the coulomb interaction between the electron and hole .
1b makes possible the particle motion in the radial plane to be faster than along the z - axis , which also allows to use adiabatic approximation just like in the soeqd case . here
5and6 , for the speqd case we have(12)(13 ) the homogenous electrical and magnetic fields are represented as   and   just as is in the previous case , while the vector potential is . for this case
we assume the wf to have a structure(14 ) following the above - mentioned adiabatic approximation , when thezcoordinate of the  slow  subsystem is fixed , the particle motion is localized in two - dimensional potential well having effective width   the schrdinger equation for the  fast  subsystem has the form(15 ) after number of transformations we shall obtain the following energy expression 1(z ) for the  fast  sub - system(16 ) while n1 and n2 are some numbers depending on the magnetic field intensity .
the expression ( 16 ) is an effective potential being incorporated in the schrdinger equation for the  slow 
another notable issue of the problem is the direct inter - band absorption of the light by soeqd in the strong sq regime . in other words ,
the relationc  { ab , ab } holds , whereabis the effective bohr radius of electron ( or hole ) .
we consider the case of a  heavy  hole , when e  h , eand hare the effective masses of electron and hole , respectively .
the absorption coefficient is given by the expression ( 20 ) where v and v are sets of quantum numbers ( qn ) corresponding to the electron and heavy hole , eg is the forbidden gap width in the bulk semiconductor ,  is the incident light frequency and a is a quantity proportional to the square of matrix element in decomposition over bloch functions .
numerical calculations were made for the qd fromgaas , with parameters : e = 0.067me , e = 0,12h , er = 5.275 mev , ab = 104  , eg = 1.43 ev .
finally , for the quantitykand for the absorption edge ( ae ) we have obtained:(21)(22 ) where , while   and   are certain integral quantities .
let us now to consider the selection rules in this case . in the absence of the fields ,
the transitions between the levels with qnn = n,n = n andm = m are allowed .
when the external fields are present the transitions are possible in all cases of magnetic quantization in the radial direction , between the levels with magnetic qnsm = m. under weak magnetic quantization , whenah  a1 , where   is magnetic length , transitions are allowed between the levels with radial qnsnr = nr , and thus between the oscillatory qnsn = n. in the case whenaha1 , the selection rules among the oscillatory qns are cancelled , which is a result of competition between size and magnetic quantization . a magnetic field of extremely high - intensity , for whichah  a1 , restores the selection rules between the oscillatory qns .
it should be also noted that the selection rules in the direction of electrical field are becoming obsolete .
consider now a direct inter - band light absorption inside speqds in the strong sq regime .
in this particular case for thekand ae we obtained the following expressions(23)(24 ) respectively ; where , while , where   and   are certain integral quantities . at absence of the fields at the speqd case , the transitions between the qnsn = n,nr = nr , andm = m are allowed .
the application of the external fields to the speqd the following alternations from the previous case can be indicated .
when the magnetic quantization is either weak or moderate , i.e. whenah  a1oraha1(herea1is the short half - axis ) , the selection rules in the radial direction are becoming completely obsolete . as a result of this ,
meanwhile , transitions between the magnetic qns remain unchanged : m = m. note that in the hypothetical case of extremely strong magnetic field ( whenah  a1 ) the selection rules for transitions between the oscillatory qns are completely restoring .
as one can infer from the obtained results , the cc energy spectrum both of soeqds and speqds is equidistant .
more correctly , each level of the  fast  subsystem has a family of equidistant  slow  subsystem levels positioned thereupon .
the obtained result is valid only for the levels close to the well bottom ( or having low qn values ) , due to the assumed adiabatic approximation .
however , the transition frequency between the equidistant levels when the fields are present is usually higher .
for example , in the absence of fields , when half - axes of soeqd are a1 = 2.5abc1 = 0.5ab , the transition frequency for the first equidistant family is obtained equal to    which falls into the ir part of spectrum . with the same half - axes , but in the presence of fields h = 10 t , f = 500 v / cm the transition frequency is almost one - and - half times higher : .
the only exception is speqd with h = 0 , f  0 . in that case
all equidistant levels are shifted for the same value depending on the intensity of the electrical field .
the inter - level distance remains the same as in the absence of electrical field ( shifted oscillator ) .
figures 2 and 3 illustrate the dependence of cc energy spectrum families inside soeqd on intensities respectively of electrical and magnetic fields , under fixed lengths of half - axes .
2 that the first family of the cc energy spectrum falls with growth of the electric field intensity , while the second family grows and then reduces .
this behavior was called  anomalous stark effect  by the authors of ref . .
they explained such field dependence of energy levels by behavior of the particle probability density || .
however , no such anomaly is revealed under correct definition of the stark effect . in other words ,
the dependence of the inter - level distance on the electrical field intensity has monotonously growing character .
dependence of the first two equidistant cc energy spectrum families inside soeqd on the electrical field intensity , under fixed values of magnetic field intensity and half - axes lengths : h = 100 t , c1 = 0.5ab , a1 = 2.5ab dependence of the first equidistant family of cc energy spectrum inside soeqd on the magnetic field intensity , under fixed values of electrical field intensity and half - axes lengths : f = 1000 v / cm , c1 = 0.5ab , a1 = 2.5ab as one can see from the fig .
3 , all levels of the cc energy spectrum family are growing when the magnetic field intensity is increased .
this is conditioned by growth of the magnetic quantization contribution into the cc energy increase .
inter level distance is increased with magnetic field intensity while the levels are remaining equidistant .
figures 4 and 5 show the dependence of ae inside soeqd on the intensities of electrical and magnetic fields , respectively , when the lengths of soeqd half - axis are fixed .
4 that ae value is shifted towards longer wavelengths when the electrical field intensity is growing . this fact is explained by a change of the quantum - well bottom caused by the external electrical field . as a result of this phenomenon ,
the energy of electron is reduced , while the energy of hole increases , which in turn reduces the forbidden gap width .
presence of the magnetic field increases the energy of the both particles , independently of the charge sign . otherwise , the ae shift is due to magnetic quantization effect ( blue shift ) .
absorption edge dependence on the electrical field intensity , at different values of the magnetic field inside soeqd having fixed half - axis lengths : , c1 = 0.5ab and a1 = 2.5ab absorption edge dependence on the magnetic field intensity , at different values of the electrical field inside soeqd having fixed half - axis lengths : , c1 = 0.5ab and a1 = 2.5ab figure 6 illustrates the three - dimensional view of the ground state energy for a cc inside the speqd as a function of electrical and magnetic field intensities under fixed lengths of half - axis .
one can see that the effects of the electrical and magnetic fields are different : increase of the magnetic field intensity increases the particle energy , while increase of the electrical field intensity reduces the particle energy .
the first phenomenon is explained by a growth of the magnetic quantization contribution into cc energy increase , while the second by a change of potential well bottom , or competition of sq and electrical fields effect on cc .
ground state energy dependence of cc inside speqd as a function of electrical and magnetic field intensities , under fixed values of half - axis : a1 = 0.3ab and c1 = 2ab similar dependence of ae on the electrical and magnetic fields in soeqd is illustrated in fig .
the behavior of this dependence can be explained similarly as in the case of speqd . absorption edge dependence on the electrical and magnetic field intensities inside speqd , under fixed values of half - axis : a1 = 0.3ab and c1 = 2ab so far we have studied the absorption of a system consisting of semiconductor qds having identical dimensions . for comparison of the obtained results with experimental data ,
one has to take into account the random character of speqd and soeqd dimensions ( or half - axis ) obtained during the qd technological growth process .
so , for making the comparison the absorption coefficient should be multiplied by concentration of qds .
then , instead of the distinct absorption lines we will obtain a series of fuzzy maximums as a result of the size dispersion by semiaxis .
for illustration of the size dispersion by the semiaxis we used two experimentally observable models . in the first model we used the lifshits - slezov distribution function : ( 25 ) where   is some average value of the semiaxis . in the second model the gaussian distribution function
is used ( see e.g. ): 
 ( 26 ) figure 8 illustrates the dependence of the absorption coefficient k on the frequency of incident light , for the ensemble of soeqds in the absence of external fields .
note that in the model of gaussian dots distribution , a single distinctly explicit maximum of absorption is observed .
when the light frequency is increasing , the second slightly notable maximum is traced . further increase of the incident light frequency results in decrease of the absorption coefficient . when the lifshits - slezov model is realized during a device growth , a number of distinctly explicit absorption maximums as a function of incident light frequency is observed .
note that weakly expressed oscillations on the first peak are seen in the second case , which are due to inter - band transitions of the first equidistant family ( see additional graphical insertion in fig .
. absorption coefficient dependence on the incident light frequency , for the ensemble of soeqds having two different distribution functions
ellipsoidal qds , especially soeqds or speqds , have various commercial applications , in particular in large two - dimensional focal plane arrays in the mid- and far infrared ( m&fir ) region , having important applications in the fields of pollution detection , thermal imaging object location and remote sensing as well as ir imaging of astronomical objects ( see e.g. us patent # 6541788 ) .
this research has been performed in the framework of the armenian state research program  semiconductor nanoelectronics  .
the authors express their gratitude to the rector of russian - armenian university , prof .
 in framework of the adiabatic approximation the energy states of electron as well as direct light absorption are investigated in strongly oblate and strongly prolate ellipsoidal quantum dots ( qds ) at presence of electric and magnetic fields . 
 analytical expressions for particle energy spectrum are obtained . 
 the dependence of energy levels configuration on qd geometrical parameters and field intensities is analytically obtained . 
 the energy levels of electrons are shown to be equidistant both for strongly oblate and prolate qds . 
 the effect of the external fields on direct light absorption of a qd was investigated . 
 the dependence of the absorption edge on geometrical parameters of qds and intensities of the electric and magnetic fields is obtained . 
 selection rules are obtained at presence as well as absence of external electric and magnetic fields . in particular 
 , it is shown that the presence of the electric field cancels the quantum numbers selection rules at the field direction , whereas in radial direction the selection rules are preserved . 
 perspectives of practical applications for device manufacturing based on ellipsoidal quantum dots are outlined . 
Introduction
Theory
SOEQD Case: Electronic States Inside the Strongly Oblate Ellipsoidal Quantum Dot in the Presence of Unidirectional Electric and Magnetic Fields
SPEQD Case: Electronic States Inside a Strongly Prolate Ellipsoidal Quantum Dot in the Presence of Unidirectional Electric and Magnetic Fields
Direct Inter-Band Light Absorption
Discussion of Results
Outline of Possible Practical Applications
Acknowledgements