due to three - dimensional confinement for electrons in the quantum - dot structure , quantum - dot infrared photodetectors ( qdips ) have attracted much attention for theoretical and experimental studies in recent years [ 1 - 3 ] . one important characteristic for qdips is the sensitivity to normal - incidence infrared radiation which is advantage to focal plane arrays . phonon interaction makes the qdips have another advantages of displaying low dark current , large detectivity , and better response . the introduction of strain may provide a facile way to fabricate various wavelength from mid - wavelength to long - wavelength multicolor infrared ( ir ) detectors via inas or ingaas quantum dot ( qd ) capped by gaas , ingaas , inp , or gainp . meanwhile , the geometry shape of qds always results in quite different responding wavelength for qdips . nowadays , more complicated nanostructures , such as qd molecules are investigated for the potential use of photoelectric devices . it is well known that the much sensitivity of qd s bound energy levels to the shape , size , and strain provides the detector greater potential to obtain the ideal responding wavelength for the application of medical or molecular application . so the study of the shape , size , and strain of qd system has been an interesting subject for the development and precious controlling of the qdips structure . much theoretical and experimental work has been done to explore the effect of the shape , size , or strain of qds on the bound energy levels or the possible optical transition . the bound energy levels in fat lenslike qd basing on the quantum - well approximate theoretical results have a bigger difference by comparing to the experimental results . in wojs work , the energy levels of lenslike in0.5ga0.5as / gaas qd were studied as a function of the dot s size , and found that the parabolic confining potential and its corresponding energy spectrum were shown to be and excellent approximation . here , we calculate the strain energy of self - assembled qds with the short - range valence - force - field ( vff ) approach to describe inter - atomic forces by using bond stretching and bending . the role of strain ( for three different shapes ) in determining the bound levels is analyzed in detail . considering three different shape qds with the same ratio of the base to the vertical aspect 3:1 , the bound energy levels are calculated by the recursion method . the theoretical results show that the difference of bound energy levels of lenslike inas qd matches with the experimental results . while the bound energy levels of pyramid - shaped qd have the biggest difference from the measured experimental data . though the bound - to - continuum transition of the truncated pyramid qd is mostly acceptable because the behavior is much similar to the structure of the well - studied quantum - well infrared photodetectors ( qwips ) , the bound ground states of electrons and holes are very far from the experimental results . sample preparations and experimental results ,  the investigated experimental device and experimental results such as afm / tem images , the photoluminescence ( pl ) , and photocurrent ( pc ) spectrums are described . in the section theoretical results and discussions ,  the exact strain distributions of pyramid , truncated pyramid , and lenslike - shaped inas / gaas qd are calculated by the short - range vff approach , and the energy levels of the bound states are calculated by the recursion method . the sample was grown on semi - insulating gaas ( 001 ) substrates by using the solid - source molecular beam epitaxy ( mbe ) . five layers of nominally 3.0 momolayer ( ml ) inas ( quantum dots ) were inserted between highly si - doped bottom and top gaas 1000 nm contact layers with doping density 1  10 cm . each layer of inas is capped by 21 ml spacer gaas material to form the inas qds , and the five layers of gaas / inas are called s - qd . in addition there is a 50 nm gaas layer inserted between the s - qd regions and bottom ( top ) si - doped gaas contact layers , respectively . typical qdip structure of gaas / inas material the typical constant - mode ambient atomic force microscopy ( afm ) data and the cross - sectional tem for the counterpart samples are present in fig . the average height of quantum dots is about 74  16  , and the quantum dot density has a range from 613/umto 733/um . the average quantum dot width with the range from 228 to 278  represents the full width at half maximum ( fwhm ) of afm scan profile . figure 2b shows that the cross - sectional transmission electron microscope images on the s - qd counterpart . it is noted that the quantum dot density in the lower layer is higher than that in the upper layer . atypical afm - determined island size distribution.bcross-sectional tem images of qdip structure the near - infrared photoluminescence ( pl ) as a function of energy at 77 k is shown in fig . a main peak corresponding to the quantum dot ground state transitions is centered at 1.058 ev and a small broad shoulder due to smaller quantum dots or inas wetting layer appears at 1.216 ev . figure 4shows the intra - band photocurrent as a function of energy at 77 k in the absence of bias . it is well known that the intra - band photocurrent can present more direct information on the quantum dot electronic states . the near infrared photoluminescence ( pl ) spectra at 77 k the intra - band photocurrent ( pc ) at 77 k and 0 bias here , we adopt the short - range vff approach to describe interatomic forces in terms of band stretching and bending . the model has been widely applied in bulk and alloys [ 10 - 14 ] , as well as low - dimensional systems . it was further developed to an enharmonic vff model by bernard and zunger for si  ge compounds , alloys and superlattices . in the vff model , the deformation of a lattice structure is completely specified when the location of every atom in a strained state is given , and the elastic energy of a bond is minimal in its three - dimensional bulk lattice structure . for small deformations , the bond energy can be written as a taylor expansion in the variations of the bond length and the angles between the bond and its nearest neighbor bonds . under the rubric of short - range contributions and by following the general notations in ref . [ 9 - 14 ] , the elastic energy of an interatomic bond ( by setting the elastic energy at equilibrium as the zero reference ) can be written in the harmonic form , 
 ( 1 ) with j = 1,2 ... 6 denoting the six nearest neighbor bonds in zinc blende structure . ri is the variation of the length of bond i , and ij is the variation of the angle between the ith and the jth bonds . the total elastic energy is the sum of all bond energies   the numerical values of k s for vff bonds of zinc blende bulk materials are easily obtained from elastic coefficients c11c12 , and c44 listed in ref . . note that k s values depend slightly on the temperature of the material due to the temperature dependence of the lattice constant . the values listed in table 1 are obtained for the materials at 100 k. values of c s   and k s ( at 100 k ) of zincblende inas and gaas bulk materials the local band edges employing the following formulas for the conduction ( cb ) , the heavy hole ( hh ) , and the light hole ( lh ) bands can be approximated as:(2 ) where the hydrostatic strain h and the biaxial strain b are defined as(3 ) vhh and vlh are the heavy - hole and light - hole bands , acav and b are the deformation potentials , and ecb / vb are the unstrained band edge energies . notice that the sheer - strain - induced hh  lh coupling and split - off contributions are ignored . in our calculation the investigated three different shaped inas qds follows pyramid - shaped ( with the four facets being ( 111 ) ,   and   inas qd with the height and base being 81 and 162  , respectively , truncated pyramid - shaped ( with the four facets being same with these of pyramid - shaped qd ) inas qd with the height and base being 80 and 168  , respectively , and lense - shaped inas qd with the height and diameter being 80 and 240  , respectively . the distribution and value of strain based on inas and gaas technology and the stranski - krastanov self - assembly technique , the qd can have different shape and symmetry , which therefore has effect on the strain distribution in qd and the corresponding band offset . the second one is the degree of anisotropy of elastic property , which is described by elastic constants . in our calculation , the qdip system is made up of inas / gaas . so , the second term has the same impact on the strain distribution . in the following , we will have a look at the relationship of the strain distribution and the shape of qds . 5 . the hydrostatic strain hmakes the height of cb lower , but the height changes are almost same for different shape of qds . while the biaxial strain bis rather complex for the three different shapes . for the lens - shaped qd , the inas lattice is compressed by gaas in the growth plane and stretched in the plane , which is vertical to the growth plane . the lattice of the top of qd is stretched in the growth plane , and in the central b = 0 , which means that there is no splitting for the valence band in the position . the compressing and stretching condition of bottom in the pyramid qd is the same as the lens - shaped qd . the difference of qd shape makes the strain distribution rather different , and the calculated strain distribution of truncated - pyramid qd resembles to that of the pyramid - shaped qd . the existence of gaas coating makes the strain distribution at the top of the three different shapes of qd to present different characters as : h < 0 and b < 0 for lenslike and truncated - pyramid qd ; and h < 0 and b > 0 for pyramid - shaped qd . figure 6shows the calculated confinement potential distribution induced with the strain for three different shaped qd . the difference of electron energy levels mainly comes from the difference of shape because there is little change in the value of hfor different shaped qds . the splitting of hole potential is determined by biaxial strain b , which changes a lot when the shape varies . the calculated strain distribution of the pyramid and lens - shaped qd at x - z plane , wherea+care hydrostatic strain from 0.075 ( blue ) to 0.07 ( red ) andb+c ) are biaxial strain from 0.13 ( blue ) to 0.12 ( red ) the calculated bandoffset ( doted + line ) and energy level ( dotted line ) of the three different shaped qd when strain is included the recursion method is used to calculate the energy levels of three different shaped qds . for the experimental data , the broadening ( fwhm ) of the pc spectrum and the first peak of the pl spectrum are 34.01 and 29.34 mev , respectively . the much greater difference in the fwhm implies the possibility that the second peak comes from wetting layer , but not from the bound levels of qds . so for the simplicity of comparing with the experimental results , we only calculate all the energy levels of electron and the bound ground energy level of hole with the corresponding results shown in fig . 6 ( dotted line ) . next , we present the change in energy level for three different shape qds . from fig . the energy difference between the ground states of hole and electron is 0.990 ev for pyramid qd , 1.215 ev for truncated pyramid qd , and 1.053 ev for lens - shaped qd , respectively . the possible bound - to - bound transition of electronic inter - subbands is 254.1 mev for pyramid qd and 156.4 mev for lens - shaped qd . there is only one bound state in conduction band , so the possible transition of different energy levels is bound - to - continuum state with the energy being 199.3 mev . from the experimental data , the pl spectrum presents the ground state transition from electron to hole with the value being 1.058 ev . compared with the experimental data , the corresponding calculated value has the difference ( dpl ) of 6.43% for pyramid qd , 14.8% for truncated - pyramid qd , and 0.47% for lens - shaped qd . also the differences ( dpc ) from the measured pc peak ( 170.675 mev ) are 48.88% , 16.77% , and 8.36% for pyramid , truncated - pyramid , and lens - shaped qd , respectively . the compared results show that the energy difference of the lens - shaped qd is the most favored for the possible qd structure . though the transition of the bound - to - continuum transition for truncated pyramid qd is mostly acceptable because the behavior is much like the well - studied qwip structure , the energy difference of the bound ground states between electron and hole is rather far from the experimental results . the pyramid is not possible to be the shape of our investigated qd for the biggest difference of 48.88% . if we define a parameter  to estimate whether the shape or size is the most favored to qd of investigated qdip structure , the most suitable expression should be with dpl and dpc being the difference of calculated and measured pl and pc spectrum , respectively . in our calculation ,  is 34.86% , 15.82% , and 5.92% for pyramid , truncated - pyramid , and lens - shaped qd , respectively . for the lens - shaped qd ,  has the least value , which means the qdip structure is constructed by the lens - shaped qd . the lens - shaped inas / gaas qd is observed by many researchers , and in this way our calculation can get a good agreement with the measured data . also the pl , pc spectrum , and  provide us a way to find out the most suitable shape and size of qd which makes  the minimum . the different shape of qds can have different response wavelength as described in our calculation . the results mean that one can obtain the ideal response wavelength of qdip structure by controlling the growth condition to change the shape of qds . here , we adopt the short - range vff approach to describe interatomic forces in terms of band stretching and bending . the model has been widely applied in bulk and alloys [ 10 - 14 ] , as well as low - dimensional systems . it was further developed to an enharmonic vff model by bernard and zunger for si  ge compounds , alloys and superlattices . in the vff model , the deformation of a lattice structure is completely specified when the location of every atom in a strained state is given , and the elastic energy of a bond is minimal in its three - dimensional bulk lattice structure . for small deformations , the bond energy can be written as a taylor expansion in the variations of the bond length and the angles between the bond and its nearest neighbor bonds . under the rubric of short - range contributions and by following the general notations in ref . [ 9 - 14 ] , the elastic energy of an interatomic bond ( by setting the elastic energy at equilibrium as the zero reference ) can be written in the harmonic form , 
 ( 1 ) with j = 1,2 ... 6 denoting the six nearest neighbor bonds in zinc blende structure . ri is the variation of the length of bond i , and ij is the variation of the angle between the ith and the jth bonds . the total elastic energy is the sum of all bond energies   the numerical values of k s for vff bonds of zinc blende bulk materials are easily obtained from elastic coefficients c11c12 , and c44 listed in ref . . note that k s values depend slightly on the temperature of the material due to the temperature dependence of the lattice constant . the values listed in table 1 are obtained for the materials at 100 k. values of c s   and k s ( at 100 k ) of zincblende inas and gaas bulk materials the local band edges employing the following formulas for the conduction ( cb ) , the heavy hole ( hh ) , and the light hole ( lh ) bands can be approximated as:(2 ) where the hydrostatic strain h and the biaxial strain b are defined as(3 ) vhh and vlh are the heavy - hole and light - hole bands , acav and b are the deformation potentials , and ecb / vb are the unstrained band edge energies . notice that the sheer - strain - induced hh  lh coupling and split - off contributions are ignored . in our calculation the investigated three different shaped inas qds follows pyramid - shaped ( with the four facets being ( 111 ) ,   and   inas qd with the height and base being 81 and 162  , respectively , truncated pyramid - shaped ( with the four facets being same with these of pyramid - shaped qd ) inas qd with the height and base being 80 and 168  , respectively , and lense - shaped inas qd with the height and diameter being 80 and 240  , respectively . the distribution and value of strain based on inas and gaas technology and the stranski - krastanov self - assembly technique , the qd can have different shape and symmetry , which therefore has effect on the strain distribution in qd and the corresponding band offset . the second one is the degree of anisotropy of elastic property , which is described by elastic constants . in our calculation , the qdip system is made up of inas / gaas . so , the second term has the same impact on the strain distribution . in the following , we will have a look at the relationship of the strain distribution and the shape of qds . 5 . the hydrostatic strain hmakes the height of cb lower , but the height changes are almost same for different shape of qds . while the biaxial strain bis rather complex for the three different shapes . for the lens - shaped qd , the inas lattice is compressed by gaas in the growth plane and stretched in the plane , which is vertical to the growth plane . the lattice of the top of qd is stretched in the growth plane , and in the central b = 0 , which means that there is no splitting for the valence band in the position . the compressing and stretching condition of bottom in the pyramid qd is the same as the lens - shaped qd . the difference of qd shape makes the strain distribution rather different , and the calculated strain distribution of truncated - pyramid qd resembles to that of the pyramid - shaped qd . the existence of gaas coating makes the strain distribution at the top of the three different shapes of qd to present different characters as : h < 0 and b < 0 for lenslike and truncated - pyramid qd ; and h < 0 and b > 0 for pyramid - shaped qd . figure 6shows the calculated confinement potential distribution induced with the strain for three different shaped qd . the difference of electron energy levels mainly comes from the difference of shape because there is little change in the value of hfor different shaped qds . the splitting of hole potential is determined by biaxial strain b , which changes a lot when the shape varies . the calculated strain distribution of the pyramid and lens - shaped qd at x - z plane , wherea+care hydrostatic strain from 0.075 ( blue ) to 0.07 ( red ) andb+c ) are biaxial strain from 0.13 ( blue ) to 0.12 ( red ) the calculated bandoffset ( doted + line ) and energy level ( dotted line ) of the three different shaped qd when strain is included the recursion method is used to calculate the energy levels of three different shaped qds . for the experimental data , the broadening ( fwhm ) of the pc spectrum and the first peak of the pl spectrum are 34.01 and 29.34 mev , respectively . the much greater difference in the fwhm implies the possibility that the second peak comes from wetting layer , but not from the bound levels of qds . so for the simplicity of comparing with the experimental results , we only calculate all the energy levels of electron and the bound ground energy level of hole with the corresponding results shown in fig . 6 ( dotted line ) . next , we present the change in energy level for three different shape qds . from fig . the energy difference between the ground states of hole and electron is 0.990 ev for pyramid qd , 1.215 ev for truncated pyramid qd , and 1.053 ev for lens - shaped qd , respectively . the possible bound - to - bound transition of electronic inter - subbands is 254.1 mev for pyramid qd and 156.4 mev for lens - shaped qd . there is only one bound state in conduction band , so the possible transition of different energy levels is bound - to - continuum state with the energy being 199.3 mev . from the experimental data , the pl spectrum presents the ground state transition from electron to hole with the value being 1.058 ev . compared with the experimental data , the corresponding calculated value has the difference ( dpl ) of 6.43% for pyramid qd , 14.8% for truncated - pyramid qd , and 0.47% for lens - shaped qd . also the differences ( dpc ) from the measured pc peak ( 170.675 mev ) are 48.88% , 16.77% , and 8.36% for pyramid , truncated - pyramid , and lens - shaped qd , respectively . the compared results show that the energy difference of the lens - shaped qd is the most favored for the possible qd structure . though the transition of the bound - to - continuum transition for truncated pyramid qd is mostly acceptable because the behavior is much like the well - studied qwip structure , the energy difference of the bound ground states between electron and hole is rather far from the experimental results . the pyramid is not possible to be the shape of our investigated qd for the biggest difference of 48.88% . if we define a parameter  to estimate whether the shape or size is the most favored to qd of investigated qdip structure , the most suitable expression should be with dpl and dpc being the difference of calculated and measured pl and pc spectrum , respectively . in our calculation ,  is 34.86% , 15.82% , and 5.92% for pyramid , truncated - pyramid , and lens - shaped qd , respectively . for the lens - shaped qd ,  has the least value , which means the qdip structure is constructed by the lens - shaped qd . the lens - shaped inas / gaas qd is observed by many researchers , and in this way our calculation can get a good agreement with the measured data . also the pl , pc spectrum , and  provide us a way to find out the most suitable shape and size of qd which makes  the minimum . the different shape of qds can have different response wavelength as described in our calculation . the results mean that one can obtain the ideal response wavelength of qdip structure by controlling the growth condition to change the shape of qds . in summary , we have studied the strain distribution of self - assembled qd by the short - range vff approach to describe inter - atomic forces in terms of bond stretching and bending . the strain - driven self - assembled process of qd based on lattice mismatch has been clearly demonstrated . the recursion method is used to calculate the bound energy levels of qd for three different shapes but at the same ratio 3:1 for the base to the vertical aspect . the results indicate that the difference of bound energy is mainly controlled by the shape . . moreover , the strain and the shape both play key role in determining the ground state of hole . the results show that the difference of bound - to - bound energy levels of lenslike inas qd matches well with the experimental data , while the pyramid - shaped qd has the biggest difference from the measured data . though the bound - to - continuum transition for truncated pyramid qd is mostly acceptable because the behavior is much like the well - studied qwip structure , the bound ground states between electron and hole value is rather far from the experimental results . also the biggest difference of 48.88% makes the pyramid an impossible shape for our investigated qd . our theoretical investigation provides a feasible method for finding the most seemly geometry and size of qdip structure by adjusting the shape / size of qd and the comparing theoretical and experimental results . the project is partially supported by the national natural science foundation of china ( grant no:10474020 ) , cnkbrsf 2006cb13921507 , and knowledge innovation program of cas .
we present a systemic theoretical study of the electronic properties of the quantum dots inserted in quantum dot infrared photodetectors ( qdips ) . 
 the strain distribution of three different shaped quantum dots ( qds ) with a same ratio of the base to the vertical aspect is calculated by using the short - range valence - force - field ( vff ) approach . 
 the calculated results show that the hydrostatic strain hvaries little with change of the shape , while the biaxial strain bchanges a lot for different shapes of qds . 
 the recursion method is used to calculate the energy levels of the bound states in qds . 
 compared with the strain , the shape plays a key role in the difference of electronic bound energy levels . 
 the numerical results show that the deference of bound energy levels of lenslike inas qd matches well with the experimental results 
 . moreover , the pyramid - shaped qd has the greatest difference from the measured experimental data .