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the search for increased functionality in semiconductor - based devices is made possible with the advancement of microfabrication and epitaxial growing techniques .
a quantum well , built from two wider - bandgap semiconductors separated by a thin layer of narrower - bandgap semiconductor , can now be designed to deviate from the conventional rectangular- or square - well profiles in order to obtain more efficient properties .
a study has shown , for example , that @xmath0-doped parabolic quantum wells absorb far - infrared radiation at the bare - harmonic - oscillator frequency independent of electron - electron interactions and the number of electrons in the well @xcite .
another device is that of a heterostructure made from a high bandgap `` spike '' placed in the middle of a rectangular quantum well which showed a reduced material gain leading to an increased threshold current @xcite .
furthermore , simulations on a diode laser based on strained non - square shaped quantum well yield enhanced radiative current performance as compared to a device based on an optimal square well of the same width and emission length @xcite .
the authors of ref .
@xcite concluded that their embedded quantum well design may be suitable for optical confinement and carrier capture .    with the advances in band - gap engineering
, it is suitable to have an easily manipulated quantum well model that yields the optimized properties prior to fabrication .
a technique was developed using supersymmetric quantum mechanics to optimize the quantum well structure in respect to maximizing the gain in optically pumped intersubband lasers @xcite .
this method adds a bound state lower than the existing ground state energy of a potential well , thereby , varying the well s initial shape in the process .
the resulting quantum well may not have the symmetric structure of the initial well used .
in contrast , this work will show that one obtains a lower ground state by an appropriate embedding of a quantum well in another quantum well while maintaining the symmetry of the initial composite potential .    here a composite quantum well is constructed through the use of modified pschl - teller ( mpt ) potentials @xcite similar to that used in ref .
these potentials are related in form to rosen - morse potentials @xcite and have been used successfully to model disordered quantum wires @xcite .
the mpt - type of potentials are chosen since they offer a high degree of control and flexibility . moreover
, different single potential wells can be joined smoothly at the edges forming one continuous potential .
therefore , the systematic numerical procedure established in solving the eigenvalue equation for one composite quantum well can readily be used even when the parameters of the constituent single wells are varied .
the effective changes in the energy spectrum and scattering properties that occur when a quantum well is embedded in another well will be investigated in this paper .
this will serve as aid to experiments in that constructed composite quantum wells of the same type and symmetry with different embedded well sizes have fundametally different features .
is in units of @xmath1 and @xmath2 is in angstrom.,width=309 ]
a quantum well - in - a - well can be constructed from a sum of three mpt potentials , that is , @xmath3}\ ; , \label{potential}\ ] ] where @xmath4 is the effective mass and @xmath5 is dirac s constant .
the well - in - a - well system consists of two left and right wells , labelled with indices @xmath6 and @xmath7 , respectively , and a center well denoted by index @xmath8 .
their location relative to the origin are @xmath9 , @xmath10 , and @xmath11 . in this work ,
the width parameter @xmath12 is the same for all wells studied and it is kept to a constant value of @xmath13 . only the value of the depth parameter @xmath14 is varied . here
the depth parameters of the side wells , @xmath15 , are set equal , that is , @xmath16 .
this is done to retain the symmetric shape of the well about the origin .
the middle well has a depth parameter @xmath17 and its value can be different from @xmath18 . by varying the values of @xmath18 , @xmath17 and the shift parameter @xmath19 , the size of the effective embedded potential well relative to the resulting main quantum well can be controlled .
figure 1 illustrates a composite well as obtained from its constituent potential wells .
note that the model represents the conduction band of a quantum well system .
the bound states of a single electron in the composite well system described above can be obtained by using eq .
( [ potential ] ) in the schrdinger equation in one dimension @xmath20\psi(z)=e\psi(z)\;. \label{se}\ ] ] it follows that the wavefunction can be determined through an iterative procedure from eq .
( [ se ] ) in the central difference form @xcite , that is , @xmath21\psi(z )   \nonumber \\
+ 2\psi(z )   - \psi(z-\delta z)\;. \label{wavefunction}\end{aligned}\ ] ] here @xmath22 is an arbitrary infinitesimal step size and the initial values of @xmath23 and @xmath24 are obtained via simple symmetry arguments .
the advantage of using the mpt potentials is that the eigenenergies of a single mpt potential is known analytically @xcite .
hence , the difference in energy states of the constructed composite well as compared to the single mpt well can be related to the potential parameters .
the shooting method @xcite is implemented in this work to obtain each energy eigenvalue in a given energy range .
each solution must satisfy the boundary conditions that @xmath23 and its derivative vanish at infinity .
in addition to the boundary conditions , the minimum tolerance set for numerical convergence of @xmath25 and its derivative is @xmath26 . to accurately obtain the bound states , the whole potential depth is scanned to check for energy intervals wherein the first derivative of the wavefunction changes sign at infinity .
this signals that within this energy range a bound state can be found .
the bisection method is then applied to this particular interval to search for the bound state with a convergence limit of @xmath27 .
another property of this well - in - a - well system that will be studied here is the scattering of a free particle from this potential landscape via the transfer matrix approach .
the transmission probability is obtained from the ratio between the amplitude of the transmitted wave ( @xmath28 ) and that of the incident wave ( @xmath29 ) @xmath30 the transfer matrix technique yields @xcite @xmath31 e^{i[\kappa(z+\delta z ) -\kappa(z)]z}\;,\ ] ] where @xmath32\right)^{1/2}/\hbar$ ] . here
@xmath33 is the incident particle s kinetic energy .
it is further assumed here that the effective mass does not vary in space .
length measurements are given in angstroms and energy measurements are in units of @xmath34 .
figure [ fig1 ] shows a quantum well - in - a - well system , whose subwell size is 24@xmath35 of the full well depth .
the properties of the composite well is studied using a single mpt well of the same depth as a benchmark .
in addition , a parabolic well having the same width is fitted to the composite well for comparison .
the same approach is used to obtain the bound states for the single mpt well and the parabolic well .    ,
@xmath36 and @xmath37 , while for the single mpt well @xmath38 .
the inset shows the eigenenergies @xmath39 where @xmath0 is the bound state index.,width=309 ]    one finds that the composite well s ground state is lower by @xmath40 relative to the ground state of the single mpt well of the same depth .
in this example , the ground state of the parabolic well is lowest as expected due to its deeper potential depth .
note that the composite well has more bound states than that of single mpt well of the same depth or the parabolic well of the same width .
further , the interval between bound states are more evenly distributed in the composite system as compared to the single mpt well .
the distribution of states for the composite well is more similar to the even distribution of energy states of a parabolic well of the same width rather than that of the single mpt potential .
next , fig .
[ fig3 ] illustrates three well - in - a - well systems of the same depth , symmetry and base width .
the latter is the potential width at @xmath41 .
what varies in this plot is the depth of the component side wells yielding systems having different depths of embedded wells .
figure [ fig3 ] shows systems with embedded wells of depths ( a ) 88@xmath35 , ( b ) 64@xmath35 and ( c ) 28@xmath35 relative to the composite well s full depth .
the corresponding bound states of the three composite wells are shown in fig .
as the depth of the component side wells increases the number of bound states also increases .
well ( c ) has the most number of bound states .
furthermore , the change in slope of the plot of eigenstate @xmath39 relative to the bound state index @xmath0 , towards a constant value indicates the shift towards an even distribution of the energy states .
this occurs for the case of well ( c ) since the shape of the edge of this well approaches that of a parabolic potential .
recall that for the case of the simple harmonic potential well , the energy levels are evenly distributed .
so as the bottom of the component side wells approaches the depth of the middle well , we expect a straight line
. however , near the base of the potential the shape retains the tail of a modified pschl - teller potential well , hence the last two bound states are nearer to each other as compared to those adjacent states in the middle and edge of the well .
another phenomenon which is affected by the potential is the transport of a particle above it .
recall that in the case of a finite rectangular well of width @xmath42 , or similarly , a potential barrier of the same width , transmission resonance is observed when the wave number takes on integral multiples of @xmath43 @xcite .
it has also been demonstrated that a low energy incident electron above a rectangular well may be captured into a bound state due to dissipation @xcite .
transmission resonance is only restored for particle kinetic energies beyond the `` captive '' energy region @xcite .    in the case of scattering above a single mpt potential , transmission resonance is observed when @xmath14 is an integer regardless of the magnitude of the kinetic energy of the incident particle @xcite . unlike in the well - in - a - well systems studied here , there are no resonant wells .
this is true even for well ( a ) in which its constituent wells , by themselves , are absolute transparent potentials .
figure [ fig5 ] shows the transmission probability of a free particle with effective mass @xmath4 above each composite well in fig .
a particle has a lower probability of transmission if the kinetic energy of the particle approaches zero .
as expected , the larger the kinetic energy of the incident particle the more likely it is to be transmitted .
the monotonic increasing trend of the transmission probability for the composite wells remains valid even when the well width is increased .
this is in contrast to the appearance of an oscillatory nature of the transmission coefficient when the width of a finite rectangular well is widened .
the behavior of interest is that well ( a ) has the lowest probability of transmission relative to systems ( b ) and ( c ) .
note that for a particle scattered in a finite rectangular well in the non - resonance regime , the transmission probability increases with decreasing depth and width .
the opposite behavior is , thus , observed here , wherein the most shallow side wells and the most narrow middle well yield a composite system that creates more disturbance to particle transmission . in the perspective of an incident particle with energies corresponding to @xmath44 , wells ( b ) and ( c ) are more slowly varying potential functions relative to well ( a ) .
the abruptness of the change in the potential in ( a ) provides a stronger force in reducing the probability of transmission in this @xmath45 regime .
this paper presented a simulation model for a composite quantum well - in - a - well system and investigated the quantum mechanical properties arising from such construction .
a superposition of modified pschl - teller potentials is chosen for the model because the constituent wells can easily be varied without increasing the numerical complexity in solving the schrdinger equation .
this work showed that deviations in quantum well structures of the same functional form and depth yield entirely different properties as shown in the differences in their ground state energies , the number and distribution of bound states and the transmission probabilities of an incident free particle above the composite wells .
tailor - made quantum well systems as presented here offer ease and flexibility that can be suited to desired features for application purposes .
c. villagonzalo is grateful for the support provided by the office of the vice president for academic affairs through the university of the philippines system grant .
the resulting stationary states and scattering properties of an effective potential brought about by embedding a quantum well in another well are investigated in this work . 
 the composite well system is constructed via a superposition of modified pschl - teller potential wells . the energy spectrum in each composite 
 well is obtained using the shooting method and the transport of a particle above this system is analyzed using the transfer matrix method . 
 it is shown that decreasing the size of the embedded middle well lowers the ground state energy of the well - in - a - well system . moreover , the bound states increase in number and become more evenly spaced . 
 in addition , the transmission probability of a free particle incident above a composite well is lowest for the system with a large embedded well as compared to well - in - a - well systems of the same depth . 
 small variations in designed potential wells yield different quantum mechanical features .    
 and    quantum wells , pschl - teller potentials , bound states , transmission probability