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Inverse scattering, Phase reconstruction for the S-matrix,
one-dimensional Schroedinger equation, variable-mass mapping,
BenDaniel and Duke's equation
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%%%%%%% definitions %%%%%%%%%%%%%
\def\rmi{{\rm i}}
\def\rme{{\rm e}}
\def\rmd{{\rm d}}
\def\mbr{{\bbox{r}}}
\def\mbq{{\bbox{q}}}
\def\Re{\hbox{\rm Re}}
\def\Im{\hbox{\rm Im}}
\def\Tr{\hbox{\rm Tr}}
\def\calD{{\cal D}}
\def\calE{{\cal{E}}}
\def\calK{{\cal K}}
\def\calL{{\cal L}}
\def\calO{{\cal O}}
\def\calP{{\cal P}}
\def\calQ{{\cal Q}}
\def\frK{{\frak K}}
\def\frL{{\frak L}}
\def\frP{{\frak P}}
\def\frQ{{\frak Q}}
\def\calR{{\cal R}}
\def\calT{{\cal T}}
\def\sfD{{\sf D}}
\def\sfN{{\sf N}}
\def\sfQ{{\sf Q}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\preprint{}%\today }
\draft
%\tighten
\title{Design of semiconductor heterostructures with preset electron
reflectance by inverse scattering techniques}
\author{Daniel Bessis,$^{1,2}$ and G. Andrei Mezincescu$^{3}$
\protect{\footnote{E-mail mezin@alpha1.infim.ro}}}
\address{$^1$ CTSPS, Clark-Atlanta University, Atlanta, GA 30314\\
$^2$ Service de Physique Th\'eorique, C.E. Saclay,
F-91191 Gif-sur-Yvette Cedex, France\\
$^3$ Institutul Na\c tional de Fizica Materialelor, C.P. MG-7,
R-76900 Bucure\c sti -- M\u agurele, Rom\^ania}
\maketitle
\begin{abstract}
We present the application of the inverse scattering method
to the design of semiconductor heterostructures having a
preset dependence of the (conduction) electrons' reflectance
on the energy. The electron dynamics are described by either
the effective mass Schr\"odinger, or by the (variable
mass) BenDaniel and Duke equations. The problem of phase (re)construction
for the complex transmission and reflection coefficients is
solved by a combination of Pad\'e approximant techniques, obtaining
reference solutions with simple analytic properties.
Reflectance-preserving transformations allow bound state and reflection
resonance management. The inverse scattering problem for the Schr\"odinger
equation is solved using an algebraic approach due to Sabatier.
This solution can be mapped unitarily onto a family of BenDaniel and
Duke type equations. The boundary value problem for the
nonlinear equation which determines the mapping is discussed in some detail.
The chemical concentration profile of heterostructures whose self consistent
potential yields the desired reflectance is solved completely in the
case of Schr\"odinger dynamics and approximately for
Ben-Daniel and Duke dynamics. The Appendix contains a brief digest of results
from scattering and inverse scattering theory for the one-dimensional Schr\"odinger
equation which are used in the paper.
\end{abstract}
\pacs{}%{\bf Draft: }
\begin{multicols}{2}
\narrowtext
{{\small \tableofcontents}}
\section{Introduction}\label{i}
A semiconductor heterostructure can be modelled by a system of
equations describing (with a certain degree of completeness and precision)
the state of the system. The equations depend on a set of
structural and compositional data (SCD). Essentially, these are
the spatial dependence of the chemical composition (including dopant
profiles), the applied external fields, etc.
The system's behavior (response) is described by functional data (FD),
such as the electric or thermal conductance, the energy dependence of the
electron transmittance, the wavelength dependence of the optical absorption
coefficient, etc.
The FD can be computed using the solution of the equations, and are thus
functionals of the SCD.
To design a heterostructure for a certain application is to find a set
of SCD, which is physically (and technologically) achievable, such that the
values of a chosen subset of FD will be within desirable ranges.
The designer solves thus an inverse problem: inverting the functional
dependence of the FD on the SCD.
This problem is rather ill-posed. The desired ranges of FP may be unachievable.
Generally speaking, even if a certain desired set of FD values is achievable, the
set of SCD which achieves it is not unique. This absence of uniqueness is
not bad in itself. If several solutions {\em can be obtained}, then
one may further optimize the design in terms of other properties which
were not included in the original specifications. The difficulty is
mathematical. Inverting a one-to-one functional dependence can be a
formidable task, which is further aggravated by the lack of uniqueness.
A brute-force approach to the problem is always possible: computing the
FD for a set of achievable SCD (ideally all) and selecting the best.
In practice, brute force optimization is restricted to rather small sets
of parameters describing the FD. This is mostly due the fact that any
conceivable penalty function will be non-convex. Its graph will have a rather
complex multi-valley shape. The search algorithm is forced to do a thorough
investigation of this landscape and will eventually fail through run time
limitations. The intelligent designer will partially avoid such restrictions
by noting trends, trying to break the design into combining manageable blocks
and locally improving promising configurations.
Thus, cases when the solution of the inverse problem is realizable by methods
which are less costly than the brute force approach can be rather useful.
Even if one has to simplify somewhat the physical model, precious insights
on new promising configurations can be obtained.
In this paper we will try to review the possible applications of inverse
scattering techniques to some aspects of heterostructure design.
Our physical model for the electron states in the heterostructure will be the
(one band) effective mass approximation for the envelope function
of the (conduction band) electrons.
The simplest approach is to assume a constant effective mass. Then,
the envelopes of electron states satisfy an effective Schr\"odinger
equation\cite{Bastard}.
Choosing the $z$-axis along the growth direction,
\begin{equation}
\rmi\hbar\frac{\partial}{\partial t}\Psi(\mbr,t) =
-\frac{\hbar^2}{2m_e}\Delta\Psi (\mbr,t)+U(z)\Psi(\mbr,t).
\label{S1}\end{equation}
Here, $m_e$ is the (conduction band) electron's effective mass.
The potential is
\begin{equation}
U(z)=\calE(z)+U_{ext}(z)+\Phi_{sc}(z),
\label{S2}\end{equation}
where $\calE(z)=\calE_{cond}[c(z)]$ is the (conduction) band offset
(assumed to depend only on the local chemical composition, $c(z)$);
$U_{ext}(z)$ is the (possibly equal to zero) external applied potential
and we lumped in $\Phi_{sc}(z)$ the potential of the ionized (donor)
impurities (dopants) and terms which will make the full potential
$U(z)$ self-consistent. Various models can be considered for $\Phi_{sc}(z)$:
a Hartree self consistent potential, exchange-correlation corrections can
be incorporated. The only requirement is that $\Phi_{sc}(z)$ has to be an
{\it explicitly defined} functional of the full potential $U(z)$.
The next step is to take into account the spatial dependence of the
effective mass. The Schr\"odinger equation with constant effective mass (SE)
for the envelope function, (\ref{S1}), is replaced by the BenDaniel and
Duke\cite{BDD66} equation (BDD):
\begin{equation}
\rmi\hbar\frac{\partial}{\partial t}\Psi(\mbr,t) =
-\bbox{\nabla}\frac{\hbar^2}{2m(z)}\bbox{\nabla}\Psi (\mbr,t)+U(z)\Psi(\mbr,t).
\label{S3}\end{equation}
Here the effective mass of the conduction band electrons
$m(z)=m_{cond}[c(z)]$ is assumed to depend only on the local chemical
composition $c(z)$. The self-consistent potential is given again by
(\ref{S2}). The $\Phi_{sc}(z)$ term is now a functional of both $m(z)$
and $U(z)$.
In the following we will consider only stationary states of the equations
(\ref{S1}) and (\ref{S3}). Furthermore, since the potential depends only on
the coordinate along the growth direction, $z$, the motion in the
perpendicular plane is free. Setting
\begin{equation}
\Psi(\mbr,t)=\psi(z)\rme^{\rmi\left(\mbq_\perp\mbr_\perp-Et/\hbar\right)},
\label{none}\end{equation}
in the Schr\"odinger (\ref{S1}) and BenDaniel and Duke (\ref{S3}) equations,
where $E$ is the energy,
$\mbr_\perp=(x,y,0)$ and $\mbq_\perp=(q_1,q_2,0)$ are, respectively,
the coordinates and components of the quasi-momentum in the directions
perpendicular to the growth axis, we obtain the
one-dimensional Schr\"odinger,
\begin{equation}
\psi^{\prime\prime}(z)
+\left[k^2-\mbq^2_\perp-V(z)\right]\psi(z) =0,
\label{0-s1}\end{equation}
and BDD
\begin{equation}
\left[\frac{m_\infty\psi^{\prime}(z)}{m(z)}\right]^\prime
+\left[k^2-\frac{m_\infty\mbq^2_\perp}{m(z)}-V(z)\right]\psi(z) =0,
\label{0-bdd1}\end{equation}
equations for $\psi(z)$. Here, and in the following, we use the
notation $m_\infty$ for the electron effective mass in the embedding
material: $m_\infty=m_e$ for SE and $m_\infty=m(\pm\infty)$ for BDD.
We introduce the notations
\begin{equation}
k=\frac{\sqrt{2m_\infty E}}{\hbar};~~V(z)=\frac{2m_\infty U(z)}{\hbar^2}.
\label{ksq}\end{equation}
Throughout this paper, the square root function is defined with non negative
imaginary part: $\Im\left(\sqrt{E}\right)\ge 0$. For real positive $E$
in (\ref{ksq}), $k>0$. The prime will often be used for derivatives.
The inverse spectral theory for the one-dimensional
Schr\"odinger equation\cite{PT,ZS} has been successfully applied to some
optimization problems for the bound states in semiconductor
quantum wells\cite{ZCh97,MI96,MI97,TMI97,TMI98}.
Inverse scattering theory\cite{CS} for the one-dimensional Schr\"odinger
equation, (\ref{0-s1}) with $\mbq_\perp=0$, shows how one can recover the
potential in (\ref{0-s1}) from the knowledge of the scattering data:
the complex transmission and reflexion to the right/left
coefficients $\{T(k),\,R_\pm(k)\}$ for all real values of the wave-number $k$.
Widely used in electric circuit modelling\cite{CS,RT95},
it has been recently applied for designing heterostructure Bloch wave
filters\cite{BMMV97}.
We want to solve the following problem: let the electron
dynamics be given by either the SE, (\ref{0-s1}), or by the BDD equation,
(\ref{0-bdd1}), with the self-consistent potential (\ref{S2}).
Find chemical composition and dopant profiles, going to constant limits
at infinity,
such that the heterostructure defined by these data has a given energy
dependence of the electron reflectance at $\mbq_\perp=0$ and a given
operating temperature:
\begin{equation}
\calR(E)=\left\vert R_\pm(k)\right\vert^2. %
\label{refl}\end{equation}
The zero of the energy scale is chosen at the conduction band minimum
for the asymptotic composition at infinity.
In the SE case, the mini-bands are parabolic in $\mbq_\perp$ and (\ref{refl}) will
hold for $\mbq_\perp\ne 0$ with $E$ changed to $E+\hbar^2\mbq_\perp^2/2m_e$.
This is no longer true in the case of the BDD equation (\ref{0-bdd1}).
For sufficiently small $|\mbq_\perp|$,
the mini-bands will be approximately parabolic only as long as the
$\frac{m_e\mbq^2_\perp}{m(z)}$ term in the effective potential in (\ref{0-bdd1})
can be treated as a first order perturbation.
One could also select a nonzero value of $\mbq^2$ at which (\ref{refl}) is valid,
such as the one corresponding to the transverse thermal energy at the desired
operating temperature.
Since only the energy dependence of the reflectance is given,
the first step in solving our problem is to find the sets of
scattering data (SD) which are compatible with $\calR(E)$, {\it i.e.}
find the phases of the scattering data.
In section \ref{pha} we show how to construct SD which correspond to real
valued potentials $V(z)$ with exponential decay at infinity from $\calR(E)$.
Physically, one might expect that the potential is determined by the
its bound state energies and its resonances.
The reflectance $\calR(E)$ embodies only information on the transmission
resonances: sharp minima of the reflectance.
There is another type of resonance, the reflection resonances,
analogous to the resonances that occur in three-dimensional potential
scattering on a spherically symmetric potential. These are sharp {\it phase}
variations of the reflection coefficients.
The information on reflection resonances and on the bound states
is not apparent in the reflectance.
The transformations of the SD, which do not change
the reflectance, will be discussed. Using these transformations,
we will define reference solutions for the recovery of the SD from the reflectance.
The reference solutions have no bound states and simpler analytic properties.
Combinations of reflectance-preserving transformations can then be used to obtain
the SD of other solutions to the phase reconstruction problem from the
reference solutions, by dressing them up with bound states and reflection
resonances.
We will use Pad\'e approximation methods\cite{baker} to represent the
scattering data and to find parameterizations for a large class of solutions,
corresponding to potentials which tend exponentially to zero at infinity.
We will find that on this type of input data, the (re)construction process
amounts essentially to finding the roots of some polynomials and grouping
them into subsets.
In section \ref{rat} we present a simple and efficient algorithm for solving
the inverse scattering problem for scattering data in the form obtained in
section \ref{pha}. We will use results due to Kay, Moses and
Sabatier\cite{K55,KM56a,KM56b,K60,S83} for the inverse problem with rational
coefficients\cite{CS}.
In section \ref{vmm} we present the variable mass unitary mapping
of the SE to the BDD equation\cite{BBM95,BBM96,BMMV97}. We formulate
the boundary value problem which must be solved for determining the
coordinate transformation which defines the mapping, given the material
relation between the effective mass and the band offset. This ill-conditioned
problem can be solved by a a shooting method. In the case when the relation between
the mass and the offset is linear, the solution takes a simpler form.
We also give an efficient perturbative method for solving the mapping equation.
In section \ref{v} we show that obtaining chemical composition
profiles and self-consistent potentials for them in the SE in the
inverse scattering approach is simpler than obtaining the self-consistent
potential for a given chemical composition profile. We
also discuss the functional equation which must be solved
for obtaining the chemical (effective mass) profile corresponding
to self-consistent potentials in the BenDaniel and Duke's equation.
For the reader's convenience, in Appendix \ref{appA} we give a brief outline
of results from scattering and inverse scattering theory, which are needed
and often referred to in the main body of the paper.
%Finally, in
%Appendix \ref{appB} we give an outline on the Darboux transformation method
%for managing the problems of bound states and reflection resonances.
\section{Phase reconstruction}\label{pha}
Inverse scattering theory for the one-dimensional Schr\"odinger equation,
(\ref{0-s1}) with $\mbq_\perp=0$,
\begin{equation}
\psi^{\prime\prime}(z)
+\left[k^2-V(z)\right]\psi(z) =0,
\label{2s} \end{equation}
on which we give a primer in Appendix \ref{appA}, shows that
one can recover a fast decaying and piecewise continuous potential in
(\ref{2s}) from the knowledge of the scattering data (SD): the complex
transmission and reflexion to the right/left coefficients
$\{T(k),\,R_\pm(k)\}$
for all real values of the wave-number $k$ {\it if there are no bound states}.
If bound states are present, knowledge of the SD is not sufficient for unique
recovery of the potential. If the number of bound states is exactly $n$,
then a $n$-parameter family of potentials gives exactly the same scattering
data. %\cite{parratio}
The SD are completely determined by one of the reflexion coefficients and
the values of the energies of the bound states.
The phase of $T(k)$ can be obtained from a logarithmic dispersion relation.
(see {\it e.g.} the book by Chadan and Sabatier\cite{CS}, XVII.1.5). The
other reflection coefficient can be obtained from (\ref{7a}).
In this section we assume that the reflectance, $\calR(E)$, is known on the
positive energy half-axis, $E>0$. We want to construct sets of SD which
satisfy (\ref{refl}).
Since the values of the SD for scattering by short-range and piecewise
continuous potentials must satisfy the constraints\cite{CS,DT79} which are
enumerated at the end of Appendix \ref{i-M}, the function $\calR(E)$ cannot
be arbitrary. It must be non-negative and smaller than unity, with the
exception of $\calR(0)$, which is generically\cite{calR0} equal to 1.
If the potential is piecewise continuous, $\calR(E)$ must go to zero no
slower than $E^{-2}$ for large values of $E$.
The transmittance $\calT(E)=\left\vert T(k)\right\vert^2$, where $T(k)$
is the complex transmission coefficient, is readily recovered from
$\calR(E)+\calT(E)=1$. Thus, we know the absolute values of the scattering
data and we need the phases.
\subsection{Reflectance-preserving transformations}\label{pha-1}
The problem of finding the phases of the scattering coefficients knowing only their
absolute values on the real axis is underdetermined and has an infinite
number of solutions.
Before considering the phase (re)construction problem, we will introduce
two types of transformations which modify the phases of the
scattering data without changing the reflectance.
Let the set of scattering data
\begin{equation}
\{T(k),\,R_+(k)\,R_-(k)\},\label{sd}
\end{equation}
be a solution of the phase reconstruction problem, {\it i.e.} the
scattering coefficients satisfy (\ref{refl}) and the conditions enumerated
at the end of Appendix \ref{i-M}. Then, as mentioned above, if the SD
(\ref{sd}), has no bound states, there is an unique solution to the inverse
scattering problem: a potential $V(z)$ in (\ref{2s}) such that the SD
calculated for this equation coincide with (\ref{sd}). If $T(k)$ has $n$
simple imaginary poles in the upper half plane, that is $n$ bound states,
then a $n$-parameter family of potentials can be constructed, such that the
SD of each potential coincides with (\ref{sd}).
Let now $\lambda>0$ be a positive number, such that $k=\rmi\lambda$ is not
a pole of $T(k)$, {\it i.e.} that $E=-\hbar^2\lambda^2/2m_e$ is not a bound
state of (\ref{sd}).
Define a new set of SD by the transformation:
\begin{eqnarray} %^{(\rmi\lambda)}
{T}(k)&\to&\frac{(k+\rmi\lambda)}{(k-\rmi\lambda)}T(k),
\label{lam1}\\
{R}_\pm(k)&\to&\frac{(\rmi\lambda+k)}{(\rmi\lambda-k)}R_\pm(k),
\label{lam3}
\end{eqnarray}
The new set of SD will also satisfy the conditions set out at the end of
Appendix \ref{i-M}, so that a $(n+1)$-parameter family of potentials can
be constructed with each potential having the SD (\ref{lam1}-\ref{lam3}).
The transformation (\ref{lam1}-\ref{lam3}) can also remove bound states.
If the initial SD, (\ref{sd}), have a bound state for $E=-\hbar^2\lambda^2/2m_e$,
then the transformation (\ref{lam1}-\ref{lam3})
with $\lambda$ changed into $-\lambda$ in the right-hand sides of the equations,
transforms that bound state into an {\em anti-bound} state -- an imaginary pole
of $T(k)$ in $\Im(k)<0$, leaving all the others in place.
Let $\zeta$ be an arbitrary complex number with nonzero real and imaginary
parts. Then, we can define a second type of reflectance preserving
transformation of the SD (\ref{sd}):
\begin{eqnarray}%^{(\zeta)} \tilde
{T}(k)&\to&T(k),\label{zet1}\\
{R}_-(k)&\to&\frac{(k-\zeta)(k+\zeta^*)}{(k-\zeta^*)(k+\zeta)}R_-(k),\\
{R}_+(k)&\to&\frac{(k-\zeta^*)(k+\zeta)}{(k-\zeta)(k+\zeta^*)}R_+(k).
\label{zet3} \end{eqnarray}
Here and in the following, we use the notation $^*$ for complex conjugation.
The transformed SD, (\ref{zet1}-\ref{zet3}), have the same reflectance and
bound states as (\ref{sd}) and satisfy the conditions enumerated at the end
of Appendix \ref{i-M}.
Using the inverse scattering method, one can construct from the SD,
(\ref{zet1}-\ref{zet3}), a new $n$-parameter family of piecewise continuous
potentials which goes to zero at infinity.
The transformation (\ref{zet1}-\ref{zet3}) has a simple interpretation.
Assume that the imaginary part of $\zeta$ is much smaller than its
real part, $|\Im(\zeta)|\ll|Re(\zeta)|$, and the initial reflection
coefficients, $R_\pm(k)$, are slowly varying on the scale
$|\Im(\zeta)|$ near $k=\pm\Re(\zeta)$.
Then, the new scattering data (\ref{zet1}-\ref{zet3}) have a
{\it reflection resonance} of width $|\Im(\zeta)|$ at $k=\pm\Re(\zeta)$.
Indeed, the phases of the new reflexion coefficients vary by $\pm 2\pi$ in a
small interval of width $2|\Im(\zeta)|$ centered on $k=\pm\Re(\zeta)$.
A third type of transformation adds purely imaginary resonances
to the reflection coefficients leaving the transmission coefficient unchanged:
\begin{eqnarray}%^{(\rmi\lambda)} \tilde
{T}(k)&\to& T(k), \label{la1}\\
{R}_+(k)&\to&\frac{(\rmi\lambda+k)}{(\rmi\lambda-k)}R_+(k),\\
{R}_-(k)&\to&\frac{(\rmi\lambda-k)}{(\rmi\lambda+k)}R_-(k),
\label{la3}
\end{eqnarray}
where $\lambda$ is an arbitrary real number. The new SD will also satisfy the
conditions enumerated at the end of Appendix \ref{i-M}. The reconstruction
of the potential is done exactly as for the second type of transformation.
A sequence of transformations of the second and third type can be written as:
\begin{eqnarray}
T(k)&\to& T(k), \label{lp1}\\
R_-(k)&\to&\frac{S_N(k)}{S_N(-k)}R_-(k),\\
R_+(k)&\to&\frac{S_N(-k)}{S_N(k)}R_+(k), \label{lp3}
\end{eqnarray}
where $S_N(k)$ is any polynomial whose zeros are invariant with respect
to reflection through the imaginary axis, {\it i.e.} if $\zeta$ is a zero, then
$-\zeta^*$ is also a zero of the polynomial. Such polynomials, normalized by
the condition $S_N(0)=1$, satisfy $S_N(k)=[S_N(-k)]^*$ for real $k$.
Let us recall that an arbitrary polynomial of degree $n$, $\Pi_n(x)$,
with $\Pi_n(0)=1$, can be expressed through its zeros:
\begin{equation}
\Pi_n(x)=\prod_{i=1}^n\left(1-\frac{x}{x_i}\right).
\label{ro-r}\end{equation}
Here $x_i, ~i=1,\ldots,n$ are the zeros of $\Pi_n(x)$ (including
multiple ones according to their algebraic multiplicity).
Thus, $S_N(k)$ is completely determined.
%In Appendix \ref{appB} we will show how one can define Darboux
%transformations of the one-dimensional Schr\"odinger equation (\ref{2s}).
%The reflectance-preserving transformations (\ref{lam1} - \ref{zet3}) can be
%implemented as simple first- or second-order differential transformations of
%the solutions of (\ref{2s}) and its potential.
\subsection{The reference solutions}\label{pha-2}
Let us assume that we have found a solution of the phase reconstruction
problem. Generically, it will have some bound states and reflection
resonances. Using suitably chosen transformations of type
(\ref{lam1}-\ref{lam3}) one can obtain from it a solution for which the
transmission coefficient has no poles in the upper complex half-plane
({\it i.e.} no bound states). Then, by a sequence of transformations of type
(\ref{zet1}-\ref{la3}) with suitably chosen parameters one can find a
solution for which the reflection to the left coefficient has no poles or
zeros in the upper half plane.
We will call this solution of the phase reconstruction problem
{\it the left reference solution}. The left reference solution's
transmission coefficient, $T^{(r-)}(k)$, and the reflection to the left
coefficient, $R_-^{(r-)}(k)$, are analytic and have no zeros in $\Im(k)>0$.
In a similar way, we can define the right reference solution,
for which $T^{(r+)}(k)$ and $R_+^{(r+)}(k)$, are analytic and have no
zeros in $\Im(k)>0$. The left and right reference solutions are connected
by a transformation of type (\ref{lp1}-\ref{lp3}):
\begin{eqnarray}
T^{(r+)}(k)&=& T^{(r-)}(k)=T^{(r)}(k), \label{ref1}\\
R_-^{(r+)}(k)&=&\frac{S_{N+M}(k)}{S_{N+M}(-k)}R_-^{(r-)}(k),\\
R_+^{(r+)}(k)&=&\frac{S_{N+M}(-k)}{S_{N+M}(k)}R_+^{(r-)}(k).
\label{ref3}\end{eqnarray}
Here, the polynomial
\begin{equation}
S_{N+M}(k)=A_N(k)B_M(-k),
\label{polAB}\end{equation}
where the zeros of the polynomial $A_N(k)\, /\, B_M(k)$ coincide (including
multiplicities) with the poles/zeros of $R_-^{(r-)}(k)$ in $\Im(k)>0$.
Taking into account(\ref{ro-r}) the polynomials are completely determined.
The reference solutions are in a certain sense the maximally non-symmetric
solutions. Indeed, as shown in the next section \ref{rat}, the potentials
corresponding to the left/right reference solutions are identically
zero for $x<0$ / $x>0$.
Other solutions of the phase reconstruction problem can be obtained from the
reference ones by adding bound states and reflection resonances with
reflectance-preserving transformations of type (\ref{lam1}-\ref{la3}).
The left reference solution's $R_-^{(r-)}(k)$ is analytic and has
no zeros in $\Im(k)>0$. Thus, the logarithm $\ln[R_-^{(r-)}(k)]$
is also analytic in $\Im(k)>0$.
The phase of the left reference solution (which is equal to the imaginary
part of $\ln[R_-^{(r-)}(k)]$) can be obtained from the logarithm of
its absolute value (which is equal to the real part of $\ln[R_-^{(r-)}(k)]$),
using a (subtracted) logarithmic dispersion relation (See {\it
e.g.} \cite{clinton,RL96}).
We will proceed in a different manner, which is more adequate with
the physical context.
The effective mass approximation is valid
only for energies within an interval not exceeding several hundred
millielectronvolts (meV) near the $\Gamma$ point minimum in the
$Al_xGa_{1-x}As$ system (or in the lattice matched $In_{1-x-y}Al_xGa_yAs$
systems). Thus, two sets of scattering data having the same (or close) low and intermediate-energy behavior, but whose exact high-energy behavior
is different, can be considered equivalent.
We need a good approximation of the SD in the physically relevant
range of energies, which obeys the high-energy constraints set forward
in the Appendix \ref{appA}. The Pad\'e approximation method\cite{baker} is
a good framework for that. An added bonus, which will be apparent
in the following section \ref{rat}, is the simplification of the calculations
needed for recovering the potential.
\subsection{Pad\'e phase reconstruction}\label{pha-3}
We start with approximating the input design data for the reflectance
by a type II [p,p+q+2] Pad\'e approximant\cite{baker}:
\begin{equation}
\calR(E)=\frac{\calP_{p}(E)}{\calQ_{p+q+2}(E)},
\label{R_E}\end{equation}
where $\calP_{p}(E)$ and $\calQ_{p+q+2}(E)$ are polynomials of
degrees $p$ and, respectively, $p+q+2$ with $p,\,q\ge 0.$
Since the reflectance must be non-negative and less than 1 for all $E\ge 0$,
the polynomials must satisfy
\begin{equation}
0\le\calP_{p}(E)<\calQ_{p+q+2}(E),
\label{R_E1}\end{equation}
on the positive half-axis. The second inequality (\ref{R_E1}) is strict
for all $E>0$ and becomes an equality only for $E=0$. This ensures that
$\calR(0)=1$ as it should be in the generic case\cite{calR0}.
We can rewrite it in the form
\begin{equation}
\calQ_{p+q+2}(E)=\calP_{p}(E)+E\calK_{p+q+1}(E),
\label{R_E2}\end{equation}
where the polynomial $\calK_{p+q+1}(E)>0$ for $E\ge0$.
The polynomial $\calP_p(E)$ is normalized by setting
\begin{equation}
\calP_{p}(0)=1.
\label{R_E3}\end{equation}
Finally, the transmittance is approximated by the [p+q+2,p+q+2] Pad\'e
approximant:
\begin{equation}
\calT(E)=1-\calR(E)=\frac{E\calK_{p+q+1}(E)}{\calP_p(E)+E\calK_{p+q+1}(E)}.
\label{R_E4}\end{equation}
We can choose the coefficients of the polynomials in (\ref{R_E})
as the parameters. These can be obtained from standard type II Pad\'e
fitting routines.
Taking into account (\ref{R_E1}) with equality at $E=0$ and
(\ref{R_E3}), the fit is obtained by solving a system of $2p+q+2$
linear equations with $2p+q+2$ unknowns (the polynomials' coefficients),
which make the fit exact at $2p+q+2$ chosen points.
Thus, the [p,p+q+2] Pad\'e (\ref{R_E}) is fully determined.
Eq. (\ref{R_E4}) does not introduce additional parameters.
As we will see further on, it is advantageous to reparameterize
in terms of the zeros and poles of (\ref{R_E}). Since the reflectance and
the transmittance are real and non-negative, the zeros and poles of
(\ref{R_E}) and (\ref{R_E4}) are either real negative or come in
complex conjugate pairs.
The only exception to this rule are eventual real positive zeros of
$\calP_p(E)$, which have an even order of degeneracy (generically =2).
The heterostructure is transparent to Bloch waves (maximal transmission
resonances) at the energies corresponding to these degenerate zeros.
Now, we want to find the left reference solution by solving eq. (\ref{refl}),
\begin{equation}
\left\vert R_-^{(r-)}(k)\right\vert^2=\calR(E),
\label{refl-}\end{equation}
with the reflectance given by (\ref{R_E}).
The reflection to the left coefficient, $R_-^{(r-)}(k)$, will be sought
as a [p,p+q+2] Pad\'e approximant in the variable $k$
\begin{equation}
R_-^{(r-)}(k)=-\frac{P_{p}(k)}{Q_{p+q+2}(k)},
\label{R-}\end{equation}
with the normalization
\begin{equation}
P_{p}(0)=Q_{p+q+2}(0)=1,
\end{equation}
which agrees with (\ref{R_E3}). As mentioned above, $R_-^{(r-)}(k)$ for
the left reference solution of (\ref{refl-}) is analytic in the upper
half-plane and has no zeros there.
Then, the zeros of both the denominator and numerator of (\ref{R-}) must
lie in the lower complex half-plane. A further constraint on the roots
follows from the relation (\ref{8}),
\begin{equation}
\left[R_-(k)\right]^*=R_-(-k).\label{c-c}\end{equation}
This relation holds only if for each $r$, which is a zero/pole of (\ref{R-}),
$-r^*$ is also a zero/pole.
Let us now use (\ref{ro-r}) to represent all the polynomials
involved in the equation (\ref{refl-}) as products:
\begin{equation}
\frac{\prod_{i=1}^p(1-k/p_i)(1+k/p_i)}
{\prod_{j=1}^{p+q+2}(1-k/q_j)(1+k/q_j)}=
\frac{\prod_{i=1}^p(1-E/{\frak{p}}_i)}
{\prod_{j=1}^{p+q+2}(1-E/{\frak{q}}_j)}.
\end{equation}
Here $p_i$ and ${\frak{p}}_i, ~i=1,\ldots,p$ are the zeros of $P_p(k)$,
respectively $\calP_p(E)$, while $q_j$ and ${\frak{q}}_j, ~j=1,\ldots,p+q+2$
are the zeros of $Q_{p+q+2}(k)$, respectively $\calQ_{p+q+2}(E)$.
Taking into account the relation (\ref{ksq}), $E=\hbar^2k^2/m_e$, we find
the following relations between the zeros of the polynomials:
\begin{eqnarray}
{\frak{p}}_i&=&\frac{\hbar^2p_i^2}{2m_e};~~i=1,\ldots,p;\label{p1}\\
{\frak{q}}_j&=&\frac{\hbar^2q_j^2}{2m_e};~~j=1,\ldots,p+q+2.\label{q1}
\end{eqnarray}
These relations solve the problem up to the ambiguity of the
signs of the square roots. Let us show that the reference solution
is unique.
As mentioned in the Introduction, the square-root function as maps the
complex plane cut along $[0,+\infty)$ onto the upper complex half-plane,
{\it i.e.} if $E$ is in the cut plane, then $\Im\left(\sqrt{E}\right)>0$.
Since the zeros and poles of the reference solution cannot lie in the upper
half-plane the sign choices in (\ref{p1}-\ref{q1}) are
unique\cite{confluence}:
\begin{eqnarray}
\hbar p_i&=&-\sqrt{2m_e{\frak{p}}_i};~~i=1,\ldots,p;\label{p2}\\
\hbar q_j&=&-\sqrt{2m_e{\frak{q}}_j};~~j=1,\ldots,p+q+2.\label{q2}
\end{eqnarray}
It remains to show that if $r$ is one of the zeros, respectively poles,
of (\ref{R-}), then $-r^*$ is also a zero, respectively pole.
This is obvious if $r$ is real\cite{confluence} or imaginary. Otherwise,
both the real and the imaginary parts of $r^2$ are nonzero.
Then, $r^{*2}$ is also a zero (pole) of (\ref{R_E}), since as noted
above these come in complex-conjugate pairs. But our definition of
the square-root function leads to
$\sqrt{r^{*2}}=-\left(\sqrt{r^2}\right)^*.$
Thus, (\ref{R-}) with the zeros and poles given by (\ref{p2}-\ref{q2}) is
the unique left reference solution for the reflection to the left coefficient
if the reflectance is given by the Pad\'e approximant (\ref{R_E}).
We can readily recover the complex transmission coefficient of the reference
solutions using the same approach.
The (complex) transmission coefficient of the reference solution is
given by the [p+q+2,p+q+2] (diagonal) Pad\'e approximant
\begin{equation}
T^{(r)}(k)=\frac{kK_{p+q+1}(k)}{Q_{p+q+2}(k)},
\label{T-pade}\end{equation}
normalized by setting the coefficient of $k^{p+q+1}$ in the polynomial
$K_{p+q+1}(k)$ equal to the coefficient of $k^{p+q+2}$ in the already
determined polynomial $Q_{p+q+2}(k)$, so that $\lim_{k\to\infty}T(k)=1.$
Let 0 and ${\frak{k}}_j, ~j=1,\ldots,p+q+1$ be the zeros of the
transmittance, (\ref{R_E4}).
\begin{equation}
\calP_p({\frak{k}})=\calQ_{p+q+2}({\frak{k}}).
\end{equation}
Repeating the reasoning that led to (\ref{p2}), we recover the zeros
$\kappa_j$ of the polynomial $K_{p+q+1}(k)$
\begin{equation}
\hbar \kappa_j=-\sqrt{2m_e{\frak{k}}_j};~~j=1,\ldots,p+q+1,\label{kt}
\end{equation}
and
\begin{equation}
K_{p+q+1}(k)=-\frac{\prod_{i=1}^{p+q+1}\left(\kappa_i-k\right)}
{\prod_{j=1}^{p+q+2}q_j}.\label{Kp+q}
\end{equation}
Finally, using the relation (\ref{7ac}), we recover the left reference
solution's $R_+^{(r-)}(k)$ as a [2p+q+1,2p+2q+3] Pad\'e approximant:
\begin{equation}
R_+^{(r-)}(k)=-\frac{K_{p+q+1}(k)}{K_{p+q+1}(-k)}
\frac{P_p(-k)}{Q_{p+q+2}(k)}.
\label{R+}\end{equation}
The first fraction in the above expression is a phase factor.
Thus, if the reflectance is given in the Pad\'e form (\ref{R_E}),
the left reference solution is uniquely determined. The right/left reflection
coefficients of the right reference solution are given by
right-hand side of (\ref{R-})/(\ref{R+}). The potentials corresponding to
the right/left reference solutions are mirror images:
$V^{(r-)}(x)=V^{(r+)}(-x)$.
Other solutions can be obtained by using transformations of type
(\ref{lam1} - \ref{lam3}) to introduce bound states into the reference
solutions.
Then, transformations of type (\ref{zet1} - \ref{lp3}) can be used for
reflection resonance management.
A distinguished class of solutions, having the same transmission
coefficient as the reference ones and involving no additional parameters, can
be obtained by redistributing reflection resonances between the reflection
coefficients to the left/right. This is achieved by factorizing the
polynomials $P_p$ and $K_{p+q+1}$
\begin{equation}
P_{p}(k)=P_{p_+}(k)P_{p_-}(k);
~~~K_{p+q+1}(k)=K_{n_+}(k)K_{n_-}(k),
\label{C-k}\end{equation}
into factors which satisfy the complex conjugation relation (\ref{c-c})
for real $k$. Here, $p_++p_-=p$, $n_++n_-=p+q+1$ and all the factors
are normalized by $P_{p_\pm}(0)=1$.
Applying the transformation (\ref{lp1}-\ref{lp3}) with
$S(k)=K_{n_-}(k)P_{p_+}(-k)$ to the left reference solution, we obtain
a solution having the same transmission coefficient $T^{(r)}(k)$, given by
(\ref{T-pade}), and the reflection coefficients are given by the
[n$_\pm$+p,n$_\pm$+p+q+2] Pad\'e approximants
\begin{eqnarray}
R_-(k)&=&-\frac{K_{n_-}(k)}{K_{n_-}(-k)}
\frac{P_{p_+}(-k)P_{p_-}(k)}{Q_{p+q+2}(k)},\label{dis2}\\
R_+(k)&=&-\frac{K_{n_+}(k)}{K_{n_+}(-k)}
\frac{P_{p_+}(k)P_{p_-}(-k)}{Q_{p+q+2}(k)}.
\label{dis3}\end{eqnarray}
\section{Solution of the inverse scattering problem for the Schr\"odinger
equation when the scattering data are rational functions}\label{rat}
In this section we will present the solution of the inverse scattering
problem for the one-dimensional Schr\"odinger equation in the case when
the scattering data are given in the Pad\'e approximant form we
obtained in the preceding section \ref{pha-3}.
The inverse scattering problem in the case of rational coefficients has
been first considered by Kay and Moses\cite{K55,KM56a,KM56b,K60}. Significant
results are due to Sabatier\cite{S83,CS}. We will follow
Sabatier's approach\cite{S83,CS} quite closely.
As shown in the Appendix \ref{i-M}, the potential in (\ref{2s}) can be
recovered from the transformation kernels.
\begin{equation}
V(x)=\mp2\frac{\rmd}{\rmd x} K_\pm(x;x\mp 0).\label{3rec}
\end{equation}
The transforming kernels $K_\pm(x;y)$ are the solutions of the Marchenko
equations, (\ref{A14}), (\ref{A14plus})
\begin{eqnarray}
K_-(x;y)+M_-(x+y)&\hbox{\hskip -1pt}=\hbox{\hskip -1pt}&
\int_{-\infty}^x\hbox{\hskip -3pt}\rmd s M_-(y+s)K_-(x;s)=0,\nonumber\\
\label{m14-}\\
K_+(x;y)+M_+(x+y)&\hbox{\hskip -1pt}=\hbox{\hskip -1pt}&
\int^{\infty}_x\hbox{\hskip -3pt}\rmd s M_+(y+s)K_+(x;s)=0.\nonumber\\
\label{m14+}
\end{eqnarray}
The Marchenko kernels, $M_\pm(u)$, are given by (\ref{A15}).
\begin{equation}
M_\pm(u)={1\over{2\pi}}\int_{-\infty}^{+\infty}\rmd\kappa
\rme^{\pm\rmi\kappa u} R_\pm(\kappa)
+\sum_{j}\left(C_j^\pm\right)^{-2}\rme^{\mp\lambda_ju}.\label{3m15}
\end{equation}
Here, $C_j^\pm$ are the normalization constants of the bound states, defined
in (\ref{b-state}).
Inspection of (\ref{m14-}) and (\ref{m14+}) shows that the first
variable, $x$, enters the equations only as a parameter. Also,
for negative/positive $x$, the integral in (\ref{m14-})/(\ref{m14+})
involves only negative/positive values of $s$. Since we need
only the contact values of the transformation kernels in (\ref{3rec}),
it is natural to use the $K_-$ version of (\ref{3rec}), obtained by
solving (\ref{m14-}) for negative $x$, and the solution of (\ref{m14+})
for positive $x$. Taking into account the definition of the
Marchenko kernels, (\ref{3m15}), we see that
we can close the integration contour into $\Im(k)>0$ in the
expression of the relevant kernel ($M_+/M_-$ for $x>0/x<0$).
In particular, if one of the reflection coefficients is analytic there,
its contribution to the corresponding Marchenko kernel is identically zero.
In the absence of bound states, this implies that the corresponding
$M_\pm=0$ and, therefore, the potential will be zero on the corresponding
half-axis.
If the reflection coefficients are rational functions, like the
solutions of the phase reconstruction problem we obtained in
section \ref{pha-3}, they have at most a finite number of poles in $\Im(k)>0$.
The corresponding Marchenko kernels (\ref{3m15}) will be given by
finite sums of exponentials, which go to zero at the corresponding infinity.
For simplicity's sake, we will assume that all the poles are simple.
The multiple pole case can be dealt with as a limiting case
of pole confluence. Then, the corresponding kernel, $M_+(u)/M_-(u)$ for
$u>0/u<0$, is
\begin{equation}
M_\pm(u)=\rmi\sum_{j\in\Omega_\pm}\varrho_j^\pm
\rme^{\pm\rmi \nu_j^\pm u}. \label{2-18}
\end{equation}
Here the sets of (all distinct) complex numbers
$\left\{\nu_j^\pm\right\}_{j\in\Omega_\pm}$ consist of the (simple) poles of
$R_\pm(k)$ in $\Im(k)>0$ and, if bound states with energies
$-\hbar^2\lambda_j^2/2m_e$ are present, $\rmi\lambda_j$.
The coefficients $\varrho_j^\pm$ are either the residues of
the corresponding reflection coefficients or, if $\nu_j^\pm$ comes
from a bound state, $-\rmi\left(C_j^\pm\right)^{-2}$.
Substituting the separable Marchenko kernels (\ref{2-18}) into
the Marchenko equation (\ref{m14-}) for $x<0$, we obtain
\begin{equation}
K_-(x;y)=\rmi\sum_{j\in\Omega_-}\rho^-_j\rme^{-\rmi\nu_j^-y}
%\left[\rme^{-\rmi\nu_j^-x}+
Y_j^-(x),%\right],
\label{2-19}\end{equation}
where
\begin{equation}
Y_j^-(x)=\rme^{-\rmi\nu_j^-x}+
\int_{-\infty}^x\rmd y K_-(x;y)\rme^{-\rmi\nu_j^-y}.\label{2-Yj}
\end{equation}
Multiplying (\ref{2-19}) by $\rme^{-\rmi\nu_m^-x}$ and integrating with
respect to $y$ we obtain a system of $\#(\Omega_-)$ linear equations for
the $\#(\Omega_-)$ unknowns $Y_j^-(x)$. Here we used the notation
$\#(\Omega)$ for the number of elements in the set $\Omega$. Substituting
the solution into (\ref{2-19}), we obtain after a little algebra
\begin{equation}
K_-(x;x-0)=-\frac{\rmd}{\rmd x} \Tr
\ln \left[1-\rme^{-\rmi x\sfN^-}\sfD^-\rme^{-\rmi x\sfN^-}\right]. %\ln \det \sfD^-(x) =
\label{2-20}\end{equation}
Here, $\sfN^-$ and $\sfD^-$ are $\#(\Omega_-)$ square matrices with
\begin{eqnarray}
N_{mj}^-&=&\delta_{mj}\nu_j^-;\\
D^-_{mj}&=&\frac{\rho_j^-}{\nu_m^-+\nu_j^-}.
\label{2-sys1}\end{eqnarray}
Solving in the same manner the equation for $K_+(x;y)$, we obtain
\begin{equation}
K_+(x;x+0)=\frac{\rmd}{\rmd x}\Tr %\ln [1-\sfQ^+(x)\sfD^+\sfQ^+(x)],
\ln \left[1-\rme^{-\rmi x\sfN^+}\sfD^+\rme^{-\rmi x\sfN^+}\right].
\label{2-22}\end{equation}
where the elements of the $\#(\Omega_+)$ square matrices $\sfN^+$
and $\sfD^+(x)$ are
\begin{eqnarray}
N_{mj}^+&=&\delta_{mj}\nu_j^+;\\
D^+_{mj}&=&\frac{\rho_j^+}{\nu_m^++\nu_j^+}.
\label{2-sys1+}\end{eqnarray}
We can substitute now (\ref{2-20}) and (\ref{2-22}) into (\ref{3rec})
to obtain
\begin{equation}
V(x)=-8\Tr \left[\sfN^\pm\frac 1{1-\sfQ^\pm(x)}
\sfN^\pm\frac{\sfQ^\pm(x)}
{1-\sfQ^\pm(x)}\right],
\label{2-23}\end{equation}
where the $\pm$ signs are for $x>0$/$x<0$ and
\begin{equation}
\sfQ^\pm(x)=\rme^{2\rmi|x|\sfN^\pm}\sfD^\pm.
\label{2-24}\end{equation}
Thus, the potential is given by (\ref{2-23}) in terms of a trace of
the inverse of finite matrices.
\section{Variable-mass mapping: Schr\"odinger's to BenDaniel and
Duke's equation}\label{vmm}
In section \ref{rat} we have presented a simple algorithm
for solving the inverse scattering problem for the one-dimensional
Schr\"odinger equation in the case of rational SD. %%, which are obtained %in our approach to the phase reconstruction problem.
If the conduction electron dynamics is described by the BDD equation, we
will not start by posing the inverse scattering problem for that equation
from scratch.
Instead of that, we will use a family of unitary
transformations\cite{BBM95,BBM96,BMMV97}. The transformations
map the one-dimensional Schr\"odinger equation,
(\ref{0-s1}) with $\mbq_\perp=0$, and potential $V_S(z)$,
\begin{equation}
\psi^{\prime\prime}(z)
+\left[k^2-V_S(z)\right]\psi(z) =0,
\label{3-1} \end{equation}
into BenDaniel and Duke equations, (\ref{0-bdd1}), with $\mbq_\perp=0$,
and potential $V_{BDD}(z)$,
\begin{equation}
\left[\frac{m_\infty}{m(z)}\psi^{\prime}(z)\right]^\prime
+\left[k^2-V_{BDD}(z)\right]\psi(z) =0,
\label{3-2}\end{equation}
with a variable effective mass $m(z)$ and a potential $V_{BDD}(z)$.
The effective mass and the new potential are functionally related
to the parameters defining the unitary transformation.
Since the mapping is unitary, the SD for the equation (\ref{3-2})
will be identical with the SD for (\ref{3-1}).
We start thus with a Schr\"odinger reference equation, (\ref{3-1}),
which is the solution of the inverse scattering problem. From it
we obtain a family of BDD equations with the same scattering
data. The problem is to choose among these transformations those
which map the solution of the inverse problem for the SE onto
acceptable BDD equations.
\subsection{Unitary mapping}\label{umaping}
Let us introduce a (nonlinear) coordinate transformation
\begin{equation}
z=X(x),
\label{3-3}\end{equation}
which maps the interval $(-\infty,+\infty)$ into $(-\infty,+\infty)$.
Here the function $X(x)$ is a smooth monotonically increasing function:
$X^\prime(x)>0$.
The monotonicity ensures that a unique inverse transformation exists:
$x=Z(z)$, with
\begin{equation}Z[X(x)]=x\end{equation}
and $X[Z(z)]=z$. The inverse function $Z(z)$ is also smooth and
monotonically increasing, with
\begin{equation}
Z^\prime(z)=1/X^\prime[Z(z)].
\label{3-4}\end{equation}
Let us associate with the coordinate transformation (\ref{3-3}) a mapping
$\hat{\sf{U}}_X$ of the space of square-integrable functions, $L^2$, which transforms
each element of the space $f\in L^2$ into $\hat{\sf{U}}_Xf$ with
\begin{equation}
(\hat{\sf{U}}_Xf)(x)=\sqrt{X^\prime(x)}f[X(x)].\label{3-5}
\end{equation}
Making the change of variables (\ref{3-3}) in the normalization integral
\begin{eqnarray}
\langle f|f\rangle&=&\int_{-\infty}^{+\infty}\rmd z |f(z)|^2=
\int_{-\infty}^{+\infty}\rmd x X^\prime(x)|f[X(x)]|^2\nonumber\\
&=&\langle \hat{\sf{U}}_Xf|\hat{\sf{U}}_Xf\rangle
=\langle f|\hat{\sf{U}}_X^*\hat{\sf{U}}_X|f\rangle,\label{3-6}
\end{eqnarray}
we see that the mapping $\hat{\sf{U}}_X$ maps the square integrable functions
into square integrable functions conserving the norm, {\it i.e.}
$\hat{\sf{U}}_X$ is isometric on $L^2$. Since the inverse transformation,
\begin{equation}
\left(\hat{\sf{U}}_X^{-1}f\right)(z)=\sqrt{Z^\prime(z)}f[Z(z)].\label{3-7}
\end{equation}
exists and is non-singular, the mapping $\hat{\sf{U}}_X$ is unitary:
\begin{equation}
\hat{\sf{U}}_X^*=\hat{\sf{U}}_X^{-1}.\label{3-U}
\end{equation}
We want to see the effect of the transformation $\hat{\sf{U}}_X$ on functions
satisfying the Schr\"odinger equation (\ref{3-1}). We will consider
only the coordinate transformations for which the function $X(x)$
is twice differentiable with a piecewise continuous third
derivative\cite{smoo3}, $X^{\prime\prime\prime}$.
Substituting (\ref{3-5}) into (\ref{3-1}), we obtain after some algebra,
\begin{equation}
\left[\frac{\chi^\prime(x)}{[X^\prime(x)]^2}\right]^\prime+
\left[k^2-W_X(x)\right]\chi(x)=0.
\label{3-8}\end{equation}
Here $\chi(x)=(\hat{\sf{U}}_X\psi)(x)$ and
\begin{equation}
W_X(x)=V_S[X(x)]+
\frac{X^{\prime\prime\prime}(x)}{2[X^\prime(x)]^3}-
\frac{5[X^{\prime\prime}(x)]^2}{4[X^\prime(x)]^4}.
\label{3-9}\end{equation}
The equation (\ref{3-8}) satisfied by the transformed function
$\chi(x)$ resembles the BDD equation (\ref{3-2}) if we set
\begin{equation}
m(x)=m_\infty\left[X^\prime(x)\right]^2=m_{cond}[c(x)].
\label{3-10}\end{equation}
The function $X(x)$ must satisfy
\begin{equation}
\lim_{x\to\pm\infty}X^\prime(x)=1.
\label{3-11}\end{equation}
This ensures that $m(x)$ tends at infinity to the constant limit $m_\infty$.
We will also require that $X^\prime(x)-1$ decays at infinity faster
than $|x|^{-2-\delta}$ for some $\delta>0$. In this case,
the limits of $X(x)-x$ at $\pm\infty$ are finite. This
ensures that the Jost solutions of the original Schr\"odinger equation
are mapped into Jost-type solutions of the transformed one.
%\input{encl2}
Let $f_\pm(x;k)$ be the Jost solutions of (\ref{3-1}),
which obey the boundary conditions (\ref{A3}):
\begin{equation}
\lim_{x\to\pm\infty}\rme^{\mp\rmi kx}f_\pm(x;k)=1.
\label{3-jbc}\end{equation}
Then, applying to them the mapping (\ref{3-5}), we obtain
near the corresponding infinities
\begin{eqnarray}
\lim_{x\to\pm\infty}\rme^{\mp\rmi kx}\bigl({\sf{\hat{U}}}_X f_\pm\bigr)(x;k)=
\rme^{\rmi kd_\pm}.
\label{s-s}\end{eqnarray}
Here,
\begin{equation}%{eqnarray}%
d_+=\int_0^{+\infty}\rmd \xi \left[X^\prime(\xi)-1\right];~~
d_-=\int^0_{-\infty}\rmd \xi \left[X^\prime(\xi)-1\right];
\label{3-dpm}\end{equation}%{eqnarray}%
where we assumed $X(0)=0$.
Let $f_\pm^{(X)}(x;k)$ be the Jost-type solutions of the transformed equation
(\ref{3-8}), defined by the same boundary conditions (\ref{3-jbc}).
Taking into account the asymptotic behaviors of the Jost functions
of the Schr\"odinger equation (\ref{3-1}), we find the relation between the
scattering data for the transformed equation and those of the original one:
\begin{eqnarray}%{equation}
T^{(X)}(k)&=&\rme^{\rmi k(d_+-d_-)}T(k),\\
R^{(X)}_\pm(k)&=&\rme^{\pm 2\rmi kd_\pm}R_\pm(k).
\label{3-newj}\end{eqnarray}%{equation}
Thus, the reflectances of the original and the
transformed equations are equal.
In the following subsection we will study the variable effective
mass mapping nonlinear differential equation in the general case
of an arbitrary dependence of the band offset on the concentration.
We will show that the physically realizable concentration profiles
for devices embedded in a material of homogeneous composition
can be obtained by solving a (nonlinear) boundary value problem.
A necessary condition for the existence of achievable solutions is that
the dependence of the conduction band offset ${\cal{E}}_c$ on the
effective mass be a {\it non-decreasing} one. If this condition
is not satisfied, it may be still possible to embed the device in
a {\it periodic} super lattice, a case which will not be discussed
in this paper. In section \ref{bv-sol} we consider in more detail
the case when ${\cal{E}}_c(m)$ is a linear function, an approximation
which seems quite reasonable in the $AlGaAs$ system. In this case, the
solution of the nonlinear boundary value problem can be then expressed
through the canonical solutions of the Cauchy (initial value)
problem for a third-order linear differential equation. We also discuss
methods for obtaining stable approximate solutions.
\subsection{Differential equation}\label{diff}
Let us now compare the potentials $W_X$, (\ref{3-9}), and $V_{BDD}$.
The latter is given by (\ref{S2}) and (\ref{ksq}):
\begin{equation}
V_{BDD}(x)=\frac{2m_\infty}{\hbar^2}\left[
\calE(x)-\calE(\infty)+\Phi_{sc}(x)\right].
\label{3-12}\end{equation}
Here, the band offset is %a function of the local chemical composition
\begin{equation}
\calE(x)=\calE_{cond}[c(x)],
\label{3-13}\end{equation}
and we set $U_{ext}(x)=0$. The zero of the energy scale is chosen at
the value of the band offset for the embedding (asymptotic) composition.
The band offset, $\calE(x)$ and the effective mass, $m(x)$
depend on the position only through their dependence on the local chemical
composition $c(x)$, (\ref{3-10}) and (\ref{3-13}).
We will consider only the case when the function $m(c)$ is invertible.
Then, we can substitute the inverse function $c(m)$ into $\calE_{cond}(c)$:
\begin{equation}
\calE_{cond}(c)=\calE_c(m).
\label{3-14}\end{equation}
In this section we consider only unbiased structures with negligible
electron density.
Then, in the square brackets in (\ref{3-12}), $\Phi_{sc}(x)$ is
equal to zero. Equating the two potentials (\ref{3-9}) and (\ref{3-12})
an using (\ref{3-10}), we obtain a third-order nonlinear differential
equation for $X(x)$:
\begin{eqnarray}
\frac{X^{\prime\prime\prime}(x)}{2[X^\prime(x)]^3}&-&
\frac{5[X^{\prime\prime}(x)]^2}{4[X^\prime(x)]^4}
+V_S[X(x)]=\label{3-15}\\
&=&\frac{2m_\infty}{\hbar^2}\left\{
\calE_c[m_\infty X^{\prime\, 2}(x)]-\calE_c(m_\infty)\right\}.\nonumber
\end{eqnarray}
The values of $X^\prime(x)$ are restricted to the physically achievable
interval
\begin{equation}
\sqrt{\frac{\underline{m}}{m_\infty}}\le\dot{X}
\le\sqrt{\frac{\overline{m}}{m_\infty}}.
\label{3-16}\end{equation}
Here $\underline{m}$ and $\overline{m}$ are the minimal and, respectively,
the maximal values of the effective mass, $m(c)$, in the physically achievable
chemical composition range.
The equation (\ref{3-15}) does not depend explicitly on $x$. Setting
\begin{equation}
X^\prime=S\left[X(x)\right],
\label{3-17}\end{equation}
substituting this into (\ref{3-15}) and replacing the derivatives
with respect to $x$ according to
\begin{equation}
\frac{\rmd}{\rmd x}=\frac{\rmd X}{\rmd x}
\frac{\rmd}{\rmd X}=S\frac{\rmd}{\rmd X},
\label{3-18}\end{equation}
we obtain a second-order equation for $S,$
\begin{equation}%\begin{eqnarray}
2\frac{S^{\prime\prime}}{S}-3\frac{\left (S^{\prime}\right )^2}{S^2}
=\frac{8m_\infty}{\hbar^2}\left[\calE_c(m_\infty S^2)-
\calE_c(m_\infty)-U(X)\right].
\label{3-19}\end{equation}%\end{eqnarray}
Here $S^\prime$ is the derivative of $S$ with respect to $X$,
$U(X)=\hbar^2V_S(X)/2m_\infty$ is the potential in energy units.
The asymptotic condition (\ref{3-11}) reads now $S(\pm\infty)=1$.
\subsection{Boundary value problem}\label{bv}
%\subsection{Behavior at infinity and achievability}\label{inf}
Let us consider potentials $V(X)$ which are identically equal to zero
outside some interval [$X_-,X_+$].
For $X$ outside this interval,
the equation (\ref{3-19}) does not depend explicitly
on $X$ so that its order may be further reduced. Setting
\begin{equation}
m(X)=m_{\infty} S^2 \label{3-20}
\end{equation}
and
\begin{equation}
S^\prime(X)=Q(m(X)),\label{3-21}
\end{equation}
the equation (\ref{3-19}) becomes
\begin{equation}
2\frac{{\rmd} Q^2}{{\rmd} m} -3\frac{Q^2}{m}=
\frac{10}{\hbar^2}\biggl[\calE_c(m)-\calE_c(m_\infty)\biggr].
\label{3-22}\end{equation}
The solution of (\ref{3-22}) is
\begin{equation}
Q^2(m)=\frac{4m^{\frac 32}}{\hbar^2}\int_{m_\infty}^m
\bigl[\calE_c(\mu )-\calE_c(m_\infty)\bigr]
\mu^{-\frac 32}\rmd\mu. \label{3-23}
\end{equation}
If $m$ is close to $m_\infty,$ the right hand side of
(\ref{3-23}) goes to zero as $(m-m_\infty)^2.$
Indeed, for small $|\mu -m_\infty|,$ the term in square
brackets in (\ref{3-23}) is approximately equal to
$\calE_c^{\prime}(m_\infty)\left (\mu -m_\infty \right)$
and
\begin{equation}
Q^2(m)\approx \frac{2\calE^\prime _c(m_\infty)}{\hbar^2}
\left (m-m_\infty\right )^2 +{\cal{O}}\left[(m-m_\infty)^3\right].
\label{3-24}\end{equation}
Since $Q^2$ is non-negative, inspection of (\ref{3-24}) shows
that the effective mass may tend to a constant limit at
infinity {\em if and only if} the band offset dependence
on the effective mass, $\calE_c(m),$ is a
{\em non-decreasing function}.
\begin{equation}
\frac{\rmd\calE_c(m)}{\rmd m}\ge 0
\label{3-25}\end{equation}
In plain words, if the effective mass does not follow the band
offset, then the potential $V(X)$ cannot be embedded in an alloy of
homogeneous composition. Noting that in this
case the device may be embedded in a periodic superlattice,
we will restrict ourselves here to the case (\ref{3-25}).
Let us note that equation (\ref{3-19}) may be solved in quadratures
on the intervals $(-\infty,X_-)$ and $(X_+,+\infty)$.
Indeed, substituting (\ref{3-23}) into (\ref{3-21})
and taking into account (\ref{3-20}) we get
\begin{equation}
\frac{\rmd S}{\rmd X}=\pm\hbox{sign}(1-S) Q(m_{\infty} S^2),
\label{3-26}\end{equation}
where $Q(m)$ is the non-negative square root of the
right hand side of (\ref{3-23}) and the sign is positive on
$(X_+,+\infty)$ and negative on $(-\infty,X_-)$.
The solution on $(X_+,+\infty)$ is given in parametric form by
\begin{eqnarray}
X &=& X_++\hbox{\rm sign}\left[1-S(X_+)\right]
\int_{S_+(X_+)}^{s} \frac{\rmd \sigma}{Q(m_{\infty}\sigma^2)},\label{3-27}\\
x &=& x_++\hbox{\rm sign}\left[1-S(X_+)\right]
\int_{S_+(X_+)}^{s} \frac{\rmd \sigma}{\sigma Q(m_{\infty} \sigma^2)},
\label{3-28}\end{eqnarray}
where $x_+$ is the value of $x$ which maps to $X_+$:
$X_+=X(x_+).$ A similar representation is valid for
the interval $(-\infty ,X_-).$
Thus, we have obtained
the solutions which are regular at infinity. There,
$S(X)=1 +{\cal{O}}\left[\rme^{-\kappa|X|}\right]$ tends
exponentially to the constant limit $1.$
The asymptotic rate of decay $\kappa$ may be obtained
from (\ref{3-24}):
\begin{equation}
\kappa =
\frac{2m_\infty}{\hbar}\sqrt{2\calE_c^\prime(m_\infty)}.
\label{3-29}\end{equation}
Let us now briefly discuss the asymptotic behavior at
infinity for the regular solutions of (\ref{3-19}) for
potentials $V(X)$ which go to zero at infinity.
One may readily see from (\ref{3-19}) that
$S(X)$ must still converge to $1$ at infinity.
As long as the potential $V(X)$ decays at infinity
faster than $\rme^{-\kappa|X|}$ the properties of the solutions
discussed above remain asymptotically valid.
For potentials with slower falloff at infinity, the regular solution
tends asymptotically to the regular solution of the equation
$S^{\prime\prime}=8m_{\infty}\calE_c^\prime(m_\infty)S/\hbar^2+V(X)/2.$
% \subsection{Boundary value problem}\label{bv}
It is important to note that any regular solution of (\ref{3-19})
satisfies at all points $X_\pm$ outside the support of the potential
the boundary conditions (\ref{3-26}).
To construct a solution of (\ref{3-19}) which is regular
on the whole axis $X,$ we will chose a value for the
embedding alloy effective mass, $m_\infty$.
The regular solutions on the interval $(-\infty ,X_-)$ can be parameterized
by the value $S(X_-)=\tau$. The value of the
derivative at $X_-$ is given by (\ref{3-26}) with
the minus sign:
\begin{equation}
S^\prime(X_-;\tau)=
-\hbox{\rm sign}\left[1-\tau\right] Q(m_{\infty} \tau^2),
\label{3-30}\end{equation}
where $Q(m)$ is the positive square root of the right-hand
side of (\ref{3-23}).
Then, we integrate the equation (\ref{3-19}) numerically with
the initial values defined above over the support of the
potential up to $X_+,$ obtaining $S(X_+;\tau)$ and
$S^\prime(X_+;\tau).$
If the solution is regular then it must satisfy (\ref{3-26})
with the plus sign at $X_+$:
\begin{equation}
S^\prime(X_+;\tau)=
\hbox{\rm sign}\left[1-\tau\right]
Q\left[m_{\infty} S^2(X_+;\tau)\right].
\label{3-31}\end{equation}
If they exist, the solutions of equation (\ref{3-31}) give the values
of $\tau$ for which we can find regular solution of the differential
equation (\ref{3-19}) on the whole real axis. To be achievable these solutions
must also satisfy the physical bounds (\ref{3-16}).
Numerically, the shooting method outlined here is rather
ill-conditioned. Since the general solution of (\ref{3-19}) is
singular, multiple-precision arithmetic has to be used
for the integration of the differential equation if the support of
the potential is not short enough.
In the following section we will examine in more detail
the important case when the dependence of the
band offset on the effective mass is linear. We will show that
in this case the solutions of the nonlinear equation (\ref{3-19})
can be found among the solutions of a third-order {\em linear}
equation.
\subsection{Solution of the boundary value problem in the case of linear
dependence of the band offset on the effective mass}\label{bv-sol}
%\subsection{Linear dependence of the band edge offset on the effective mass}\label{lin}
For some alloys like $Al_cGa_{1-c}As$ in the concentration range
$0\le c\le .45$ the conduction band minimum is at the center of the
Brillouin zone ($\Gamma$) and the dependence of $m_{cond}(c)$ and
$\calE_{cond}(c)$ on the concentration $c$ approximately
linear. In the $Al_cGa_{1-c}As$ system, for $0\le c\le .45,$ the offset
from the position at $GaAs$ is
\begin{equation}
\calE_c(m)={B}(m-\underline{m}).
\label{3-32}\end{equation}
The constant ${B}$ is
\begin{equation}
{B}= \frac{\Delta E}{\overline{m}-\underline{m}}=9.41 eV/m_0,
\label{3-33}\end{equation}
Here $\underline{m}=.067m_0 $ is the conduction band effective mass for $GaAs$;
$\overline{m}=.104m_0 $ and $\Delta E$ are, respectively, the mass and
band offset for $Al_{.45}Ga_{.55}As$. $m_{0} $ is the electron mass.
Substituting this into (\ref{3-19}) we obtain
\begin{equation}
2{S^{\prime\prime}}{S}-3{\left (S^{\prime}\right )^2}
=\kappa_\infty^2{S^2}\left[S^2-1-v(X)\right].
\label{3-34}\end{equation}
Here,
\begin{eqnarray}%{equation}
\kappa_\infty^2&=&{8{B}m_\infty^2}/{\hbar^2},\label{3-35}\\%~~
v(X)&=&{4V(X)}/{\kappa_\infty^2}=U(X)/({B}m_\infty),\label{3-36}
\end{eqnarray}%{equation}
and $U(X)=\hbar^2V(X)/2m_\infty$ is the potential measured in energy units.
%\subsection{Linear equation}
Let us define a new unknown function %$T(X)$ --- the inverse of $S(X)$:
\begin{equation}
T(X)= {3-3}/{S(X)}.
\label{3-39}\end{equation}
Substituting (\ref{3-39}) into (\ref{3-34}) we obtain the equation
satisfied by $T(X)$:
\begin{equation}
2TT^{\prime\prime}-T^{\prime 2}+
\kappa_\infty^2\left\{1-\bigl[1+v\bigr]T^2\right\}=0.
\label{3-40}\end{equation}
Here and whenever it does not lead to ambiguities we will omit
the arguments of the functions. Taking the derivative of (\ref{3-40})
we obtain a third-order linear equation for $T$:
\begin{equation}
T^{\prime\prime\prime}-\kappa_\infty^2\left(1+v\right)T^{\prime}
-\frac 12\kappa_\infty^2v^\prime T=0.
\label{3-41}\end{equation}
An arbitrary solution of (\ref{3-41}) will satisfy the second-order
nonlinear equation (\ref{3-40}) with the zero in the right hand side
replaced by some constant ${\calK}$:
\begin{equation}
2TT^{\prime\prime}-T^{\prime 2}-
\kappa_\infty^2\bigl[1+v\bigr]T^2={\cal K}.
\label{3-42}\end{equation}
If $\calK=-\kappa_\infty^2$, the solution satisfies also (\ref{3-40}).
Let $F_{\alpha\beta\gamma}(X;X_-)$ be the solution of the initial value
(Cauchy) problem for the linear equation (\ref{3-41}) satisfying the
initial conditions
\begin{eqnarray}
F_{\alpha\beta\gamma}(X_-;X_-)&=&\alpha,\nonumber\\
F_{\alpha\beta\gamma}^\prime(X_-;X_-)&=&\beta,\label{3-43}\\
F_{\alpha\beta\gamma}^{\prime\prime}(X_-;X_-)&=&\gamma,\nonumber
\end{eqnarray}
at the point $X_-$, which is a point of continuity of the potential $v(X)$.
Since (\ref{3-41}) is linear,
\begin{equation}
F_{\alpha\beta\gamma}=\alpha F_{100}+\beta F_{010}+\gamma F_{001}.
\label{3-44}\end{equation}
The function $F_{\alpha\beta\gamma}(X;X_-)$ also satisfies the second-order
equation (\ref{3-42}) with
\begin{equation}
{\cal K}_{\alpha\beta\gamma}=2\alpha\gamma -
\beta^2 -\kappa_\infty^2\alpha^2\left[1+v(X_-)\right],
\label{3-45}\end{equation}
in the right hand side.
%\subsection{Solution of the boundary value problem in the case of linear dependence of the band offset on the effective mass}\label{bv-sol}
In \ref{bv} we outlined the shooting method for solving the boundary
value problem which leads to the regular (acceptable) solutions of
(\ref{3-19}). It is well known that shooting methods
are prone to numerical instabilities even for linear boundary
value problems. Another unpleasant feature is the fact that we have to
integrate the equation numerically over the support of the potential
for each value of $\tau$.
Let us state the boundary value problem for the function $T(X)$,
assuming the potential $v(X)$ to be identically zero outside the
interval $(X_-,X_+)$. The solutions regular on $(-\infty,X_-)$
and $(X_+,+\infty)$ are
\begin{eqnarray}
T_-(X)&=&1+\left[T(X_-)-1\right]\rme^{\kappa_\infty(X-X_-)},\label{3-46}\\
T_+(X)&=&1+\left[T(X_+)-1\right]\rme^{-\kappa_\infty(X-X_+)}.
\label{3-47}\end{eqnarray}
Thus, the boundary conditions at $X_\pm$ which must be satisfied by
the regular solution are
\begin{eqnarray}
T^\prime(X_-)&=&\kappa_\infty\left[T(X_-)-1\right],\label{3-48}\\
T^\prime(X_+)&=&-\kappa_\infty\left[T(X_+)-1\right].
\label{3-49}\end{eqnarray}
Thus, we can parameterize the regular solutions (\ref{3-46}) by the value
of $T(X_-)=\alpha$. Then, $T^\prime(X_-)$ is given by (\ref{3-48}).
Then we integrate (\ref{3-40}) up to $X_+$, where $T$ and its derivative
must satisfy (\ref{3-49}), whence the acceptable values of
$\alpha$ are determined.
Now, we can use the linear equation satisfied by $T$ to express the
solution of the initial value problem through the canonical solutions
$F_{100},~ F_{010}$ and $F_{001}$ of the initial value problem
for (\ref{3-41}). Using (\ref{3-40}) with $v(X_-)=0$ to find
$T^{\prime\prime}(X_-)$, we find the solution satisfying (\ref{3-48})
\begin{eqnarray}
T_\alpha (X)&=&\alpha F_{100}(X;X_-)+\kappa_\infty (\alpha-1) F_{010}(X;X_-)
\nonumber\\&+&\kappa^2_\infty (\alpha-1) F_{001}(X;X_-).
\label{3-50}\end{eqnarray}
At $X_+$, $T_\alpha(X)$ must satisfy the boundary condition (\ref{3-49}).
Whence we find $\alpha$:
\begin{equation}
\alpha=1-\frac{F^\prime_{100}(X_+;X_-)
+\kappa_\infty\left[F_{100}(X_+;X_-)-1\right]}
{F^\prime_{1\kappa_\infty\kappa_\infty^2}(X_+;X_-)+
\kappa_\infty F_{1\kappa_\infty\kappa_\infty^2}(X_+;X_-)},
\label{3-51}\end{equation}
where $F_{\alpha\beta\gamma}$ was defined in (\ref{3-43}).
Thus, the solution of the boundary value problem for
the nonlinear equation (\ref{3-40}) is expressed through the
canonical solutions of the initial value problem at $X_-$ for the linear
equation (\ref{3-41}).
The value of $X_-$ can be chosen arbitrarily as long as the potential
$v(X_-)=0$. We may safely assume that $v(X)=0$ in a neighborhood of $X_-.$
The functions $F_{100}$ and $F_{1\kappa_\infty\kappa_\infty^2}$
are also solutions of
the second-order nonlinear equation (\ref{3-42}) with
${\cal K}$ determined from (\ref{3-45}).
We have found the value of $\alpha$, which determines
the solution of the boundary value problem, through the solutions of two
second order (albeit nonlinear) equations. Although the integration
must be performed only once from $X_-$ to $X_+$, it is still numerically
unstable.
\subsection{Approximate solutions}\label{map-ap}
As mentioned above the numerical integration of the differential
equation over typical device lengths 20-40$nm,$ having the order of
magnitude of the electron mean free path, is rather ill conditioned.
Taking the values of the parameters for the $AlGaAs$ system, above (\ref{3-33}),
and a value $m_\infty\approx .1m_{\infty} $ for the effective mass of the
embedding material, we obtain that the natural length scale for the
differential equation (\ref{3-34}) is $1/\kappa_\infty\approx .3nm,$
which is comparable to the lattice period.
The equation (\ref{3-34}) can be rewritten as
\begin{equation}
S^2-1-v(X)=\frac{2{S^{\prime\prime}}{S}-3{\left (S^{\prime}\right )^2} }
{\kappa_\infty^2S^2}.
\label{3-60}\end{equation}
Since the right-hand side of (\ref{3-60}) has the small factor
$\kappa_\infty^{-2}$, one is tempted to proceed in a
a {\it na\"\i ve} "quasi-classical" way and neglect the right-hand side
entirely. Then,
\begin{equation}
S(X)\approx \sqrt{1+v(X)},
\label{3-61}\end{equation}
and finding the coordinate transformation $X(x)$ reduces to a simple
quadrature. This gives surprisingly reasonable results.
Finding corrections to (\ref{3-61}) seems a rather tedious task,
especially for potentials which have discontinuities.
We will note instead that from (\ref{3-36}),
\begin{equation}
v(X)=\frac{U(X)}{{B}m_{\infty}} \approx
\frac{U(X)}{.94eV},\label{3-B}
\end{equation}
for the $AlGaAs$ system. Since typical potential values are $\pm$100-200$meV$,
the potential $v(X)$ (measured in the natural units of the problem)
is small compared to 1. A "small potential" perturbative approach to
solving the boundary value problem is thus indicated.
Let us assume that $v\sim \alpha $ and seek $S$ as a series
\begin{equation}
S(X)=S_{(0)}(X)+\alpha S_{(1)}(X)+\alpha^2 S_{(2)}(X)+\ldots,
\label{3-62}\end{equation}
substitute $v\to\alpha v$ and (\ref{3-34}), rewritten as
\begin{equation}
S^{\prime\prime}=3S^{\prime\,\, 2}/2S\, +\,
\kappa_\infty^2S\left[S^2 -1-v\right]/2\, ,
\label{3-63}\end{equation}
and expand into a power series in $\alpha$. Then, after equating the terms with
the same power of $\alpha$ and setting $\alpha=1$, we obtain
a hierarchy of linear differential equations for the functions
$S_{(\ell)}(X),$ $\ell=0,1,\ldots$.
The first equation from (\ref{3-62}) is $S_{(0)}^{\prime\prime}(X)=0$.
The solution must go to $1$ when $X$ goes to infinity so that
$S_{(0)}(X)\to 1$, while all the other $S_{(\ell)}(X)\to 0$ as
$X\to\pm\infty.$ Then,
\begin{equation}
S_{(0)}(X)=1.
\label{3-64}\end{equation}
For $\ell\ge 1$, the hierarchy has the form
\begin{eqnarray}
S_{(1)}^{\prime\prime}&-&\kappa_\infty^2S_{(1)}=
-\frac 12 \kappa_\infty^2v;\label{3-65}\\
S_{(2)}^{\prime\prime}&-&\kappa_\infty^2S_{(2)}=
\frac 32 S_{(1)}^{\prime\, 2}%\nonumber\\&+&
+\frac 12\kappa_\infty^2S_{(1)}\left[3S_{(1)}-v\right];\label{3-66}\\
&\ldots&\nonumber\\
S_{(\ell)}^{\prime\prime}&-&\kappa_\infty^2S_{(\ell)}= %\nonumber\\&\phantom{-}&
{\cal{F}}_\ell\left(X;S_{(1)},\ldots ,S_{(\ell-1)},v\right);
\label{3-67}\\
&\ldots&\nonumber
\end{eqnarray}
\par Taking into account the boundary conditions for $X\to\pm\infty$,
the solution of the $\ell$-th equation in the hierarchy is
\begin{eqnarray}
S_{(\ell)}(X)&=&-\frac 1{2\kappa_\infty}\int_{-\infty}^\infty
{\rm d}\xi {\rm e}^{-\kappa_\infty|X-\xi|}\nonumber\\
&\phantom{=}&\phantom{-\frac{3-3}{2\kappa_\infty}\int_{-\infty}^\infty}
{\cal{F}}_\ell\left(\xi;S_{(1)},\ldots ,S_{(\ell-1)},v\right),
\label{3-68}\end{eqnarray}
which can be verified by direct substitution.
The large $\kappa_\infty$ limit of the first terms in the
perturbative expansion (\ref{3-62}) is
\begin{eqnarray}
S_{(0)}(X)+S_{(1)}(X)&=&1+\frac{\kappa_\infty}{2}\int_{-\infty}^\infty
{\rm d}\xi {\rm e}^{-\kappa_\infty|X-\xi|}v(\xi)\nonumber\\
%&\phantom{+}&\phantom{S_{(1)}}
&\to& 1+\frac 12 v(X),\label{3-69}
\end{eqnarray}
which coincides with the first terms in the perturbative expansion of the
"quasi-classical solution" (\ref{3-61}).
Several terms we checked also have this property.
Having found $S(X)$, we can now find the coordinate transformation
by a simple quadrature:
\begin{equation}
x=\int_0^X \frac{\rmd Y}{S(Y)}.
\label{3-quad}\end{equation}
In fact, this yields the inverse transformation, (\ref{3-4}).
Finally, the mass profile in the BDD equation is given in parametric form
by (\ref{3-quad}) and
\begin{equation}
m(X)=m_\infty S(X).
\label{3-masp}\end{equation}
The concentration profile can now be readily recovered.
%\input{encl1}
\subsection{Example}
We will illustrate the method by finding the specifications
for a filter with two narrow transmission resonances, centered
at $E_1E_2$.
This can be achieved by choosing
\begin{equation}
\calQ_3(E)=1+\frac{\delta}2
\left[1+\Bigl(\frac{2E}{\overline{E}}-1\Bigr)^3\right].
\label{q3-1} \end{equation}
Then, if $\delta\ll 1$, the background reflectance is monotonically
decreasing with $\calR_b(0)=1$ and $\calR_b(\overline{E})\approx 1-\delta$.
It has a horizontal inflection point at $E=\overline{E}/2$ with
$\calR_b(\overline{E})\approx 1-\delta/2$.
Neglecting $\delta$ with respect to 1, the full widths at half maximum (FWHM)
of the transmittance maxima are, respectively, $2\sqrt{E_1^3/F_1}$ and
$2\sqrt{E_2^3/F_2}$. Thus, $F_1$ and $F_2$ can be obtained from the FWHM
of the corresponding resonance.
We chose $E_1=40meV$ and $E_2=100meV$, with FWHMs equal to $9meV$ and,
respectively, $3meV$ as initial data. Proceeding as explained in sections
\ref{pha}, \ref{rat}, \ref{bv-sol} we obtained an effective-mass profile
which yielded the same reflectance as the input one. This continuous profile
was digitized manually into 12 steps with lenghths which are integer numbers
of lattice constants and heighths which are a combination
of three concentrations, as described in Table 1. The $Al$ concentration
of the embedding alloy is 11.4\%.
In Fig. 1 we present
the transmittance calculated for the resulting configuration.
The digitization has introduced a few artefacts (the shoulder of the
low energy line and a low amplitude broad maxima, one of which is visible
in Fig. 1). The maxima were broadened and shifted a little from the design data.
Nevertheless, the ratio of FWHM of the
high/low energy maxima of transmittance is better than 2:1.
\section{Determination of chemical composition and dopant concentration
profiles of heterostructures with preset reflectance}\label{v}
In the section \ref{rat} we have shown how, given rational expressions
for the scattering data (SD), one can construct the piecewise continuous
potential $V(x)$, decaying exponentially at $+\infty$.
The one-dimensional Schr\"odinger with $V(x)$ has the given SD.
In section \ref{vmm} we mapped unitarily the one-dimensional Schr\"odinger
equation onto a family of BenDaniel and Duke type equations depending on
the function $X(x)$ which defines the transformation.
We also discussed in some detail the solution of the nonlinear
differential equation, which determines the effective mass profile
in the case of an undoped heterostructure, with vanishing density of
conduction electrons.
Now, we want to deal with the case when a nonzero density of conduction
electrons is present and the potentials, (\ref{S2}) in the SE, (\ref{S1}),
or in the BDD equation, (\ref{S3}), are self-consistent.
Let the external potential $U_{ext}(z)=0$ so that the potential
in the BDD equation is
\begin{equation}
U(z)=\calE_{cond}[c(z)]+\Phi_{sc}(z;m;U),
\label{4-1}\end{equation}
where we evidenced the functional dependence of $\Phi_{sc}(z;m;U)$ on
the effective mass profile $m(z)$ and the full potential energy $U(z)$.
In the SE case the effective mass is constant and $\Phi_{sc}$ depends only
on $U(z)$.
Inspection of (\ref{4-1}) suggests the idea that for a given full potential
$U(z)$, we can determine the chemical composition profile by moving the
selfconsistent potential to the left hand side of (\ref{4-1}).
We will see that in the SE case this is relatively easy to do
at a given operating temperature $T_o$. The situation is trickier in
the case of BDD dynamics, where we will present a perturbative approach
to solving the functional equation which replaces the nonlinear
differential equation (\ref{3-15}).
Let us start by considering the Hartree approximation for the self-consistent
electrostatic potential. Then, $\Phi_{sc}(z;m;U)$ is the solution of Poisson's
equation
\begin{equation}
\left[\varepsilon(z)\Phi_{sc}^\prime(z)\right]^\prime= 4\pi e\rho_{ch}(z).
\label{4-2}\end{equation}
Here $-e$ is the electron charge, $\varepsilon(z)$ is the dielectric constant,
$\varepsilon(z)=\varepsilon[c(z)]$, and $\rho_{ch}(z)$ is the full charge
density, the difference between the ionized donor charge density and the
electron one.
\begin{equation}
\rho_{ch}(z)=e\left[n_d(z)-n_{el}(z)\right].
\label{44-2a}\end{equation}
The density of donor dopant ions is made of a uniform
background density $n_b$ and the local variation of the
density of donors in the heterostructure $n_\ell(z)$, which goes to
zero for large $|z|$:
\begin{equation}
n_d(z)=n_b + n_\ell(z).
\label{44-2b}\end{equation}
We assume that the doping and temperature are such that all the donors are
ionized and that the density of holes is negligible compared to the density
of donors. Then, we can neglect the valence bands. Otherwise, a multi-band
treatment is needed.
The electron dynamics is described either by the SE, (\ref{S1}), or by the BDD
equation, (\ref{S3}). The equilibrium electron density, $n_{el}(z)$, can be
calculated from the density of states for the corresponding equation,
(\ref{S1}) or (\ref{S3}),
\begin{equation}
n_{el}(z)=\int_{-\infty}^{+\infty}\frac{\rmd E\,\, \nu(z,E)}
{1+\rme^{\beta(E-\mu)}},
\label{4-3}\end{equation}
where $\beta$ is the inverse temperature (in energy units) and $\mu$ is
the chemical potential of the electrons.
A necessary condition for the stability of the system is that the full
charge is equal to zero,
\begin{equation}
\int_{-\infty}^{+\infty}\rmd z\rho_{ch}(z) =0.
\label{4-4}\end{equation}
In particular, the limiting value of electron density at infinity
equals the background ion density, $n_{el}(\pm\infty)=n_b$.
The limiting values of the potential at $\pm\infty$ will be equal only if
dipolar moment of the charge density is zero,
\begin{equation}
\int_{-\infty}^{+\infty}\rmd z\, z\, \rho_{ch}(z) =0.
\label{4-5}\end{equation}
Thus, $z^2\rho_{ch}(z)$ must go to zero at infinity and the solution
of (\ref{4-2}) is
\begin{equation}
\Phi_{sc}(z;m;U)=2\pi e\int_{-\infty}^{+\infty}\rmd u\rho_{ch}(u)
\left\vert\int_u^x\frac{\rmd v}{\varepsilon(v)}\right\vert.
\label{4-6}\end{equation}
For position-independent $\varepsilon$, which we will consider in the case
of SE dynamics, (\ref{4-6}) becomes the well-known
\begin{equation}
\Phi_{sc}(z;U)=\frac{2\pi e}{\varepsilon}
\int_{-\infty}^{+\infty}\rmd u |z-u|\rho_{ch}(u).
\label{4-7}\end{equation}
We will consider separately the cases when the electron dynamics
is described by the Schr\"odinger equation, (\ref{S1}), and by the
BenDaniel and Duke equation, (\ref{S3}).
\subsection{Schr\"odinger's equation}\label{scSch}
To compute $\Phi_{sc}(z;U)$ using (\ref{4-7}), we need the electron density
of states in (\ref{4-3}). Since the transverse degrees of freedom
separate, (\ref{none}-\ref{0-s1}), we can integrate over the transverse
quasimomenta and express the three dimensional density of states $\nu_S(z,E)$
through the density of states $\nu_{0}(z,E)$ of the one-dimensional SE
(\ref{3-1}):
\begin{equation}
\nu_S(z,E)=\frac{m_e}{\pi\hbar^2}\int_{0}^{\infty}\rmd \eta \nu_{0}(z,E-\eta).
\label{4-9}\end{equation}
The one dimensional density of states $\nu_{0}(z,E)$ is proportional to
the imaginary part of the Green function $G_0(z,z^\prime;E)$ of the
one-dimensional SE (\ref{3-1})
\begin{equation}
\nu_{0}(z,E)=
-\frac 1\pi \lim_{\delta\downarrow 0}\Im\left[G_0(z,z;E+i\delta\right].
\label{4-10}\end{equation}
This allows a standardized treatment of the bound and continuum states.
For all $E$ with $\Im(E)\ne 0$, the Green function is the solution of the equation
\begin{equation}
\left[E+\frac{\hbar^2}{2m_e}\frac{\rmd^2}{\rmd z^2}-U(z)\right]
G_0(z,z^\prime;E)=\delta(z-z^\prime),
\label{4-11}\end{equation}
which is continuous at $z=z^\prime$ and goes to zero for $z\to\pm\infty$.
In terms of the Jost functions, defined in Appendix \ref{appa-1},
the solution is
\begin{eqnarray}
G_0(z,z^\prime;E)&=&\frac{m_eT(k)}{\rmi k\hbar^2}
f_+(z;k)f_-(z^\prime;k); ~~z>z^\prime,\nonumber\\
&=&G(z^\prime,z;E);~~z0$ the spectrum is absolutely continuous
and doubly degenerate.
\subsection{Jost functions, scattering data and their properties}\label{appa-1}
The scattering is best described in terms of the Jost solutions
of (\ref{A1}), $f_\pm(z;k)$,
which for $k=0$ behave like outgoing waves near $\pm\infty$:
\begin{eqnarray}
\lim_{z\to +\infty}&f&_+(z;k)\rme^{-\rmi kz}=1;\nonumber\\
\lim_{z\to -\infty}&f&_-(z;k)\rme^{\rmi kz}=1.\label{A3}
\end{eqnarray}
Let us summarize some properties of the Jost functions (JF) which are
relevant for the scattering problem:
\begin{itemize}\label{jostf}
\item[{\it a.}] For real $k$ the JF are continuous in $z$. $f_+(z;\pm k)$
are a pair of linearly independent solutions of (\ref{A1}). The same
holds for $f_-(z;\pm k)$.
\item[{\it b.}] The JF can be continued analytically from the positive half-axis ($k>0$)
to the upper complex half-plane ${\rm Im}(k)>0$. Here, the JF are
{\it analytic} in $k$ with values which are continuous functions of $z$.
They have no zeros in $z$ for ${\rm Im}(k)>0$.
\item[{\it c.}] For large complex $|k|$ in the upper half-plane, the JF behave like
outgoing waves for all real $z$:
\begin{equation}
\rme^{\mp\rmi kz}f_\pm(z;k)=1+{\cal{O}}(k^{-1});~~ |k|\to\infty.
\label{A4}\end{equation}
\end{itemize}
The above propositions can be readily proved using the integral equation
of Volterra type which is satisfied by the JF:
\begin{equation}
f_\pm(z;k)=\rme^{\pm\rmi kz} +\int_z^{\pm\infty}\rmd y
\frac{\sin k(y-z)}k\, V(y)f_\pm(y;k). \label{volt}
\end{equation}
Now, we have two pairs of linearly independent solutions of (\ref{A1}),
$f_\pm(z;\pm k)$. One of each pair behaves like an outgoing/ingoing
near the corresponding infinity. Since (\ref{A1}) can have only two
linearly independent solutions, the outgoing wave JF can be expressed
in terms of the ingoing wave ones:
\begin{eqnarray}
T(k)f_+(z;k)= f_-(z;-k)+R_-(k)f_-(z;k);\label{6p}\\
T(k)f_-(z;k)= f_+(z;-k)+R_+(k)f_+(z;k).\label{6m}
\end{eqnarray}
Instead of seeking the asymptotic behaviors, the transmission, $T(k)$ and
the reflection coefficients to the right/left, $R_\pm(k)$, can be expressed
in terms of Wronskian determinants of the JF:
\begin{eqnarray}
T(k)&=&\frac{2\rmi k}{W[f_+(z;k),f_-(z;k)]};\label{tr}\\
R_\pm(k)&=&\frac{W[f_-(z;\pm k),f_+(z;\mp k)]}{W[f_+(z;k),f_-(z;k)]};\label{rpm}
\end{eqnarray}
where we use the notation:
\begin{equation}
W[f(z),g(z)]=f^\prime(z) g(z)-f(z)g^\prime(z),
\end{equation}
for the Wronskian of the functions $f(z)$ and $g(z)$. We remind the reader
that the Wronskian of any two solutions of (\ref{A1}) does not depend on
$z$ and is equal to zero if and only if the solutions are linearly dependent.
For real $k$ the scattering coefficients $T(k)$ and $R_\pm(k)$
satisfy the following relations:
\begin{eqnarray}
R_+(k)T(-k)+R_-(-k)T(k)&=&0,\label{7a}\\
T(k)T(-k)+R_\pm(k)R_\pm(-k)&=&1,\label{7b}
\end{eqnarray}
which express the unitarity of the $S$-matrix.
Reality of the potential implies also that
\begin{equation}
T(k)=[T(-k)]^*;~~~~R_\pm(k)=[R_\pm(-k)]^*.\label{8}
\end{equation}
Generically, $T(0)=0$ and $R\pm(0)=-1$ (if there are no "zero energy bound
states" --- bounded, but not square integrable, solutions of the
Schr\"odinger equation for $k=0$).
From (\ref{tr}) and the remark {\it c.} above we see that $T^{-1}(k)$ can be
continued analytically to the upper half-plane, Im$(k)>0$. Its zeros, if
present, are the only possible (simple pole) singularities of $T(k)$
in the upper half plane. At such a zero, the two JF $f_\pm(z;k)$ are not
linearly independent and decay exponentially for $z\to\pm\infty$. Thus,
the poles of $T(k)$ in the upper half plane can occur only for
$k=\rmi\lambda_j$, where $\hbar\lambda_j=\sqrt{-2m_eE_j}>0$ and $E_j$ are
the energies of the bound states of (\ref{A1}). The corresponding
eigenfunctions $\psi_j(z)$ are real and are normalized by
$\int_{-\infty}^{+\infty}\rmd x |\psi_j(x)|^2=1$. The two JF are proportional
to the bound-state wave function:
\begin{equation}
f_\pm(z;\rmi\lambda_j)=C_j^\pm\psi_j(z),\label{b-state}
\end{equation}
where $C_j^\pm$ are real constants.
The asymptotic expansion of $T(k)$ near the
bound-state pole $k=\rmi \lambda_j$ is
\begin{equation}
T(k)\approx \frac {\rmi }{C_j^+C_j^-}\frac 1{k-\lambda_j}
+\calO(1),\label{b-res}
\end{equation}
where $C_j^\pm$ are the constants in (\ref{b-state}).
From here we can see that the bound-state poles of $T(k)$ must be simple.
Otherwise, the product of the normalization constants is zero.
Generally speaking, the domains of analiticity of
$R_\pm(k)$ will be smaller.
If the potential $V(z)$ is zero on a half-axis and there are no bound states, then
the corresponding reflection coefficient is analytic in the upper half-plane.
Let us check this for the reflection to the left coefficient, $R_-(k)$, and
potentials which vanish for $z<0$. The denominator in (\ref{rpm}) is
analytic and has no zeros since there are no bound states. In the numerator,
$f_+(z;k)$ is always analytic and $f_-(z;-k)=\rme^{\rmi kz}$ for $z<0$. The
other case can be dealt with in a similar manner.
From (\ref{7a} - \ref{8}) we obtain relations between the analytic
continuations of the scattering data, which are valid whenever
the arguments of the functions are within the domain of analiticity:
\begin{eqnarray}
R_+(k)T(-k)+R_-(-k)T(k)=0;\label{7ac}\\
T(k)T(-k)+R_\pm(k)R_\pm(-k)=1;\label{7bc}\\
{[T(k)]}^{*} = T(-k^*); ~~~
{[R_\pm(k)]}^{*} = R_\pm(-k^*).\label{8c}
\end{eqnarray}
For positive energies, $E=\hbar^2k^2/2m_e$, (\ref{7b}-\ref{8}) imply
that the sum of the transmittance of the heterostructure and its reflectance,
\begin{equation}
\calT(E)+\calR(E)=
\left\vert T(k)\right\vert^2+\left\vert R_\pm(k)\right\vert^2=1
\end{equation}
equals unity.
\subsection{Inverse scattering. The Marchenko equation.}\label{i-M}
If the potential in the Schr\"odinger equation (\ref{A1}) is known, then,
solving (\ref{A1}) and using (\ref{6p}-\ref{6m}) one can obtain
the scattering data (\ref{sd}).
We want to explore the possibility of recovering the potential in the
Schr\"odinger equation (\ref{A1}) from the SD (\ref{sd}).
Let
\begin{equation}
F_\pm(z;k)=\rme^{\mp\rmi kz}f_\pm(z;k).\label{Fpm}
\end{equation}
The functions $F_\pm(z;k)$ are
analytic in the upper half-plane. As a function of $k$, $F_\pm(z;k)-1$ decays
at infinity no slower than $k^{-1}$, (\ref{A4}). This means that the Fourier
transformation with respect to $k$ will exist at least in the $L^2$ sense.
We define the transformation kernels
of the Schr\"odinger equation by the Fourier transforms
\begin{eqnarray}
K_+(x;y)&=&\frac 1{2\pi}\int_{-\infty}^{+\infty}\rmd k
\rme^{-\rmi k(y-x)}\left[F_+(x;k)-1\right],\label{A9-}\\
K_-(x;y)&=&\frac 1{2\pi}\int_{-\infty}^{+\infty}\rmd k
\rme^{\rmi k(y-x)}\left[F_-(x;k)-1\right].\label{A9}
\end{eqnarray}
Closing the integration contour in the upper half-plane, we see that
\begin{equation}%{eqnarray}
K_+(x,y)=0 \hbox{\rm ~for~} x>y;~~~\\
K_-(x,y)=0 \hbox{\rm ~for~} x0$ there exists $c(a)>0$ such that
\begin{equation}
\int_{-\infty}^{+\infty}
\rmd x\theta[\pm (x-a)] (1+x^2) |M^\prime_\pm(x)| < c(a).\label{D-T}
\end{equation}
\end{itemize}
The problem becomes a little trickier in the presence of bound states.
Then, the Marchenko kernels $M_\pm(u)$, (\ref{A15}), depend now not only on
the energy of the bound state, which is given by the corresponding
pole of $T(k)$, but also on the constants $C_j^\pm$, (\ref{b-state}).
The scattering data contain information only on the product $C_j^-C_j^+$,
which can be recovered from the residue of $T(k)$, (\ref{b-res}).
For each bound state % $j$
we can choose
one of the parameters $C_j^\pm$ arbitrarily. The other is fixed by
the (analytically continued) scattering data.
Thus, assuming full knowledge of the scattering data, in the case
when there are $n\ge 1$ bound states the solution of
the inverse problem is not unique. There is a $n$-parameter
family of potentials which correspond to the same scattering data.
From a physical point of view, one cannot recover the full information
on the bound states in scattering experiments, which study
only the behavior of the solutions at large distances, where
the relevant information on the bound states is exponentially vanishing.
The numerical solution of the Marchenko equation for potential reconstruction
is expensive from the computational point of view. To find a value
for $V(x)$ one has to solve (\ref{A14}) with high enough precision for
the subsequent numerical differentiation. A lot of useless data is generated
in the process, since we need only $\lim_{y\uparrow x}K_-(x;y)$ for using
(\ref{A-VK}) (or $\lim_{y\downarrow x}K_+(x;y)$).
We will solve the Marchenko equation in the manner explained in
section \ref{rat}, which is closer to the way we solve the
phase reconstruction problem in section \ref{pha}.
The resource management compares rather favorably to that of the
codes\cite{FJ90,FJ92} which have been written for the
direct solution.
\vbox{\vskip 2em}
{\bf Acknowledgments:} We thank Tom Gaylord and Elias Glytsis for introducing us
to the fascinating subject of heterostructure design and for fruitful
discussions. Illuminating discussions with Roger Balian,
Pierre Sabatier and Giorgio Mantica are gratefully acknowledged.
% Daniel Vrinceanu's assistance with computations has been invaluable.
G.A.M. thanks Alfred Msezane and Carlos Handy for support and hospitality
at C.T.S.P.S., Clark-Atlanta University, where significant parts of the
work presented here have been done.
%$^{*}$ E-mail: mezin@alpha1.infim.ro
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%\bibitem{}
\end{thebibliography}
%\newpage
\begin{table}%[h]
\caption{12 layer digitized Al$_c$Ga$_{1-c}$As filter. The
Al concentrations are $c_1=0.05714,~c_2=2c_1,~c_3=4c_1;$ the bulk
Al concentration is $c_2=11.4\%.$\hfill}
\label{table1}
\begin{tabular}{cccc}
Layer&Width&Width&Al\cr
\# &(Atomic layers) &(nm) &concentration \cr
\hline
1&6&1.696&$c_1+c_2+c_3$\\ \hline
2&9&2.543&0\\ \hline
3&18&5.088&$c_3$\\ \hline
4&5&1.413&$c_2$\\ \hline
5&10&2.827&$c_1+c_2$\\ \hline
6&10&2.827&$c_1$\\ \hline
7&14&3.957&$c_1+c_2$\\ \hline
8&14&3.957&$c_1$\\ \hline
9&13&3.675&$c_1+c_2$\\ \hline
10&13&3.675&$c_1$\\ \hline
11&14&3.957&$c_1+c_2$\\ \hline
12&11&3.109&$c1$\\ \hline
\end{tabular}
\end{table}
%\vbox{\vskip 1.0truecm}
%\newpage
%\widetext
%\protect{\vbox{\vskip 2.0truecm}}
\begin{figure}[h]
\special{psfile=Fig-1.ps hscale=50 vscale=50 voffset=-450 hoffset=-20 }
\caption{Energy dependence of the transmittance of 12 layer digitized
$Al_cGa_{1-c}As$ filter: continuous line --- Eq.(\protect{\ref{3-2}});
dotted line --- constant mass approximation Eq.(\protect{\ref{3-1}})
with $m_0=m_\infty$. Insert: potential energy profile.\label{fig1}}
\end{figure}
\end{multicols}
\end{document}
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end
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6 -1 V
6 0 V
6 -1 V
6 0 V
6 -1 V
6 0 V
6 -1 V
6 0 V
6 -1 V
6 0 V
6 0 V
6 -1 V
6 0 V
5 -1 V
6 0 V
6 0 V
6 0 V
6 -1 V
6 0 V
6 0 V
6 0 V
6 -1 V
6 0 V
6 0 V
6 0 V
6 0 V
6 -1 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 -1 V
6 0 V
6 0 V
6 0 V
6 0 V
6 1 V
5 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 0 V
6 1 V
6 0 V
6 0 V
6 0 V
6 0 V
6 1 V
6 0 V
6 0 V
6 0 V
6 1 V
6 0 V
6 0 V
6 0 V
6 1 V
6 0 V
6 0 V
6 1 V
6 0 V
6 0 V
5 1 V
6 0 V
6 1 V
6 0 V
6 1 V
6 0 V
6 1 V
6 0 V
6 1 V
6 0 V
6 1 V
6 0 V
6 1 V
6 1 V
6 0 V
6 1 V
6 1 V
6 0 V
6 1 V
6 1 V
6 1 V
6 1 V
6 0 V
6 1 V
6 1 V
6 1 V
6 1 V
6 1 V
6 1 V
6 1 V
5 1 V
6 1 V
6 2 V
currentpoint stroke M
6 1 V
6 1 V
6 2 V
6 1 V
6 1 V
6 2 V
6 1 V
6 2 V
6 2 V
6 1 V
6 2 V
6 2 V
6 2 V
6 2 V
6 2 V
6 2 V
6 2 V
6 2 V
6 3 V
6 2 V
6 3 V
6 2 V
6 3 V
6 3 V
6 3 V
6 3 V
6 3 V
5 4 V
6 3 V
6 4 V
6 4 V
6 4 V
6 4 V
6 4 V
6 5 V
6 4 V
6 5 V
6 6 V
6 5 V
6 6 V
6 6 V
6 6 V
6 7 V
6 7 V
6 7 V
6 7 V
6 8 V
6 9 V
6 9 V
6 9 V
6 10 V
6 10 V
6 11 V
6 12 V
6 12 V
6 13 V
6 14 V
6 14 V
5 16 V
6 16 V
6 17 V
6 19 V
6 19 V
6 21 V
6 22 V
6 24 V
6 25 V
6 27 V
6 29 V
6 31 V
6 33 V
6 35 V
6 38 V
6 41 V
6 43 V
6 47 V
6 51 V
6 54 V
6 58 V
6 63 V
6 68 V
6 73 V
6 78 V
6 85 V
6 91 V
6 98 V
6 105 V
6 113 V
5 120 V
6 129 V
6 137 V
6 145 V
6 153 V
6 160 V
6 166 V
6 172 V
6 175 V
6 176 V
6 174 V
6 169 V
6 161 V
6 149 V
6 133 V
6 113 V
6 91 V
6 66 V
6 38 V
6 12 V
6 -15 V
6 -41 V
6 -64 V
6 -85 V
6 -101 V
6 -116 V
6 -126 V
6 -133 V
6 -138 V
6 -140 V
5 -140 V
6 -138 V
6 -135 V
6 -131 V
6 -127 V
6 -121 V
6 -116 V
6 -110 V
6 -105 V
6 -99 V
6 -93 V
6 -88 V
6 -83 V
6 -78 V
6 -74 V
6 -69 V
6 -65 V
6 -62 V
6 -57 V
6 -55 V
6 -51 V
6 -48 V
6 -45 V
6 -43 V
6 -40 V
6 -38 V
6 -36 V
6 -34 V
6 -32 V
6 -30 V
5 -29 V
6 -27 V
6 -25 V
6 -25 V
6 -23 V
6 -22 V
6 -20 V
6 -20 V
6 -18 V
6 -18 V
6 -17 V
6 -16 V
6 -15 V
6 -14 V
6 -14 V
6 -13 V
6 -12 V
6 -12 V
6 -11 V
6 -10 V
6 -10 V
6 -10 V
6 -9 V
6 -8 V
6 -8 V
6 -8 V
6 -7 V
6 -7 V
6 -7 V
6 -6 V
5 -6 V
6 -5 V
6 -5 V
6 -5 V
6 -4 V
6 -5 V
6 -4 V
6 -3 V
6 -4 V
6 -3 V
6 -3 V
6 -2 V
6 -3 V
6 -2 V
6 -2 V
6 -2 V
6 -1 V
6 -2 V
6 -1 V
6 -1 V
6 -1 V
6 0 V
6 0 V
6 -1 V
6 0 V
6 1 V
6 0 V
6 1 V
6 0 V
6 1 V
6 1 V
5 2 V
6 1 V
6 2 V
6 1 V
6 2 V
6 3 V
6 2 V
6 2 V
6 3 V
6 3 V
6 3 V
6 3 V
6 3 V
6 3 V
6 4 V
6 4 V
6 4 V
6 4 V
6 4 V
6 5 V
6 4 V
6 5 V
6 5 V
6 5 V
6 5 V
6 6 V
6 5 V
6 6 V
6 6 V
6 6 V
5 7 V
6 6 V
6 7 V
6 7 V
6 7 V
6 7 V
6 7 V
6 8 V
6 8 V
6 8 V
6 8 V
6 8 V
6 8 V
6 9 V
6 9 V
6 9 V
6 9 V
6 9 V
6 10 V
6 9 V
6 10 V
6 10 V
6 10 V
6 10 V
6 11 V
6 10 V
6 11 V
6 11 V
6 11 V
6 11 V
5 11 V
6 11 V
6 12 V
6 11 V
6 12 V
6 11 V
6 12 V
6 12 V
6 12 V
6 11 V
6 12 V
6 12 V
6 12 V
6 12 V
6 12 V
6 12 V
6 11 V
6 12 V
6 12 V
6 11 V
6 12 V
6 11 V
6 11 V
6 11 V
6 11 V
6 10 V
6 11 V
6 10 V
6 10 V
6 10 V
5 9 V
6 9 V
6 9 V
6 8 V
6 9 V
6 7 V
6 8 V
6 7 V
6 7 V
6 6 V
6 6 V
6 6 V
6 5 V
6 5 V
6 4 V
6 4 V
6 3 V
6 3 V
6 3 V
6 2 V
6 2 V
6 1 V
6 1 V
6 1 V
6 0 V
6 -1 V
6 0 V
6 -1 V
6 -2 V
6 -2 V
5 -2 V
6 -3 V
6 -3 V
6 -3 V
6 -3 V
6 -4 V
6 -5 V
6 -4 V
6 -5 V
6 -5 V
6 -5 V
6 -5 V
6 -6 V
6 -6 V
6 -6 V
6 -6 V
6 -7 V
6 -6 V
6 -7 V
6 -7 V
6 -7 V
6 -7 V
6 -7 V
6 -7 V
6 -7 V
6 -7 V
6 -8 V
6 -7 V
6 -8 V
6 -7 V
5 -7 V
6 -8 V
6 -7 V
6 -7 V
6 -8 V
6 -7 V
6 -7 V
6 -8 V
6 -7 V
6 -7 V
6 -7 V
6 -7 V
6 -7 V
6 -7 V
6 -6 V
6 -7 V
6 -7 V
6 -6 V
6 -6 V
6 -7 V
6 -6 V
6 -6 V
6 -6 V
6 -6 V
6 -6 V
6 -5 V
6 -6 V
6 -5 V
6 -6 V
6 -5 V
6 -5 V
5 -5 V
6 -5 V
6 -4 V
6 -5 V
6 -5 V
6 -4 V
6 -4 V
6 -4 V
6 -4 V
6 -4 V
currentpoint stroke M
6 -4 V
6 -4 V
6 -3 V
6 -4 V
6 -3 V
6 -3 V
6 -4 V
6 -3 V
6 -2 V
6 -3 V
6 -3 V
6 -2 V
6 -3 V
6 -2 V
6 -2 V
6 -2 V
6 -2 V
6 -2 V
6 -2 V
6 -1 V
5 -2 V
6 -1 V
6 -2 V
6 -1 V
6 -1 V
6 -1 V
6 -1 V
6 -1 V
6 0 V
6 -1 V
6 0 V
6 0 V
6 -1 V
6 0 V
6 0 V
6 0 V
6 0 V
6 1 V
6 0 V
6 1 V
6 0 V
6 1 V
6 1 V
6 1 V
6 1 V
6 1 V
6 1 V
6 1 V
6 2 V
6 1 V
5 2 V
6 2 V
6 2 V
6 1 V
6 2 V
6 3 V
6 2 V
6 2 V
6 3 V
6 2 V
6 3 V
6 2 V
6 3 V
6 3 V
6 3 V
6 3 V
6 3 V
6 4 V
6 3 V
6 4 V
6 3 V
6 4 V
6 4 V
6 4 V
6 4 V
6 4 V
6 4 V
6 4 V
6 5 V
6 4 V
5 5 V
6 4 V
6 5 V
6 5 V
6 5 V
6 5 V
6 5 V
6 5 V
6 6 V
6 5 V
6 6 V
6 5 V
6 6 V
6 6 V
6 6 V
6 6 V
6 6 V
6 6 V
6 7 V
6 6 V
6 6 V
6 7 V
6 7 V
6 6 V
6 7 V
6 7 V
6 7 V
6 7 V
6 7 V
6 8 V
5 7 V
6 7 V
6 8 V
6 8 V
6 7 V
6 8 V
6 8 V
6 8 V
6 8 V
6 8 V
6 8 V
6 8 V
6 8 V
6 9 V
6 8 V
6 9 V
6 8 V
6 9 V
6 8 V
6 9 V
6 9 V
6 9 V
6 8 V
6 9 V
6 9 V
6 9 V
6 9 V
6 9 V
6 9 V
6 10 V
5 9 V
6 9 V
6 9 V
6 10 V
6 9 V
6 9 V
6 9 V
6 10 V
6 9 V
6 10 V
6 9 V
6 9 V
6 10 V
6 9 V
6 9 V
6 10 V
stroke
grestore
%insertion
gsave
59 487 translate
0.038 0.036 scale
90 rotate
0 -5040 translate
0 setgray
/Helvetica findfont 350 scalefont setfont
newpath
LTa
LTb
1200 501 M
63 0 V
5634 0 R
-63 0 V
-5700 0 R
(-100) Rshow
1200 1135 M
63 0 V
5634 0 R
-63 0 V
-5700 0 R
(-50) Rshow
1200 1769 M
63 0 V
5634 0 R
-63 0 V
-5700 0 R
(0) Rshow
1200 2403 M
63 0 V
5634 0 R
-63 0 V
-5700 0 R
(50) Rshow
1200 3037 M
63 0 V
5634 0 R
-63 0 V
-5700 0 R
(100) Rshow
1200 3671 M
63 0 V
5634 0 R
-63 0 V
-5700 0 R
(150) Rshow
1200 4305 M
63 0 V
5634 0 R
-63 0 V
-5700 0 R
(250) Rshow
1200 4939 M
63 0 V
5634 0 R
-63 0 V
-5700 0 R
%(250) Rshow
1200 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
(-5) Cshow
1770 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
(0) Cshow
2339 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
%(5) Cshow
2909 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
(10) Cshow
3479 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
%(15) Cshow
4049 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
(20) Cshow
4618 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
%(25) Cshow
5188 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
(30) Cshow
5758 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
%(35) Cshow
6327 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
(40) Cshow
6897 501 M
0 63 V
0 4375 R
0 -63 V
0 -4675 R
%(45) Cshow
1200 501 M
5697 0 V
0 4438 V
-5697 0 V
0 -4438 V
401 2720 M
currentpoint gsave translate 90 rotate 0 0 M
(V\(x\) [meV]) Cshow
grestore
4048 -200 M
(x [nm]) Cshow
6327 3291 M
(high energy peak) Rshow
6327 2530 M
(low energy peak) Rshow
LT6
1576 2289 M
4800 0 V
1576 3151 M
4800 0 V
LT0
gnulinewidth 9 mul setlinewidth
1576 1769 M
194 0 V
0 2801 V
193 0 V
0 -3921 V
290 0 V
0 2240 V
580 0 V
0 -1120 V
161 0 V
0 560 V
322 0 V
0 -1120 V
322 0 V
0 1120 V
451 0 V
0 -1120 V
450 0 V
0 1120 V
419 0 V
0 -1120 V
419 0 V
0 1120 V
451 0 V
0 -1120 V
354 0 V
0 560 V
193 0 V
LT6
1576 2289 M
4800 0 V
1576 3151 M
4800 0 V
stroke
grestore
end
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%%Trailer
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