Accepted for publication in Reports on Math. Phys. 1997.
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\title{Transmission coefficient for ballistic transport
through quantum resonator}
\author{V.A.Geyler$^1$, I.Yu.Popov$^2$, S.L.Popova$^2$\\
$^1$ Department of Mathematics,\\
Mordovian State University, 68 Bolshevistskaya,\\
Saransk, 430000, Russia, USSR.\\
$^2$ Department of Higher Mathematics,\\
Leningrad Institute of Fine Mechanics and Optics,\\
14 Sablinskaya, 197101 Leningrad, Russia, USSR.}
\begin{document}
\maketitle
\begin{abstract}
A model for the description of ballistic transport in 1D-2D-1D
system based on the theory of self-adjoint extensions of
symmetric operators is suggested. Formula for transmission
coefficient for the system, particurlarly, in the case of presense
of a magnetic field, is obtained.
\end{abstract}
\newpage
\section{Introduction}
While studying transport properties of low-dimensional
systems (particurlarly, nanoelectronic devices), one use,
usually, the approach based on Landauer-Buttiker theory ~\cite{L},
~\cite{B1}, ~\cite{B2}.
Following this approach, one obtains direct relation between
conductivity and transmission coefficient for ballistic channels
forming the system. Sufficiently general scheme of construction of
solvable model for nanoelectronic device with two contacts was
suggested in \cite{GP1}. General expression for the transmission
coefficient for the model is determined. The
basic idea is using zero-range potentials with internal structure
\cite{AGHH}, \cite{Pav}, \cite{Pav1}. The approach is a
generalization of some examples of the models which are
described in ~\cite{GMC}, ~\cite{PP}, ~\cite{GP}.
The expression obtained in \cite{GP1} is rather general and
contains some complicated mathematical constructions (Krein
functions etc.). That is why it is interesting, especially for
physicists, to consider concrete realization of this abstract
approach. It is the main goal of the present paper. Let us
describe briefly the general scheme.
Consider a nanoelectronic device with two leads, shown on figure
1.
The state space $\cal H$ divides into a direct sum
$$
{\cal H} = {\cal H}_- \oplus {\cal H}_d \oplus {\cal H}_+,
$$
where ${\cal H}_{\pm} = L^2 ({\bf R}_{\pm}), {\cal H}_d$ is the state
space of the device. At first, we need not the concrete form of
${\cal H}_d$
(as for as a form of the Hamiltonian $H_d$ which is introduced below).
If "the channel is closed", then the system Hamiltonian $H^0$ can be
represented in a form of a direct sum
$$
H^0 = H_- \oplus H_d \oplus H_+.
$$
Here $H_{\pm}$ is the operator $- \frac{d^2}{dx^2}$ in $L^2 ({\bf
R}_{\pm})$
with the Neumann boundary condition at the point 0 (it corresponds to
the
absence of current through contacts $\pm$), $H_d$ is the Hamiltonian
of a
charged particle in the device. Let us denote as $ R_- (z), R_d (z),
R_+ (z), R^0 (z)$ the resolvents of the operators $H_- , H_d ,H_+ ,
H^0$
correspondingly. It is evident that
$R^0 = R_- \oplus R_d \oplus R_+$. To switch on the interaction we use
conventional "restriction- extension" procedure ~\cite{Pav}.
Namely, we start from a self-adjoint operator, come to a
symmetric operator by a restriction. The obtained operator has
self-adjoint extensions which gives us the model in question. Note
that among these extensions there is initial self-adjoint operator,
but it corresponds to the case of absence of interaction.
Let $S_{\pm}$ be a symmetric operator being a restriction of the
operator
$H_{\pm}$ on the domain
$$
D(S_{\pm}) = \{\varphi \in D(H_{\pm}): \varphi (0) = 0 \}.
$$
Consider, simultaneously, a symmetric operator $S_d$ with the
deficiency
indices (2,2) such that $S_d = H_d \mid _{D(S_d )}.$ We shall identify
the deficiency elements of the operator with the contacts $\pm$ in the
device. A concrete form of the elements depends, naturally, on the
physical
interpretation of the space ${\cal H}_d$ and the Hamiltonian $H_d$.
In each case it is a subject of a special analysis. An example is
given
below. It is important to note that the operator $S_d$ may not be
densely defined. Particurlarly, if ${\cal H}_d$ is a finite-
dimensional
space, then $D(S_d )$, evidently, is not dense in ${\cal H}_d$
(examples
of such operators are in ~\cite{Pav}).
We shall use below the Krein formula for resolvents of self-adjoint
operators
~\cite{AGHH}. For this purpose let us introduce standard deficiency
subspaces:
$${\cal N}_- = {\cal N}_+ = {\bf C}, {\cal N}_d = {\bf C}^2.$$
A contact of the lead with the device "is realized" by means of the
basic
elements 1 and (1,0) in ${\cal N}_-$ and ${\cal N}_d$ correspondingly,
and a contact of ${\bf R}_+$ with the device- by means of the basic
elements
1 and (0,1) in ${\cal N}_+$ and ${\cal N}_d$.Introduce Krein $\Gamma -
$
and Q- functions for the operator $S_+ , S_d , S_-$. Denote as
$G_{\pm}(x,y;\zeta )$ the Green function of the operator $H_{\pm}$.
Then the map
$\Gamma _{\pm}(\zeta): {\cal N}_{\pm} \rightarrow L^2 ({\bf R}_{\pm})$
is defined by the formula
$$
{\bf C} \ni \xi \mapsto G_{\pm}(x,0;\zeta)\cdot \xi \in L^2 ({\bf
R}_{\pm}).
$$
Let $\Gamma _d (\zeta)$ and $Q_d (\zeta)$ be Krein $\Gamma -$
function and
Q- function of the operator $S_d$:
$$\Gamma_d (\zeta): {\bf C}^2 \rightarrow {\cal H}_d,
Q_d (\zeta): {\bf C}^2 \rightarrow {\bf C}^2 .$$
$\Gamma$-function and Q-function for the operator
$S, S = S_- \oplus S_d \oplus S_+$, is a direct sum
$$
\Gamma (\zeta) = \Gamma _- (\zeta) \oplus \Gamma_d (\zeta)
\oplus \Gamma _+ (\zeta),
$$
$$
Q(\zeta) = Q _- (\zeta) \oplus Q_d (\zeta) \oplus Q_+ (\zeta).
$$
The Hamiltonian H of the system with "open channels" is a self-
adjoint
extension of the operator S.
For the resolvent of the extension we have Krein formula \cite{KL}:
\begin{equation}\label{03}
R(\zeta) = R^0 (\zeta) - \Gamma (\zeta) (Q(\zeta) + A)^{-1}
\Gamma^*(\overline{\zeta}).
\end{equation}
The operator matrix $Q(z) + A$ has a structure:
\begin{equation}\label{05}
\left( \begin{array}{cccc}
\stackrel{~}{Q}_-(\zeta) & \alpha_- & 0 & 0 \\
\alpha_- & Q^{11}_d & Q^{12}_d & 0 \\
0 & Q^{21}_d & Q^{22}_d & \alpha_+ \\
0 & 0 & \alpha_+ & \stackrel{~}{Q}_+ (\zeta)
\end{array} \right)
\end{equation}
Here $\stackrel{~}{Q}_{\pm} (\zeta) = Q_{\pm} (\zeta) + \mu_{\pm},$
where $\alpha_-$ and $\alpha_+$ are values which
characterize the "ideal" contacts "-" and "+", value
$\mu_- (\mu_+)$ characterizes the degree of
difference between the contact "-" ("+") and the "ideal" one (for
instance, if $\mu_+ > 0$ then moving particle in the wire
$\bf{R}_+$ is
repulsed by the contact "+", and if $\mu_+ < 0$ then it is
attracted).
We shall write $Q_{ij}$ instead of $Q_{ij}^d$ for simplicity.
Using this approach, general formula for the transmission
coefficient $T(E)$ may be obtained (see \cite{GP1}):
\begin{equation}\label{15}
T(E) = \frac{\mid \alpha_-\alpha_+ Q_{12}(E) \mid ^2}
{E \mid \det (Q(E) + A) \mid ^2}
\end{equation}
In particular case of ideal contacts $\mu_{\pm} = 0$ one has:
\begin{equation}\label{16}
T(E) = \frac{E \alpha_-^2 \alpha_+^2 \mid Q^{21}_d (E) \mid ^2}
{E (\alpha_-^2 Q^{22}_d (E) + \alpha_+^2 Q^{11}_d (E))^2 +
(\det Q_d (E) - E \alpha_-^2 \alpha_+^2 )^2 }
\end{equation}
\section{Model of transport through quantum resonator}
Consider a system of two semi-infinite quantum waveguides which
are connected through a quantum dot (resonator $\Omega$). We
shall deal with so-called one-mode regime. In this situation
waveguides may be simulated by semi-axes ${\bf R}_-$ and
${\bf R}_+$ (fig. 1). Then an element from the domain of the
operator $S^*$ has the form:
\begin{equation}\label{1}
f(x) =
\left\{ \begin{array}{ll}
u_-(x), & x \in {\bf R}_-;\\
a_1 G(x,x_1;k_0) + a_2 G(x,x_2;k_0) + u(x), & x \in \Omega ;\\
u_+(x), & x \in {\bf R}_+,
\end{array} \right.
\end{equation}
where $G(x,y;k)$ is the Green function for the Neumann problem
in $\Omega$, $k_0^2$ is a regular value for the operator. A choice
of a self-adjoint extension is related with a choice of "boundary
conditions" at the points $x_1, x_2$. We shall consider one type
of the extensions only, namely, those used by Kiselev and Pavlov
\cite{KP} (in their work the choice of the extension is justified
by a comparison of the model solution with a solution of the problem
in which there are narrow channels instead of lines in the model).
The corresponding conditions are:
\begin{equation}\label{2}
\begin{array}{c}
a_1 = -\alpha u'_-(0), a_2 = \alpha u'_+(0),\\
u_-(0) = \alpha (f - a_1 G(x,x_1;k_0))\vert_{x=x_1},\\
u_+(0) = \alpha (f - a_2 G(x,x_2;k_0))\vert_{x=x_2},
\end{array}
\end{equation}
where $\alpha$ is a model parameter.
The Green function for the model self-adjoint operator with
a source at a point $x_0 \in \Omega$ can be sought in a form:
\begin{equation}\label{3}
G_{\alpha}(x,x_0;k) =
\left\{ \begin{array}{ll}
a_-i k^{-1}\exp{(-i k x)}, & x \in {\bf R}_-;\\
a_1 G(x,x_1;k) + a_2 G(x,x_2;k) + G(x,x_0;k), & x \in \Omega ;\\
a_+i k^{-1}\exp{(i k x)}, & x \in {\bf R}_+.
\end{array} \right.
\end{equation}
Taking into account (\ref{2}), one obtains
\begin{equation}\label{4}
\begin{array}{c}
a_1 = -\alpha a_-, a_2 = -\alpha a_+,\\
i k^{-1}a_- =\alpha (a_1g_1(k,k_0) + a_2 G(x_1,x_2;k) +
G(x_1,x_0;k)),\\
i k^{-1}a_+ =\alpha (a_2g_2(k,k_0) + a_1 G(x_1,x_2;k) +
G(x_2,x_0;k)),
\end{array}
\end{equation}
where
\begin{equation}\label{5}
\begin{array}{c}
g_i(k,k_0) = (G(x,x_i;k) - G(x,x_i;k_0))\vert_{x=x_i} =\\
(k^2 - k_0^2)\sum_{n}\vert\phi_n(x_i)\vert^2
(\lambda_n - k^2)^{-1}(\lambda_n - k_0^2)^{-1}, i = 1, 2,
\end{array}
\end{equation}
$\lambda_n$ is an eigenvalue of the Neumann problem for the
Laplace operator in $\Omega$ associated with the eigenfunction
$\phi_n$. System (\ref{4}) reduces to
$$
a_1 = i k \alpha^2 (a_1g_1(k,k_0) + a_2 G(x_1,x_2;k) +
G(x_1,x_0;k)),
$$
$$
a_2 = i k \alpha^2 (a_2g_2(k,k_0) + a_1 G(x_1,x_2;k) +
G(x_2,x_0;k)).
$$
Hence,
$$
a_1 = D_{-1}(-i k\alpha^2G(x,x_1;k)(i k\alpha^2 g_2(k,k_0) - 1)-
k^2\alpha^4 G(x_1,x_2;k) G(x_2,x_0;k),
$$
$$
a_2 = D_{-1}(-i k\alpha^2G(x,x_2;k)(i k\alpha^2 g_1(k,k_0) - 1)-
k^2\alpha^4 G(x_1,x_2;k) G(x_1,x_0;k),
$$
where
$$
D = (i k\alpha^2 g_1(k,k_0) - 1)(i k\alpha^2 g_2(k,k_0) - 1) +
k^2\alpha^4 (G(x_1,x_2;k))^2.
$$
Therefore, the corresponding part of the matrix
$(Q(\zeta) + A)^{-1}$ in the Krein formula has the form:
$$
D^{-1}
\left(\begin{array}{cc}
-i k\alpha^2(i k\alpha^2 g_1(k,k_0) - 1) & k^2\alpha^4 G(x_1,x_2;k)\\
k^2\alpha^4 G(x_1,x_2;k) & -i k\alpha^2 (i k\alpha^2 g_2(k,k_0) - 1)
\end{array}\right).
$$
To obtain other blocks of the matrix it is necessary to
consider a situation when the source $x_0$ is in ${\bf R}_{\pm}$.
Let $x_0 \in {\bf R}_-.$ Then the Green function has the form
$$
G_{\alpha}(x,x_0;k) =
\left\{ \begin{array}{ll}
a_-i k^{-1}\exp{(-i k x)} + G_-(x,x_0;k), & x \in {\bf R}_-;\\
a_1 G(x,x_1;k) + a_2 G(x,x_2;k), & x \in \Omega ;\\
a_+i k^{-1}\exp{(i k x)}, & x \in {\bf R}_+.
\end{array} \right.
$$
Hence, system (\ref{4}) is replaced by
$$
\begin{array}{c}
a_1 = -\alpha a_-, a_2 = -\alpha a_+,\\
i k^{-1}(a_- + \exp{(-i k x_0)}) =
\alpha (a_1g_1(k,k_0) + a_2 G(x_1,x_2;k)),\\
i k^{-1}a_+ =\alpha (a_2g_2(k,k_0) + a_1 G(x_1,x_2;k)).
\end{array}
$$
Consequently,
$$
a_- = -\exp{(-i k x_0)} - i k\alpha (a_1g_1(k,k_0) + a_2
G(x_1,x_2;k)),
$$
$$
a_1 (i k\alpha^2 g_1(k,k_0) - 1) +
i k\alpha^2 a_2 G(x_1,x_2;k) = \alpha \exp{(-i k x_0)},
$$
$$
a_2 (i k\alpha^2 g_2(k,k_0) - 1) +
i k\alpha^2 a_1 G(x_1,x_2;k) = 0.
$$
Thus,
$$
a_1 = D^{-1} (i k\alpha^2 g_2(k,k_0) - 1)\alpha \exp{(-i k x_0)},
$$
$$
a_2 = -D^{-1} i k\alpha^3 \exp{(-i k x_0)} G(x_1,x_2;k),
$$
$$
a_- = - \exp{(-i k x_0)}(2 + D^{-1} (i k\alpha^2 g_2(k,k_0) - 1)).
$$
As a result, one obtains the following expression for the
corresponding (1x1) matrix block:
$$
2 + D^{-1} (i k\alpha^2 g_2(k,k_0) - 1)).
$$
The block corresponding to ${\bf R}_+$ may be obtained in a
similar way.
The model for the case of the Dirichlet condition at the
boundary of $\Omega$ is more complicated. In this situation we
have to deal with the operator extension theory in an
indefinite metrics space. As for details, see \cite{P}, \cite{P1}.
Nevertheless, the structure of the model is analogous to one
described above. We must make the following alterations.
Representation (\ref{1}) should be replaced by
\begin{equation}\label{6}
f(x) =
\left\{ \begin{array}{ll}
u_-(x), & x \in {\bf R}_-;\\
a_1 \frac{\partial G_D(x,x_1;k_0)}{\partial n} +
a_2 \frac{\partial G_D(x,x_2;k_0)}{\partial n} + u(x), & x \in
\Omega ;\\
u_+(x), & x \in {\bf R}_+,
\end{array} \right.
\end{equation}
where $G_D(x,y;k)$ is the Green function for the Dirichlet problem
in $\Omega$, $n$ is a normal. Further, system (\ref{4}) slightly
changes:
\begin{equation}\label{7}
\begin{array}{c}
a_1 = \alpha a_-, a_2 = \alpha a_+,\\
i k^{-1}a_- =-\alpha (a_1g_1^D(k,k_0) +
a_2 \frac{\partial G_D(x_1,x_2;k)}{\partial n} +
\frac{\partial G_D(x_1,x_0;k)}{\partial n},\\
i k^{-1}a_+ =-\alpha (a_2g_2^D(k,k_0) +
a_1 \frac{\partial G_D(x_1,x_2;k)}{\partial n} +
\frac{\partial G_D(x_2,x_0;k)}{\partial n},
\end{array}
\end{equation}
where
\begin{equation}\label{8}
\begin{array}{c}
g_i^D(k,k_0) =
(k^2 - k_0^2)\sum_{m}\vert\frac{\partial\phi_m^D(x_i)}{\partial
n}\vert^2
(\lambda_m - k^2)^{-1}(\lambda_m - k_0^2)^{-1}, i = 1, 2,
\end{array}
\end{equation}
$\lambda_m$ is an eigenvalue of the Dirichlet problem for the
Laplace operator in $\Omega$ associated with the eigenfunction
$\phi_m^D$. The following treatment is analogous to the previous one.
\section{The system in a magnetic field}
The model allows one to consider two-dimensional system in
a constant magnetic field ${\bf B}$ which is orthogonal
to the system plane.
Here we must deal with the Green function for the Landau operator
$$
H = \frac{1}{2 m}({\bf p} - \frac{e}{c}{\bf A})^2.
$$
Here $m, e$ are mass and charge of an electron correspondingly,
$c$ is the speed of light, ${\bf p} = -i \hbar \nabla, {\bf A}
= 2^{-1}{\bf B} {\rm x} {\bf r}$. Let $\Omega$ be a disk $\Omega =
\left\{ (r,\varphi): r \leq r_0 \right\}.$ Introduce the followimg
notation
$$
\omega = \frac{\vert e {\bf B}\vert}{c m},
\lambda = \left(\frac{\hbar c}{\vert e {\bf B}\vert}\right)^{1/2},
x_0 = \frac{r_0}{2 \lambda^2}.
$$
The spectrum of the operator $H$ with the Dirichlet boundary
conditions consists of the eigenvalues $E_{ln}$,
$$
E_{ln} = \hbar \omega (2^{-1}(l +\vert l \vert + 1) -
\epsilon_{ln}), l = 0, \pm 1, \pm 2, ..., n = 1, 2, ...,
$$
$\epsilon_{ln}$ is a root of the equation
$\Phi (\epsilon,\vert l \vert + 1,x_0) = 0.$ Here $\Phi (a,c,x) =
M(a,c,x)$ is the Kummer function. The corresponding
eigenfunction is
$$
\Psi_{ln}(r,\varphi) =
\left( \frac{m \omega}{2\pi\hbar c_{ln}}\right)^{1/2}
\frac{r^{\vert l \vert}}{(2\lambda^2)^{\vert l \vert/2}}
\exp{(i l \varphi - \frac{r^2}{4\pi \lambda^2})}
\Phi (\epsilon_{ln},\vert l \vert + 1,\frac{r^2}{2\lambda^2}),
$$
where $c_{ln}$ is normalization constant,
$$
c_{ln} = \int^{x_0}_{0} \exp{(-x)} x^{\vert l \vert}
(\Phi (\epsilon_{ln},\vert l \vert + 1,x))^2 d x.
$$
The Green function has a form
$$
\begin{array}{c}
G(r,\varphi,r',\varphi';\zeta) =
\frac{m \omega}{2\pi\hbar}\exp{(-\frac{r^2 + r'^2}{4\lambda^2})}\\
\sum^{\infty}_{l=-\infty} \frac{(r r')^{\vert l \vert}}
{(2\lambda^2)^{\vert l \vert}}\exp{(i l (\varphi - \varphi')}\\
\sum^{\infty}_{n=1}\frac
{\Phi (\epsilon_{ln},\vert l \vert + 1,\frac{r^2}{2\lambda^2})
\Phi (\epsilon_{ln},\vert l \vert + 1,\frac{r'^2}{2\lambda^2})}
{c_{ln}(E_{ln} - \zeta)}.
\end{array}
$$
One sum can be calculated. Let
$$
F_l(s,t,\zeta) = \left\{
\begin{array}{c}
\Phi (\zeta,\vert l \vert + 1,s)(\Psi(\zeta,\vert l \vert +1,t) -
\alpha \Phi (\zeta,\vert l \vert + 1,t)), s \leq t;\\
\Phi (\zeta,\vert l \vert + 1,t)(\Psi(\zeta,\vert l \vert +1,s) -
\alpha \Phi (\zeta,\vert l \vert + 1,s)), t \leq s.
\end{array} \right.
$$
Here
$$
\alpha = \frac{\Psi(\zeta,\vert l \vert +1,x_0)}
{\Phi (\zeta,\vert l \vert + 1,x_0)}, \zeta \neq \epsilon_{ln},
$$
$\Psi(a,c,x)$ is the Trikomi function. Then the Green function
takes the form
$$
\begin{array}{c}
G(r,\varphi,r',\varphi';\zeta) =
\frac{m \omega}{2\pi\hbar}\exp{(-\frac{r^2 + r'^2}{4\lambda^2})}\\
\sum^{\infty}_{l=-\infty} \frac{(r r')^{\vert l \vert}}
{(2\lambda^2)^{\vert l \vert}}\exp{(i l (\varphi - \varphi')}\\
\Gamma(\frac{1 + l + \vert l \vert}{2}-\frac{\zeta}{\hbar \omega})
F_l(\frac{r^2}{2\lambda^2},\frac{r'^2}{2\lambda^2},
\frac{1 + l + \vert l \vert}{2}-\frac{\zeta}{\hbar \omega}).
\end{array}
$$
Substitution of this expression into the formula for the
transmission coefficient (\ref{16}), described above gives us
the required result.
\section{Conclusion}
Quantum waveguide-resonator-waveguide system is considered.
Solvable model based on the theory of self-adjoint extensions
of symmetric operators is suggested. The model is concrete
realization of general construction suggested in \cite{GP1}
and its composition with so-called zero-width slit model.
For proper choice of the model parameters we use the results
of \cite{KP}.
The advantage is that the approach allows one to reveal some
general features of the physical system. We construct model
self-adjoint operator which is similar (in some sense) to that
for physical system. Green function, transmission coefficient,
etc. for the model operator is obtained in an explicit form.
As a result, one can get the dispersion equation and investigate
effectively spectral properties. The model is constructed both
for the case of absence and presence of a magnetic field.
The latter is especially interesting for physical applications,
namely, for the investigation of transport properties of
quantum dots.
\section{Acknowledgements}
The work is financially supported by the
Commission of the European Community under EC-Russia
Collaboration Contract ESPRIT P9282
ACTCS, by grants No 95-01-00439, No 96-01-00074 of RFBR,
Soros Foundation and ANS RF.
\newpage
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\end{document}