#1
N.M. Chtchelkatchev$^{\ast}$
Conductance $G$ of a 2DEG-Superconductor (S) device in a high magnetic field is studied: $G(\nu)$ is calculated. When the cyclotron diameter in 2DEG is larger than the width of the 2DEG-S surface then $G(\nu)$ becomes nonmonotonous function due to the Aharonov--Bohm type interference of quasiparticles at the surface. At certain parameters of the junction the conductance oscillates with $\nu$.
article.tex
\ifx\jetpl\undefined
  \typeout{}
  \typeout{>>>JETPL macros 30.08.00 (version for outer use)<<<}
  \def\jetpl{JETPL}
 \else
  \typeout{}
  \expandafter\endinput
 \fi

\documentclass[twoside]{article}

%-------------- General format----------------------------------------------

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%----------------------------------------------------------------------------

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%\def\rus{\Russian}
%\def\lat{\English}
\def\rus{\language=1 \lefthyphenmin=2 \righthyphenmin=2}
\def\lat{\language=0 \lefthyphenmin=2 \righthyphenmin=3}

\usepackage{latexsym}
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%---------------Maket macros--------------------------------------------------

\def\tom#1{\def\tomnum{#1}}
\def\tomnum{}

\def\maketdate#1{\def\issdate{\mdata #1 }}
\def\issdate{}

\newcount\countart
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\newwrite\sodf
\newif\if@maket \@maketfalse
\newif\if@vypusk
\newif\if@vypfp

\def\maket{\@makettrue
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    JETP Letters, Vol. 73, No. 2, 2001, pp. 94�97. }
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%------------Macros to include articles into maket-----

\def\artrus#1{\rus\artinc{#1}}
\def\artlat#1{\lat\artinc{#1}}

\newif\if@brp \@brptrue
\def\podbor{\@brpfalse}

\def\artinc#1{\def\artname{#1}
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%------------Macros to write contents-----------
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\newif\if@filesod \@filesodfalse

\def\thlat{\lat\thart}
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   \if@maket \immediate\write\@sodfile{\string\def\string\pacs\artname{#1}}
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\def\@wrthart#1{\let\thepage\relax
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 \immediate\write\@sodfile{ ���.\vypnum}
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\def\vypusksod#1{\if@vypusk \endvypusk \else \clearpage \fi
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%------------Page style macros------------------

\newcount\markpage
\newcount\relpage
\newcount\firstpage
\newcount\marknum
\newcount\markcount

\marknum=32
\def\startpage#1{\setcounter{page}{#1}
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\def\firstmark#1{\global\markpage=#1
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\def\marktext{������ � ����, ��� \tomnum, ���.\ \vypnum}
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\def\@startvyptext{\vspace*{-12mm}\begin{tabbing}
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          ����� ������ � ����������,\\ \>�������� ���������� �������
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    \def\inimark{ \if@maket
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\ps@myfooters\pagenumbering{arabic}\onecolumn

%---Macros to write text in three columns--------

\def\begintricolumns{\begingroup

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    \pagewidth=\textwidth \pageheight=\textheight

    \def\fullline{\hbox to\pagewidth}
    \newbox\partialpage

    \parindent=0pt
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    \baselineskip=10ptplus.2ptminus.2pt

    \output={\global\setbox\partialpage=\vbox{\unvbox255\bigskip}}\eject
    \output={\tricolumnout}
    \divide\hsize by 3 \advance\hsize by -1.5mm
    \multiply\vsize by 3}

\def\endtricolumns{\output={\balancetricolumns}\eject
  \endgroup \pagegoal=\vsize}

%\def\endtricolumns{\paagesofartricol\eject\endgroup \pagegoal=\vsize}

\def\tricolumnout{\splittopskip=\topskip \splitmaxdepth=\maxdepth
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  \setbox0=\vsplit255 to\dimen@
  \setbox1=\vsplit255 to\dimen@
  \setbox2=\vsplit255 to\dimen@
  \onepageout\paagesofartricol
  \unvbox255 \penalty\outputpenalty}

\def\paagesofartricol{\unvbox\partialpage
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  \hbox to\pagewidth{\box0\hfil\box1\hfil\box2}}

\def\balancetricolumns{\setbox0=\vbox{\unvbox255} \dimen@=\ht0
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  \divide\dimen@ by3 \splittopskip=\topskip

  {%\vbadness=10000
    \global\setbox5=\copy0
    \global\setbox4=\vsplit5 to\dimen@
    \global\setbox3=\vsplit5 to\dimen@}

  \setbox1=\vbox to\dimen@{\unvbox3}           %%%%%%%%%%% ??????????
  \setbox0=\vbox to\dimen@{\unvbox4}           %%%%%%%%%%% ??????????
  \setbox2=\vbox to\dimen@{\unvbox5}
  \paagesofartricol}

\def\onepageout#1{\begingroup
    \ifodd\count\z@
      \let\@thehead\@oddhead \let\@thefoot\@oddfoot
      \let\@themargin\oddsidemargin
    \else \let\@thehead\@evenhead
       \let\@thefoot\@evenfoot \let\@themargin\evensidemargin
    \fi
    \shipout\vbox{\vbox to \pageheight{#1 % now insert the main information
      \boxmaxdepth=\maxdepth
      } % this completes the \vbox to \pageheight
      \baselineskip\footskip
      \hbox to\textwidth{\@thefoot}}
    \endgroup\stepcounter{page}}

%---Macros to write text in two columns--------

\def\begintwocolumns{\begingroup

    \newdimen\pagewidth \newdimen\pageheight
    \pagewidth=\textwidth \pageheight=\textheight

    \def\fullline{\hbox to\pagewidth}
    \newbox\partialpage

    %\parindent=0pt
    %\raggedright
    %\maxdepth=2.2pt
    %\baselineskip=10ptplus.2ptminus.2pt

    \output={\global\setbox\partialpage=\vbox{\unvbox255\bigskip}}\eject
    \output={\twocolumnout}
    \divide\hsize by 2
    \multiply\vsize by 2}

\def\endtwocolumns{\output={\balancetwocolumns}\eject
  \endgroup \pagegoal=\vsize}

\def\twocolumnout{\splittopskip=\topskip \splitmaxdepth=\maxdepth
  \dimen@=\pageheight \advance\dimen@ by-\ht\partialpage
  \setbox0=\vsplit255 to\dimen@
  \setbox2=\vsplit255 to\dimen@
  \onepageout\paagesofartwocol
  \unvbox255 \penalty\outputpenalty}

\def\paagesofartwocol{\unvbox\partialpage
  \wd0=\hsize \wd2=\hsize
  \hbox to\pagewidth{\box0\hfill\box2}}

\def\balancetwocolumns{\setbox0=\vbox{\unvbox255} \dimen@=\ht0
  \advance\dimen@ by \topskip \advance\dimen@ by-\baselineskip
  \divide\dimen@ by2 \splittopskip=\topskip

  {\vbadness=10000
    \global\setbox3=\copy0
    \global\setbox1=\vsplit3 to\dimen@}

  \setbox0=\vbox to\dimen@{\unvbox1}
  \setbox2=\vbox to\dimen@{\unvbox3}
  \paagesofartwocol}

% \onepageout macro the same as for 3 columns

% Index macros

\newcount\avcount
\def\sa{0}
\def\NewLtS#1{\vglue .5cm plus .3cm minus .3cm}

\def\AventS#1#2#3#4{\def\sb{#4}
  \egtrm\par#1 {\ninbf #2}, #3\ifx\sa\sb\else\hskip1mm(\sb)\fi\avcount=1}

\def\AventfS#1#2#3#4{\def\sb{#4}
  \egtrm #1 {\ninbf #2}, #3\ifx\sa\sb\else\hskip1mm(\sb)\fi\avcount=1}

\def\SAventS#1#2#3{\egtrm\def\sb{#3}
  \ifnum\avcount=1\hskip-1.5mm;\hfill\break\else\hskip-1.5mm; \fi
  {\ninbf #1},~#2\ifx\sa\sb\else\hskip1mm(\sb)\fi\avcount=0}

% ����� ������� ��� �������� ���������� ���������
\def\AvtVypi#1#2#3#4{\def\sb{#4}
  \egtrm\par#1 {\ninbf #2}, #3\ifx\sa\sb\else\,(\sb)\fi}
\def\AvtVypic#1#2#3#4{\AvtVypi{#1}{#2}{#3}{#4};}
\def\AvtVypiii#1#2#3#4{\egtrm\def\sb{#4}
   {\ninbf #2},~#3\ifx\sa\sb\else\,(\sb)\fi}
\def\AvtVypii#1#2#3#4{\hfill\break\AvtVypiii{#1}{#2}{#3}{#4}}
\def\AvtVypiic#1#2#3#4{\AvtVypii{#1}{#2}{#3}{#4};}
\def\AvtVypiiic#1#2#3#4{\AvtVypiii{#1}{#2}{#3}{#4};}

\def\indexheading#1{\twlbf\hfill #1 \hfill\break\vskip 5mm}
%  \hbox to\pagewidth

% Theme index macros
\newcommand{\NewTL}[2]{\section*{\twlbf#1 #2}}
\newcommand{\NewTM}[2]{\vspace{-1.2mm}\subsection*{\elvbf#1 #2}}
\newcommand{\NewTS}[2]{\vspace{-1.2mm}\subsubsection*{\tenbf#1 #2}}
\newcommand{\NewTT}[2]{\vspace{-1.2mm}\subsubsection*{\tenbf#1 #2}}
\newcommand{\ArtentL}[4]{\par\lat\tenrm{~#3 -- #4 {\bf#1}, #2.}\pagebreak[0]}
\newcommand{\ArtentR}[4]{\par\rus\tenrm{~#3 -- #4 {\bf#1}, #2.}\pagebreak[0]}
\newcommand{\Artent}[4]{\par\tenrm{~#3 -- #4 {\bf#1},~#2.}\pagebreak[0]}

% ����� ������� ���������������� ���������
\def\IndLargeHed#1#2{\vspace{-1mm}\NewTM{#1}{#2}\vspace{1mm}}
\def\IndSmallHed#1#2{\NewTS{#1}{#2}}
\def\IndArt#1#2#3#4{\Artent{#2}{#3}{#1}{#4}}

% Author index macros
\def\enpage#1{\def\sb{#1}%
\ifx\sa\sb\else\hskip1mm(\sb)\fi}
%\newcommand{\mpar}{\hspace*{5mm}}
\newcommand{\mpar}{\par\hspace*{5mm}}
\newcommand{\AventFB}[6]{\mpar{\bf #1} -- #5; #6 {\bf#2},~#3\enpage{#4}.}
\newcommand{\AventNB}[5]{\mpar{\bf #1} -- ��.\ #5 {\bf#2},~#3\enpage{#4}.}
\newcommand{\SAventFB}[5]{\mpar\rule[.1cm]{3mm}{.03cm} #4; #5
{\bf#1},~#2\enpage{#3}.}
\newcommand{\SAventNB}[4]{\mpar\rule[.1cm]{3mm}{.03cm} ��. #4
{\bf#1},~#2\enpage{#3}.}
\newcommand{\NewLtB}[1]{\vspace*{-2mm}\section*{\centerline{\large\bf #1}}}
%\newcommand{\NewLtB}[1]{}

%------------------Figure macros------------------

%%%%%%
% \long\def\@caption#1#2{%
%   \if@side\else\par\fi%
%   \begingroup%
%     \@parboxrestore%
%     \if@side%
%       \addtolength{\hsize}{-\flwid}%
%     \fi%
%     \footnotesize %\normalsize
%   \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #2}%
%   \if@side\else\par\fi%
% \endgroup%
% }
%%%%%%

%\newcounter{figure}
%\def\thefigure{\@arabic\c@figure}

%%%%%%
% \def\fps@figure{tbh}
% \def\ftype@figure{1}
% \def\ext@figure{lof}
% \def\fnum@figure{Ris.~\thefigure}
% \def\figure{\@float{figure}}
% \newlength{\flwid}
% \newlength{\flh}
% \newif\if@side
% \newsavebox{\@capbox}
% \def\@sizedef#1#2{
%   \setlength{\flwid}{#1}
%   \setlength{\flh}{#2}
%   \ifdim #1>.8\textwidth
%     \@sidefalse
%   \else
%     \@sidetrue
%   \fi
% }
%
% \def\risunok#1#2#3{
%   \@sizedef{#1}{#2}%
%   %\@float{figure}
%   \@float{figure}[htb]
%   \@podpis{#3}%
%   \end@float%
% }
% \def\@podpis#1{
%   \refstepcounter\@captype
%   \setbox\@capbox\vbox{\@caption{\@captype}{#1}}
%   \if@side
%     \ifdim \ht\@capbox > \flh
%       \@sidefalse
%     \fi
%   \fi
%   \if@side
%     \leavevmode
% %\hbox to 0 pt{\lower\flh\vbox to \flh{}}\hskip\flwid\lower\ht\@capbox\box\@capbox
%  \hbox to 0 pt{\vbox to \flh{}}\hskip\flwid\raise\dp\@capbox\box\@capbox
%   \else
%     \vskip\flh
%     \@caption{\@captype}{#1}
%   \fi
% }
% %\let\endrisunok\end@float
% \let\endfigure\end@float
% \@namedef{figure*}{\@dblfloat{figure}}
% \@namedef{endfigure*}{\end@dblfloat}
%%%%%%

%%%%%% inserted for outer use
\renewcommand{\figurename}{���.}

\long\def\@makecaption#1#2{%
  \vskip\abovecaptionskip
  \sbox\@tempboxa{#1. #2}%
  \ifdim \wd\@tempboxa >\hsize
    \unhbox\@tempboxa\par
  \else
    \global \@minipagefalse
    \hbox to\hsize{\hfil\box\@tempboxa\hfil}%
  \fi
  \vskip\belowcaptionskip}

\long\def\@caption#1[#2]#3{\par\addcontentsline{\csname
  ext@#1\endcsname}{#1}{\protect\numberline{\csname
  the#1\endcsname.}{\ignorespaces #2}}\begingroup
    \@parboxrestore
    \normalsize
    \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par
  \endgroup}
%%%%%% till here

%-------- Article titles -------------

% try to uppercase title reasonably
\newtoks\@title \newcount\upc@mark \newif\ifupc@cont
\def\upc@part#1\noup#2#3\upc@end{%
\edef\upc@{#1}\uppercase\expandafter{\upc@}%
\ifx#2\upc@mark\upc@contfalse\else#2\@title={#3}\fi}
\def\upc@title{\upc@conttrue%
\loop\ifupc@cont\expandafter\upc@part\the\@title\upc@end\repeat}
\def\noup{\relax}

\newcommand{\zaglavie}[4]{\begin{list}{}{\topsep0pt\rightmargin0.5cm\leftmargin\rightmargin
   \partopsep0pt\item[]}
   \par \vskip 1 mm \begin{center}
   {\bf #1}
   \vskip 1 mm {\sl #2}
   \par\noindent \noindent{\footnotesize\sl #3}
   \vskip 1mm {\footnotesize
     \ifnum\language=0 Submitted #4 \else  ��������� � �������� #4 \fi}
 \end{center} \end{list} }

\def\Zaglavie#1#2#3#4{\title{#1}\zaglavie{\upc@title}{#2}{#3}{#4}}

\def\title#1{\@title={#1\noup{\upc@mark}}}
\title{{\tt$\backslash$title} --- ?????}
\def\@author{{\tt$\backslash$author} --- ?????}
\def\thanks#1{\footnote{#1}}
\def\and{\unskip,~}
\def\@instit{{\tt$\backslash$address} --- ?????}
\def\address#1{\def\@instit{#1}}
\def\@date{{\tt$\backslash$date} --- ?????}

\def\maketitle{\shapka\par\noindent%
        \zaglavie{\upc@title}{\@author}{\@instit}{\@date}}

%---------------Other macros--------------------

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\begin{document}
\sod{Conductance of a Semiconductor-Superconductor junction in high magnetic field}
{Chtchelkatchev N.M.}

\title{Conductance of a Semiconductor(2DEG) - Superconductor junction in high magnetic field}
\author{N.M. Chtchelkatchev$^{\ast}$}
\address{$^{\ast}$L.D.Landau Institute for Theoretical Physics RAS,
117940 Moscow, Russia \\ ~~ \\ e-mail: nms@landau.ac.ru \\}
\date{December 14, 2000}

\maketitle

\begin{abstract}
Conductance $G$ of a 2DEG-Superconductor (S) device in a high magnetic field is studied: $G(\nu)$ is calculated. When the
cyclotron diameter in 2DEG is larger than the width of the 2DEG-S surface then $G(\nu)$ becomes  nonmonotonous function due
to the Aharonov--Bohm type interference of quasiparticles at the surface.  At certain parameters of the junction the
conductance oscillates with $\nu$.
\end{abstract}

\PACS{74.80.Fp, 71.70.Di, 73.20.-r, 73.40.-c}

In recent years, the study of hybrid systems consisting of superconductors in contact with normal metals in strong magnetic
field has attracted considerable interest \cite{Akasaki} - \cite{Asano}. Investigation of physical phenomena in S-2DEG
devices in high magnetic field may help to establish a link between mesoscopic superconductivity and quantum - Hall physics.
It was found experimentally \cite{Akasaki} that zero-bias conductance $G$ of a ballistic S - 2DEG - S junction in Integer
Quantum Hall (IQH) regime exhibits quantization under variation of magnetic field. The quantum of $G$ was not equal to a
universal value in this experiment, as for instance in IQH or in a quantum point contact \cite{Glazman}, but it was an
oscillating function of the field $H$. Numerical simulations \cite{Takagaki}, \cite{Asano} showed that the conductance of a
2DEG-S contact in IQH regime is a nonmonotonous function of the filling factor $\nu$; there is nonuniversal quantization of
$G$  when 2DEG-S boundary is perfect \cite{expl_1}; at specific range of magnetic field  $G(\nu)$  oscillates. A
phenomenological theory of the conductance oscillations was suggested in  \cite{Asano}. But, it is still unclear when the
conductance becomes sensitive to $H$, why it exhibits oscillations, how one can analytically describe $G(H)$.  The
analytical form of $G(\nu)$ is found in this paper. It is shown  that the conductance becomes sensitive to $H$ when
$2R_c\gtrsim L$, where $R_c$ is a cyclotron radius in 2DEG, $L$ characterizes the length of the 2DEG-S boundary;
nonlinearities of $G(\nu)$ result from Aharonov--Bohm type interference of quasiparticles at the boundary.

We consider a junction consisting of a superconductor, 2DEG  and a normal conductor segments (see Fig.1). Magnetic field $H$
is applied along  $z$ direction, perpendicular to the plain of 2DEG. It is supposed that quasiparticle transport is
ballistic (the mean free path of an electron $l_{tr}\gg L$, where $L$ is the length of the 2DEG-S boundary). The current $I$
is supposed to flow between normal (N) and superconducting (S) terminals (the voltage $V$ is applied between them). The
conductance $G(H,L)=I/V,V\to 0$ is studied in the paper.

Following \cite{Takane_Ebisava}, we shall describe  transport properties of the junction in terms of electron and hole
quasiparticle scattering states, which satisfy Bogoliubov-de Gennes (BdG) equations. Then the conductance
\begin{gather}
\label{Takane_Ebisava} G=\left.\frac{\partial I}{\partial V}\right|_{V\to
0}=\frac{2e^2}{h}\sum_{l_o,n_i}R_{he,l_on_i}=\frac{4e^2}{h}{\cal R},
\end{gather}
where  $R_{he,l_on_i}$ is  probability of  Andreev reflection of an electron with the energy $E=0$ (with the respect to
$E_f$) incident on the superconductor in the channel with quantum number $n_i$ to a hole going from the superconductor in
the channel $l_o$.
\begin{figure}[htb]
\epsfxsize=130mm  \epsffile{fig1.eps} \caption{The device, which we investigate, consists of a superconductor, 2DEG and a
normal conductor. An electron injected from the normal conductor in IQH regime goes through an edge state to the
superconductor, reflects into a hole and an electron  which return to the normal contact through the other edge states.}
\end{figure}
Before explicit calculation of $G$, we shall discuss on a qualitative level how $G$ should depend on $H$. When $H$ is small
($R_c\ll L$) then $R_{he,l_on_i}\simeq R_{he}\delta_{l_o,n_i}$, with $R_{he}$ weakly depending on $H$. So,
\begin{gather}
\label{H_eq_0} {\cal R}\simeq R_{he}N, \; N=Lp_f/2\pi.
\end{gather}
If $2R_c\gtrsim L$, quasiparticles reflected from the superconductor (S) due to normal and Andreev reflection of the
electron  return to  S again due to bending of the trajectories by magnetic field.
\begin{figure}[htb]
\epsfxsize=130mm \epsffile{fig2.eps} \caption{ A quasiparticle wave in quasiclassical approximation can be treated as a
beam of rays.  The figure displays reflection of a ray (solid curve), corresponding to an electron injected from N, from the
superconductor. Dashed curves correspond to hole rays.}
\end{figure}
Then  $G(H)$ dependence is not  weak. We
shall investigate the conductance using semiclassical approximation when $\nu\gg 1$. An electron (hole) quasiparticle in
2DEG can be viewed in semiclassics as a  beam of rays (in a similar way propagation of light is described by beams of rays
in classical optics \cite{Born}). Trajectories of the quasiparticle rays can be found from the equations of classical
mechanics. If $R_c\gg L$ ($\nu\ll L/\lambda_f$), the edge states at  2DEG  do not overlap. Then the quantum numbers
$n_o,l_i$ (\ref{Takane_Ebisava}) of the incident electron and reflected hole correspond to the edge states. Reflection of an
electron from the superconductor is schematically shown in Fig.1. The electron ray (Fig.2) reflects into electron and hole
rays from S at $y_0$.  These rays reflect into other hole and electron rays at $y_1$. So, two hole and two electron rays
propagate between $y_2$ and $y_3$. Then eight hole rays come from S to N propagating along the same hole path beginning at
$y_3$.  One can approximate the probability $P$ of this hole path by the expression:
\begin{gather}
\label{estimate} P(y_0,n_i,n_o)\simeq
\left|r_{he}r_{ee}r_{ee}r_{ee}e^{3iS_e-i3\pi/2}+r_{hh}r_{he}r_{ee}r_{ee}e^{iS_h+2iS_e-i\pi/2}+\ldots\right|^2,
\end{gather}
where $r_{ba}$ is the amplitude of  reflection of a quasiparticle $a$ into a quasiparticle $b$ from the superconductor.
$S_{e(h)}$ is the quasiclassical action of an electron (hole) taken along the part of the trajectory connecting adjacent
points of reflection. Then $R_{eh,l_o n_i}\simeq \langle P(y_0)\rangle \delta_{n_i,l_0}$, where the average is taken over
$0<y_0<d(n_i)$, with $d$ being the length of the quasiparticle "jump" along the edge of 2DEG (see Fig. 2). Expression
(\ref{estimate}) includes interference terms which depend on $S_e-S_h$.  As $S_e-S_h=2\pi(\nu-1/2)$,  one can expect the
conductance to be a nonlinear function of $\nu$ due to the interference terms. The nature of this nonlinearity indeed
resembles Aharonov--Bohm effect, as it was supposed in \cite{Asano}, where the conductance oscillates with $H$ because
vector potential changes phases of electrons going from source to drain along different paths. It will be seen below that at
certain conditions S-2DEG conductance  oscillates with $\nu$.

Semiclassical estimates used above supposed that there is spin degeneracy; $T, eV=0$; diffraction is small: difference of
hole and electron momentum at $E>0$ was neglected. This approximation is valid when  $\max\{T, |eV|, g\mu_B H\}/\mu\ll
\lambda_F/L$, where $\lambda_F$ is Fermi wavelength in 2DEG. Calculation of the conductance below also supposes these
conditions to be satisfied.

The conductance of 2DEG-S structure will be calculated below as a semiclassical asymptotic ($\nu\gg 1$) of
(\ref{Takane_Ebisava}). If the S-2DEG surface is flat then the approach of \cite{Takane_Ebisava}, \cite{Baranger_Stone}
gives an idea how one can  express $R_{eh,ln}$ via semiclassical asymptotic of greens functions of BdG equations. Doing this
calculation we confirm, that above naive estimates of $G$ really lead to semiclassical asymptotic of the conductance:
\begin{gather}
\label{cal_R} {\cal R} =  \sum_{n_i} \int_0^{d(n_i)}\left\{\rho(n_i,y_0) \left|\sum_a t_a
\exp\left(iS_a-i\frac{\pi}{2}\mu_a\right)\right|^2\right\}dy_0,
\end{gather}
where  $n_i$ is the index of an edge state of an electron incident on the superconductor, $d(n_i)$ is the length of a
quasiparticle jump; $t_a$ is the probability amplitude for the classical quasiparticle trajectory from $y=y_0$ to
$y=y_{n_o}$ -- the coordinate of the last reflection from the superconductor. The amplitude $t_a$ is a product of Andreev
and normal reflection amplitudes; $S_a$ is the action taken from $y_0$ -- the coordinate of the first quasiparticle
reflection   to $y_{n_o}$ -- the coordinate of the last reflection (see Fig. 2);  $\mu_a$ is Maslov index of the trajectory.
For example, $t_a=r_{he}r_{ee}r_{eh}r_{he}$ and $S_a=S_h+2S_e$ for the trajectory distinguished by a thick line in Fig. 2.
Summation over $a$ means the sum over all paths connecting  $y_0$  with $y_{n_o}$ at 2DEG-S boundary. The wight function
$\rho(n_i,y_0)$, where $\int_0^d \rho(n_i,y_0) dy_0=1$, generally depends on the shape of 2DEG-S contact. If 2DEG spreads
over the region $x>0$, $y>2R_c+L, y<-2R_c$, as it is in Fig. 2, then $\rho=1/d$. Formula (\ref{cal_R}) is a central result
of the paper.

The sum over trajectories in (\ref{cal_R}) could be converted into an analytical expression:
\begin{gather}
\label{G_Ch} G = \frac{4e^2}{h} \sum_{n_i}\sum_s P_s
\frac{R_{eh}\sin^2(s\arccos(\sqrt{R_{ee}}\cos(\Omega)))}{1-R_{ee} \cos^2(\Omega)},
\end{gather}
where $\Omega=\pi \nu + \theta -2\lambda p_{\perp}$; $\theta=arg(r_{ee})$ is the phase of the amplitude of electron --
electron reflection from the superconductor, $R_{ee}=|r_{ee}|^2$ ; $p_{\perp}=p_{\perp}(n_i)$ is the perpendicular component
of momentum of a quasiparticle when it reflects from the superconductor; $\lambda$ is equal to the penetration length of the
superconductor; $\nu=E_f/(\hbar w_c)-1/2$. Function $P_s$ is the probability to have $s$ reflections from the surface of the
superconductor. When $\rho=1/d$ this function could be expressed through the maximum number of jumps $g=[L/d]$ over the
S-2DEG surface, where $[\ldots]$ denotes the integer part:
\begin{gather}
\label{P_s} P_{s}=
   \begin{cases}
     \frac{L-g d}{d}  & \text{if $s=g+1$}, \\
     1-\frac{L-g d}{d}& \text{if $s=g$}, \\
     0 & \text{otherwise}.
  \end{cases}
\end{gather}

Expressions (\ref{cal_R}-\ref{G_Ch}) are the central result of the paper; they show how the conductance depends on magnetic
field and parameters of the contact. At small magnetic field  (\ref{G_Ch}) reduces to (\ref{H_eq_0}). If $2R_c \gtrsim L$,
it follows from (\ref{G_Ch}) that the conductance becomes sensitive to  H. Few limiting cases of (\ref{G_Ch}) will be
considered below in this regime.  The Aharonov--Bohm type conductance oscillations are the most interesting property of
$G(\nu)$. It follows from (\ref{G_Ch}) that oscillations are visible when $\lambda/L\ll R_c^2/L^2$ and $R_{eh}\lesssim 1/2$.
A typical contact where S terminal is prepared with superconductor of first type, 2DEG formed in GaAs,  has $\lambda
\thicksim\lambda_F\backsim 10^{-6} cm$, $L\thicksim 10^{-3}cm$. It follows from given above conditions that if $R_c\thicksim
L$, oscillations can be seen in the contact. (These oscillations were numerically investigated in \cite{Takagaki},
\cite{Asano}. It was checked that there is consistence between the theory presented in our paper and the numerical
calculations.) When $R_{ee}\ll 1$ the conductance shows steps. It is interesting to investigate the regime $R_{eh}\ll 1$,
$L/R_c\gg 1$. Then functional dependence $G(\nu)$  resembles  light intensity distribution $I(\delta)$ seen in optics with
Lummer - Gerike interferometer \cite{Born}. If one considers $s$ as the number of beams in the interferometer,
$\delta=2\Omega$ as the phase difference between successive beams. The probability $R_{he}$ will correspond to the
transmission probability through the mirror of the interferometer. Examination of quasiparticle trajectories in
(\ref{cal_R}), which give the main contribution to the conductance, shows that they are similar to trajectories of light
beams in the interferometer.

Fig. 3 illustrates how the conductance depends on the magnetic field according to (\ref{G_Ch}).
\begin{figure}[htb]
\epsfxsize=130mm \epsffile{fig3.eps} \caption{The curve drawn with the solid line represents the conductance for $\lambda
L=3$, with dots -- for $\lambda L=1$. The curves drawn in the figure approach $G(H=0)$ (\ref{H_eq_0}) for large $\nu$.}
\end{figure}
One curve corresponds to $\lambda p_f=1$ (dots), another -- $\lambda p_f = 3$ (solid line). The solid line parallel to the X
axes represents the conductance (\ref{H_eq_0}). If  $2R_c\lesssim L$ the conductance oscillates because of quasiparticle
interference, but when $\nu$ becomes larger ($2R_c\gtrsim L$) then interference phenomena become less probable and the
conductance approaches (\ref{H_eq_0}). The graph also illustrates  that increasing the ratio $\lambda / L$ leads to smearing
of the oscillations. The following parameters of the contact were used: $Lp_f=80$, $Z=5$. Parameter $Z$ \cite{Blonder}
characterizes normal scattering from the superconductor due to Shottki barriers, differences of the effective masses in
2DEG, S and so on. Amplitudes $r_{ee},r_{eh}$ of normal and Andreev reflection from the superconductor were calculated for
zero magnetic field by matching quasiparticle wave functions in 2DEG to the wave functions in S. This procedure is true
while $(\lambda/R_c)^2\ll 1$, where $R_c$ is a cyclotron radius in the superconductor. Magnetic fields used for making the
plot satisfy this condition. The ratio $\Delta_0/E_f$ were equal to $0.02$, with the gap $\Delta(x)=\Delta_0$ in the
superconductor and zero in 2DEG.

It seems to be important to determine how a disorder at S-2DEG surface can influence on $G(\nu)$. The disorder could be
represented by the roughness of the surface, impurities and so on. Disorder at the surface can induce fluctuations of $S_a$,
$t_a$ in (\ref{cal_R}), break the interference of quasiparticles. We will characterize disorder by mean free path $l_{tr}$
of electron elastic scattering on impurities, by mean square root deviations $\delta n=\sqrt{\langle({\bf n-n_0})^2\rangle}$
of the normal unit vector to  S-2DEG boundary from the direction ${\bf n_0}$ of the $X$ axes, where $\langle({\bf
n-n_0})^2\rangle =\int_0^L dy ({\bf n}(y)-{\bf n_0})^2/L $. Then one can deduce that formula (\ref{G_Ch}) is correct if
$R_c\ll l_{tr}$ and $\delta n\ll \min\{l_{H}^2/L\lambda, 1/p_f L\}$.

When the surface of the superconductor is diffusive, i.e. $\delta n \gtrsim \max\{1/\lambda p_f, 1/\nu\}$ or $l_{tr}\lesssim
R_c$, then there would be no interference between different paths in (\ref{cal_R}). One can estimate the reflection
probability in this regime neglecting interference terms in (\ref{G_Ch}).  Then
\begin{gather}
\label{G_dirty} {\cal R}\backsimeq\sum_{n_i} \int_0^{d(n_i)} dy_0\sum_a|t_a|^2/d(n_i)
\end{gather}
The number  of quasiparticle reflections from the surface of the superconductor would be about
$s_0=[L/2R_c]$ then
\begin{gather}
\label{G_approx_1} G = \frac{4e^2}{h} {\cal R}\backsimeq[\nu]\frac{4e^2}{\hbar} \langle R_{eh}\rangle(1-2\langle
R_{eh}\rangle)^{1-s_0/2}U_{s_0}(\langle R_{ee}\rangle)/\sqrt{1-2\langle R_{eh}\rangle},
\end{gather}
where $U_s(x)=\sin(s\arccos(x))/\sin(\arccos(x))$ is the Chebishev polinomial of the second kind \cite{Gradshtein}. When
$s_0\to \infty$ then the conductance (\ref{G_approx_1}) will approach $[\nu]2e^2/h$.

One can suppose that the deviations from ideal conductance quantization (with the universal step $2e^2/h$) observed in the
experiment \cite{Akasaki} originate from the interference of quasiparticles studied above. The S-2DEG boundary of the device
used in \cite{Akasaki} was hardly flat, so contributions from harmonics with large $s$ (see eq. \ref{G_Ch}) should be
suppressed. Then it is reasonable to approximate the conductance $G$ by $(2e^2/h)f(\nu)(1+a\cos(\pi\nu+\varphi_0))$, where
$a\ll 1$ characterizes disorder at the surface, $\varphi_0$ is a phase - shift, $f(\nu)\sim [\nu]$ describes the shape of
the "quantum" of the conductance. It was checked that this formula is a good fit to the experimental data.

We thank G.B.Lesovik, M.V.Feigelman, Yu.V.Nazarov, A.Ioselevich, A.Shytov for helpful discussions. It is the Lesovik�s
interest that stimulated the appearance of the article. This work was supported by the Russian Foundation for Basic
Research, project no. 00-02-16617. The author is grateful to The Netherlands Organization for Scientific Research (NWO)
for support provided in the course of Dutch-Russian collaboration.

\begin{thebibliography}{99}
\bibitem{Akasaki}
H.Takayanagi, T.Akazaki, Physica (Amsterdam) {\bf 249-251B}, 462 (1998).

\bibitem{Williams}
T.D.Moore, D.A.Williams, Phys.Rev.B {\bf 59}, 7308 (1999)

\bibitem{Hoppe}
H.Hoppe, U.Z\"{u}like, and G.Sch\"{o}n, Phys.Rev.Lett. {\bf 84}, 1804 (2000).

\bibitem{Takagaki}
Y.Takagaki, Phys.Rev.B {\bf 57}, 4009 (1998).

\bibitem{Asano}
Y.Asano, Phys.Rev.B {\bf 61}, 1732 (2000); Y.Asano, T.Yuito, Phys.Rev.B {\bf 62}, 7477 (2000).

\bibitem{expl_1}
We say that S-2DEG boundary is "perfect" when the probability of Andreev reflection of an electron or hole quasiparticle
with zero energy and  momentum directed perpindicular to the surface  is close to unity.

\bibitem{Glazman}
L.I.Glazman, G.B.Lesovik, D.E.Khmelnitskii {\it et al.,} Pisma Zh.Eksp.Teor.Fiz. {\bf 48}, 218 (1998)[JETP Lett. {\bf 48},
238 (1988)].

\bibitem{Takane_Ebisava} C.J.Lambert, J.Phys.:Condens.Matter {\bf 3}, 6579(1991); Y.Takane and H.Ebisawa, J.Phys.Soc.Jpn.
{\bf 61}, 1685 (1992).

\bibitem{Born}
M.Born, E.Wolf, "Principles of optics", Pergamon Press, 1986, p. 341.

\bibitem{Baranger_Stone}
H.Baranger, D.DiVincentzo, R.Jalabert, {\it et al}, Phys.Rev.B {\bf 44}, 10637 (1991).

K.Richter, "Semiclassical theory of mesoscopic quantum systems", Springer-Verlag Berlin Heidelberg,
2000, (Springer tracts in modern physics; Vol. 161), pp. 63-68.

%\bibitem{Datta}
%S.Datta, P.F.Bagwell,M.P.Anatram, Phys.Low-Dim.Struct., {\bf 3}, 1 (1996).

\bibitem{Blonder}
G.E.Blonder, M.Tinkham, and T.M.Klapwijk, Phys.Rev.B {\bf 25}, 4515 (1982).

\bibitem{Gradshtein}
I.S.Gradshtein, I.M.Ryzhik, "Table of integrals, series, and products", Academic Press, Inc, 1980.

\end{thebibliography}

\end{document}

\begin{figure}[htb]
\epsfxsize=80mm  \epsffile{graph13.eps} \caption{The device, which we investigate, consists of a superconductor, 2DEG and a
normal conductor. An electron injected from the normal conductor goes through edge states to the superconductor, reflects
into a hole and an electron  which return to the normal contact at the other edge states.}
\end{figure}


\begin{figure}[htb]
\epsfxsize=85mm \epsffile{Graph25.eps} \caption{ An quasiparticle wave in quasiclassical approximation can be treated as a
beam of rays. The state of a ray can be found from the equations of classical mechanics. The figure displays reflection of
an electron ray, corresponding to an electron injected from N, from the superconductor. The ray reflects from S into an
electron and hole rays at $y_0$.  They reflect  at $y_1$ generating two electron and two hole rays. To find the conductance,
it is necessary to calculate the probability to find a hole ray propagating along the trajectory beginning at $y_3$.  }
\end{figure}

\begin{figure}[htb]
\epsfxsize=85mm \epsffile{fig3.eps} \caption{The curve drawn with the solid line represents the conductance for $\lambda
L=3$, with dots -- for $\lambda L=1$. In both cases the conductance exhibits oscillations with $\nu$ asymptotically
approaching (\ref{H_eq_0}) plotted with solid line parallel to the X axes.}
\end{figure}

cond-mat0102191.extracted_bib
\begin{thebibliography}{99}
\bibitem{Akasaki}
H.Takayanagi, T.Akazaki, Physica (Amsterdam) {\bf 249-251B}, 462 (1998).

\bibitem{Williams}
T.D.Moore, D.A.Williams, Phys.Rev.B {\bf 59}, 7308 (1999)

\bibitem{Hoppe}
H.Hoppe, U.Z\"{u}like, and G.Sch\"{o}n, Phys.Rev.Lett. {\bf 84}, 1804 (2000).

\bibitem{Takagaki}
Y.Takagaki, Phys.Rev.B {\bf 57}, 4009 (1998).

\bibitem{Asano}
Y.Asano, Phys.Rev.B {\bf 61}, 1732 (2000); Y.Asano, T.Yuito, Phys.Rev.B {\bf 62}, 7477 (2000).

\bibitem{expl_1}
We say that S-2DEG boundary is "perfect" when the probability of Andreev reflection of an electron or hole quasiparticle
with zero energy and  momentum directed perpindicular to the surface  is close to unity.

\bibitem{Glazman}
L.I.Glazman, G.B.Lesovik, D.E.Khmelnitskii {\it et al.,} Pisma Zh.Eksp.Teor.Fiz. {\bf 48}, 218 (1998)[JETP Lett. {\bf 48},
238 (1988)].

\bibitem{Takane_Ebisava} C.J.Lambert, J.Phys.:Condens.Matter {\bf 3}, 6579(1991); Y.Takane and H.Ebisawa, J.Phys.Soc.Jpn.
{\bf 61}, 1685 (1992).

\bibitem{Born}
M.Born, E.Wolf, "Principles of optics", Pergamon Press, 1986, p. 341.

\bibitem{Baranger_Stone}
H.Baranger, D.DiVincentzo, R.Jalabert, {\it et al}, Phys.Rev.B {\bf 44}, 10637 (1991).

K.Richter, "Semiclassical theory of mesoscopic quantum systems", Springer-Verlag Berlin Heidelberg,
2000, (Springer tracts in modern physics; Vol. 161), pp. 63-68.

%\bibitem{Datta}
%S.Datta, P.F.Bagwell,M.P.Anatram, Phys.Low-Dim.Struct., {\bf 3}, 1 (1996).

\bibitem{Blonder}
G.E.Blonder, M.Tinkham, and T.M.Klapwijk, Phys.Rev.B {\bf 25}, 4515 (1982).

\bibitem{Gradshtein}
I.S.Gradshtein, I.M.Ryzhik, "Table of integrals, series, and products", Academic Press, Inc, 1980.

\end{thebibliography}