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\title{Spin Hall Effect in a 2DEG in the presence of magnetic couplings}
%\titlegraphic{\pgfuseimage{logorm3}\hskip 8mm\pgfuseimage{logocnism}}
%\titlegraphic{\pgfuseimage{logocnism}}
\author{Roberto Raimondi}

%\subtitle{or: \textbf{Spin transport in a 2DEG}}
%\institute{Dipartimento di Fisica, Universit\`a di Roma Tre \\ \url{http://www.fis.uniroma3.it}}
\date{}


\begin{document}
\transduration{1}

\frame{
	\transsplitverticalin
	\frametitle{\tiny RTN Nano Meeting 2008 - {\sl Fundamentals of Nanoelectronics}  \hskip 0.5cm  April 7, 2008
Bremen, Germany  \hskip 0.5cm Jacobs University}
\begin{flushleft}
\pgfuseimage{logorm3}\hskip 10mm\pgfuseimage{logocnism}
\hskip10mm\pgfuseimage{logoaugsburg}\\
\end{flushleft}	
\titlepage 

\begin{columns}[T]
\begin{column}{6cm}
{\small
Coworkers:

Peter Schwab (Augsburg)

Cosimo Gorini (Augsburg)

Michael Dzierzawa (Augsburg)

Mirco Milletar\`i (Lancaster)
}
\end{column}

\begin{column}{6cm}
{\tiny 
\begin{itemize}
\item Phys. Rev. B {\bf 74}, 035340 (2006)

\item Physica E {\bf 40} , 1078 (2008)

\item arXiv:0801.1786
\end{itemize}
}
\end{column}

\end{columns}

}
	





\frame{
  \transsplitverticalin
  \nameslide{outline}
  \frametitle{Outline}
{\Large
  \begin{block}{}
  \begin{enumerate}
  \item SHE in a 2DEG: why vanishes   
  \item Finite SHE: in-plane magnetic field
   \item Finite SHE: Magnetic impurities
  \item Sketch of the derivation
  \item Conclusions
  \end{enumerate}
 \end{block}
%\begin{block}{}
% \begin{enumerate}
%  \item<1-| alert@1>SHE in a 2DEG: why vanishes 
%  \item<2-| alert@2>How to get a SHE  
%  \item<3-| alert@3>Magnetic impurities
%  \item<4-| alert@4>In-plane magnetic field
% \item<5-| alert@5>Sketch of the derivation
%  \item<6-| alert@6>Conclusions
% \end{enumerate}
% \end{block}
}

}

%Define the initial point of the slide
\frame{
  \transsplitverticalin
  \nameslide{shereview1}
  \frametitle{SHE in a 2DEG: why is zero}
 %Divide the space in two columns 
\begin{columns}[T]
\begin{column}{5cm}
\vskip 1cm
\includegraphics[width=5cm]{spinhall}
\end{column}
\begin{column}{6cm}
\begin{exampleblock}{Definition}
{\Large
\begin{equation}
\nonumber
j^z_{s,y}=\sigma_{sH}{\cal E}_x
\end{equation}}
\begin{equation}
\nonumber
H_{so}=\alpha  {\bf p}\times {\bf \hat e}_z \cdot {\boldsymbol \sigma}, \
 \sigma_{sH}^0=\frac{|e|}{8\pi}
\end{equation}
{\tiny 
\begin{itemize}
  \item J. Sinova et al., Phys. Rev. Lett.   {\bf 92}, 126603  (2004).

  \end{itemize}
}

\end{exampleblock}
\end{column}
\end{columns}
 \vskip 1cm


 %Divide the space in two columns 
\begin{columns}[t]
\begin{column}{5cm}
\begin{alertblock}{Continuity-like equation}
\begin{equation}
\nonumber
\frac{\partial s_y}{\partial t}+\frac{\partial }{\partial \mathbf{ x}}\cdot \mathbf{ j}^y_{s}
=-2m\alpha j_{s,y}^z
%+2\omega_s s_z-\frac{4}{3\tau_{sf}}s_y,
\end{equation}
\end{alertblock}

\end{column}
\begin{column}{6cm}
\vskip 1cm
\setbeamercolor{references}{fg=black,bg=white}
\begin{beamercolorbox}{references}
{\tiny 
\begin{itemize}
  \item O. V. Dimitrova, Phys. Rev. B \textbf{71}, 245327 (2005).
  \item  O. Chalaev and D. Loss, Phys. Rev. B \textbf{71}, 245318 (2005).

  \item E. I. Rashba, Phys. Rev. B \textbf{\textbf{70}}, 201309(R) (2004).
  \end{itemize}
}
\end{beamercolorbox}
%To draw a postit
%\setbeamercolor{postit}{fg=red,bg=yellow}
%\begin{beamercolorbox}[sep=1em,wd=5cm]{postit}
%Questo \`e un postit
%\end{beamercolorbox}

\end{column}
\end{columns}  
 }

\frame{
  \transsplitverticalin
 \nameslide{shereview2}
\frametitle{SHE in a 2DEG: how it happens}

\begin{columns}[T]
\begin{column}{3cm}
\includegraphics[width=2cm]{bubble}
\end{column}

\begin{column}{3cm}
\includegraphics[width=2cm]{vertex}
\end{column}
\begin{column}{6cm}
{\small Vertex corrections cancel  bubble }

{\tiny 
\begin{itemize}
  \item J. I. Inoue, et al.  Phys. Rev. B  \textbf{70}, 041303(R) (2004).
\item  E. G. Mishchenko, et al., Phys. Rev. Lett. \textbf{93}, 226602 (2004).
\item R. Raimondi and P. Schwab, Phys.  Rev. B {\bf 71}, 033311 (2005).
\item A. Khaetskii, Phys. Rev. Lett. \textbf{96}, 056602 (2006).

  \end{itemize}
}
\end{column}

\end{columns}

\vskip 1cm 
\begin{block}{Connection with in-plane polarization}

Physical insight from the quasiclassical Green function approach

\begin{equation}
\nonumber
j^z_{s,y}\propto \sigma_{sH}^0{\cal E}_x+\frac{1}{8\pi \alpha N_0\tau}s_y^0,   \    \     \     \sigma_{sH}^0=\frac{|e|}{8\pi}
\end{equation}

\setbeamercolor{references}{}
\begin{beamercolorbox}{references}
{\tiny 
\begin{itemize}
  \item R.  Raimondi,  et al.,
 Phys. Rev. B {\bf 74}, 035340 (2006).
    \end{itemize}
}
\end{beamercolorbox}
\end{block}




}

%Define the initial point of the slide
\frame{
  \transsplitverticalin
  \nameslide{shereview3}
  \frametitle{Electrically-induced in-plane spin polarization}
 %Divide the space in two columns 
 \includegraphics[width=8cm]{edelstein4}
\begin{columns}[T]

\begin{column}{6cm}
\begin{exampleblock}{Effective spin-orbit field}
\begin{equation}
\nonumber
s_y=-N_0\alpha \delta p
\end{equation}
\begin{equation}
\nonumber
\delta p=|e| {\cal E}_x \tau
\end{equation}
\end{exampleblock}
\end{column}
\begin{column}{5cm}
\begin{alertblock}{Edelstein's result}
\begin{equation}
\nonumber
s_0=-N_0 |e| \alpha\tau {\cal E}_x 
\end{equation}
\end{alertblock}
\setbeamercolor{references}{fg=black,bg=white}
\begin{beamercolorbox}{references}
{\tiny 
\begin{itemize}
  \item V. M. Edelstein, Solid State Commun. {\bf 73}, 233 (1990).

  \end{itemize}
}
\end{beamercolorbox}


\end{column}
\end{columns}

 }


\frame{
 \transsplitverticalin
\frametitle{Spin relaxation}

\begin{columns}[T]

\begin{column}{5cm}
\begin{block}{Bloch's equations}
\begin{eqnarray}
\partial_t s_x&=&-\frac{1}{\tau_s}s_x \nonumber \\
\partial_t s_y&=&-\frac{1}{\tau_s}  \left( s_y +  s_{0} \right) \nonumber\\
\partial_t s_z&=&-\frac{2}{\tau_s}s_z\nonumber \end{eqnarray}
\end{block}
\begin{equation}
\nonumber
\tau_s^{-1}= 2 \alpha p_F (\alpha p_F \tau )
\end{equation}

{\small Dyakonov-Perel relaxation}
\end{column}
\begin{column}{6cm}
\vskip 1cm
\setbeamercolor{evidence}{fg=red ,bg=white}
\begin{beamercolorbox}{evidence}
 The  static  solution allows 
 
 in-plane spin polarization only
 \end{beamercolorbox}
 \begin{equation}
\nonumber
s_y=-s_0, \   \   s_z=0
\end{equation}
 

\vskip 1cm
\begin{alertblock}{Question}
What changes in order 

to have a finite SHE?
\end{alertblock}
\end{column}
\end{columns}

}




\frame{
 \transsplitverticalin
   \nameslide{magnetic}
  \frametitle{In-plane magnetic field  ({\tiny  Cf. Milletar\`i's talk})}
  
  \begin{columns}[T]
  \begin{column}{5cm}
    \begin{equation}
\nonumber
H_{mf}=\frac{1}{2}g_L \mu_B {\cal B}_x {\bf \hat e}_x
\cdot {\boldsymbol \sigma}
\end{equation}

{\small Zeeman energy $2\omega_s\equiv g_L \mu_B {\cal B}_x$}
\vskip 0.5cm 
{\tiny Related work: 

\begin{itemize}
\item Q. Lin, et al.,  Appl. Phys. Lett. \textbf{88}, 122105 (2006).
\item  H. A. Engel, et al., Phys. Rev. Lett.  {\bf 98}, 036602 (2007).
\end{itemize}
}
 
    \end{column}
   \begin{column}{7cm}
    \begin{block}{Continuity-like equation with extra  term} 
\begin{equation}
\nonumber
\frac{\partial s_y}{\partial t}+\frac{\partial }{\partial \mathbf{ x}}\cdot \mathbf{ j}^y_{s}
=-2m\alpha j_{s,y}^z+2\omega_s s_z
\end{equation}
\end{block}


    \end{column}
  \end{columns}
  \vskip 0.5cm

 \vskip 0.5cm
 \begin{columns}[T]

\begin{column}{6cm}

\begin{eqnarray}
\partial_t s_x&=&-\frac{1}{\tau_s}(s_x-s_x^{eq}) \nonumber \\
\partial_t s_y&=&-\frac{1}{\tau_s} \left[ s_y +  s_1 \right]+2\omega_s s_z \nonumber\\
\partial_t s_z&=&-\frac{2}{\tau_s}s_z-2\omega_s \left[s_y+s_2
  \right]\nonumber
\end{eqnarray}
\end{column}

\begin{column}{6cm}
\vskip 0.5cm
{\small $s_x^{eq}=N_0\omega_s$ depends on magnetic field}
\vskip 0.5cm
{\small $s_1$ and $s_2$ depend on electric field}
\vskip 0.5cm
\setbeamercolor{evidence}{fg=red ,bg=white}
\begin{beamercolorbox}{evidence}
{\small If $s_1=s_2$ no out-of-plane polarization!}
\end{beamercolorbox}
\end{column}


\end{columns}
{\small\bf  With short-range disorder $s_1=s_2=s_0=-|e|N_0 \alpha \tau{\cal E}_x$}
 
 
}


\frame{
 \transsplitverticalin
\frametitle{Magnetic field with long-range disorder}

\begin{block}{Scattering kernel}
\begin{equation}
\nonumber
W(\theta =\phi -\phi' )=V_0+2V_1\cos (\phi -\phi ')+2V_2 \cos (2\phi -2\phi ')+\dots.
\end{equation}
\end{block}
\vskip 1cm
{\small Naively, one would use the transport time:}

 \begin{equation}
  \nonumber
  \tau \rightarrow \tau_{tr} \    \     \
   \tau^{-1}_{tr} =  N_0 \int_0^{2\pi} {\rm d}\theta W (\theta)(1-\cos (\theta ))= \frac{1}{\tau}\frac{V_0-V_1} {V_0}
 \end{equation}
  
However:

\begin{itemize}
\item It would give no effect, since $s_1=s_2=-s_0\tau_{tr}/\tau$
\item But it is also  wrong!
\end{itemize}


}

\frame{
 \transsplitverticalin
\frametitle{The correct generalization: more scattering times}


\begin{columns}[T]
\begin{column}{5cm}
\begin{equation}
\nonumber
s_1=-s_0\frac{\tau_E}{\tau}, \  \ s_2=-s_0\frac{\tau_{tr}}{\tau}
\end{equation}

\end{column}

\begin{column}{7cm}
\begin{equation}
\nonumber
\frac{1}{\tau_{E}}=N_0 \int_0^{2\pi} {\rm d}\theta W (\theta) (1-\cos (2\theta ))=\frac{V_0-V_2}{\tau V_0}
\end{equation}

\end{column}
\end{columns}
\begin{alertblock}{Consequence from Bloch's equations: magnetic field dependence}
\begin{eqnarray}
s_y&  =& -\frac{V_0 \, s_0 }{V_0-V_2}
\frac{1+2\omega_s^2\tau_s^2(V_0-V_2)(V_0-V_1)^{-1}}{1+2\omega_s^2\tau_s^2}\nonumber \\
s_z & =&  -\frac{V_0 \, s_0 }{V_0-V_2} \frac{V_1-V_2}{V_0-V_1}
\frac{\omega_s\tau_s}{1+2\omega_s^2\tau_s^2}\nonumber
\end{eqnarray}

{\tiny For SHE see Milletar\`i's talk}
\end{alertblock}

{\small Note also the Dyakonov-Perel relaxation time renormalization}
\begin{equation}
\nonumber
\tau_s^{-1}= 2 \alpha p_F (\alpha p_F \tau_{tr} )
\end{equation}

}


\frame{
 \transsplitverticalin
 \frametitle{Physical interpretation}

\begin{equation}
\nonumber
s_y \sim \delta n_+-\delta n_-   \   \   \
\delta n_{\pm} =\frac{\partial \epsilon_{F\pm}}{\partial p_x}\delta p_{\pm x} N_{\pm}
\     \     \delta p_{\pm x} = -|e|{\cal E}_x \tau_{tr, \pm}
\end{equation}

\setbeamercolor{evidence}{fg=red ,bg=white}
\begin{beamercolorbox}{evidence}
The new time $\tau_E$ means a modification of the Edelstein result
due to different DOS and transport times in the spin-orbit splitted bands
\end{beamercolorbox}
\begin{equation}
\nonumber
s_y=\frac{v_F}{4}(N_+\tau_{tr,+}-N_-\tau_{tr,-})|e|{\cal E}_x =-s_0 \frac{V_0}{V_0-V_2}
\end{equation}

}


\frame{
\transsplitverticalin
\frametitle{Experimental relevance}


\begin{columns}[T]
\begin{column}{8cm}
 \includegraphics[width=8cm]{kato_fig_2004}
 
 {\tiny 
\begin{itemize}
\item Y.K. Kato, R.C. Myers, A.C. Gossard, and D.D. Awshalom, 
Phys.\ Rev.\ Lett. {\bf 93}, 176601 (2004).
\end{itemize}
}
\end{column}

\begin{column}{4cm}
$InGaAs$

{\small Faraday rotation spectroscopy $\theta_F \propto s_z$

$w=60 \mu m$  and $l=200 \mu m$

Some estimates:
\begin{itemize}
\item $\tau_s \sim 10 ns$ $\tau_{tr}\sim 1 ps$
\item $\alpha p_F \sim 10^{-5} eV$
\item $\omega_s \sim 10^{-6} eV$
\end{itemize}}
\end{column}
\end{columns}


 
 
}



\frame{
 \transsplitverticalin
   \nameslide{magnetic2}
  \frametitle{Magnetic impurities with short-range correlations}
  
  \begin{columns}[t]
  \begin{column}{5cm}
 
  \begin{equation}
\nonumber
H_{mi}=V_m ({\bf x}){\bf S}\cdot {\boldsymbol \sigma}
\end{equation}

{\tiny Related work
\begin{itemize}
\item J.\ Inoue, et al.,
Phys.\ Rev.\ Lett. {\bf 97}, 046604 (2006).
\item  P.\ Wang,  et al., Phys.\ Rev.\ B {\bf 75}, 075326 (2007).
\end{itemize}
}


   \end{column}
   \begin{column}{5cm}
    \begin{alertblock}{Spin-flip time}
  \begin{equation}
\nonumber
\tau_{sf}^{-1} =2\pi N_0 V_m^2 S(S+1)
\end{equation}
\end{alertblock}
   
    
  \end{column}
  \end{columns}
  \vskip 0.5cm
 \begin{block}{Continuity-like equation with extra source terms} 
\begin{equation}
\nonumber
\frac{\partial s_y}{\partial t}+\frac{\partial }{\partial \mathbf{ x}}\cdot \mathbf{ j}^y_{s}
=-2m\alpha j_{s,y}^z-\frac{4}{3\tau_{sf}}s_y
\end{equation}
\end{block}
 \vskip 0.5cm
 \begin{columns}[T]
  \begin{column}{5cm}
 {\small For weak magnetic scattering $\tau \ll \tau_{sf}$}
  \begin{equation}
\nonumber
\sigma_{s H}=\frac{|e|}{3\pi}\frac{\tau}{\tau_{sf}}
\end{equation}
   \end{column}
   \begin{column}{5cm}
 However, one can do better!
 
 What about $\tau \sim \tau_{sf}$?
  \end{column}
  \end{columns}

}




\frame{
 \transsplitverticalin
 \frametitle{Let us go back to Bloch's equations}

\begin{columns}[T]
\begin{column}{6cm}
\begin{eqnarray}
\partial_t s_x&=&-\left(\frac{1}{\tau_s}+\frac{4}{3\tau_{sf}}\right)s_x \nonumber \\
\partial_t s_y&=&-\left(\frac{1}{\tau_s}+\frac{4}{3\tau_{sf}}\right)  \left[ s_y +  {\tilde s}_0 \right] \nonumber\\
\partial_t s_z&=&-\left(\frac{2}{\tau_s}+\frac{4}{3\tau_{sf}}\right)s_z\nonumber
\end{eqnarray}


\includegraphics[width=6cm]{mag_imp}

{\tiny Different colors for different value of $\alpha p_F \tau$ and $V_1=V_2=0$}
\end{column}

\begin{column}{6cm}
\begin{equation}
\nonumber
{\tilde s_0}=-s_0\frac{\tilde \tau_E}{\tau} \   \   \  \tau_s^{-1}= 2 \alpha p_F (\alpha p_F \tau_{tr} )
\end{equation}
\begin{equation}
\nonumber
\frac{1}{\tilde \tau_E}=\left(\frac{1}{ \tau_E}+\frac{1}{\tau_{sf}}\right)\left(1+\frac{4\tau_s}{3\tau_{sf}}\right)+
\frac{4}{3\tau_{sf}}
\end{equation}
\begin{equation}
\nonumber
\tau_{tr}^{-1}=\frac{1}{\tau}\frac{V_0-V_1}{V_0}+\frac{1}{\tau_{sf}}
\end{equation}


\begin{alertblock}{SHE}
\begin{equation}
\nonumber
\sigma_{s H}=\frac{|e|}{3\pi}\frac{\tilde \tau_E}{\tau_{sf}}
\end{equation}

{\small when $\tau_{sf}\sim \tau$, $\sigma_{sH}\rightarrow$ the universal value}
\vskip 0.3cm
{\tiny Relevant for mercury telluride-cadmium telluride quantum wells

\begin{itemize}
\item Konig et al. Science {\bf 318}, 766 (2007)
\item Wen Yang, Kai Chang and Shou-Cheng Zhang,  PRL {\bf 100}, 056602 (2008) 
\item Bernevig, Hughes, Shou-Cheng Zhang, Science {\bf 314}, 1757 (2006)
\end{itemize}
}

\end{alertblock}
\end{column}

\end{columns}

}





\frame{
 \transsplitverticalin
\frametitle{Sketch of calculation}

\begin{exampleblock}{Quasiclassical Green function}
\begin{equation}
\nonumber
\check{ g } \equiv \check{g}_{t_1t_2}(\mathbf{ \hat{p}};\mathbf{ x}) =
\frac{\rm i}{\pi} \int {\rm d } \xi \, \check G_{t_1t_2}(\mathbf{ p},\mathbf{ x}), \
\check G = 
\left( \begin{array}{cc} 
G^R & G \\
0   & G^A 
\end{array}
\right)
\end{equation}
\end{exampleblock}

\begin{block}{Eilenberger-like equation ({\tiny Phys. Rev. B {\bf 74}, 035340 (2006)})}
\begin{equation}
\nonumber
 \partial_{t} \check g  =
  -\frac{1}{2}\sum_{\mu = \pm } \left\{
    \frac{\bf p_\mu}{m}  \right.
   +\left. \frac{\partial}{\partial \bf p}({\bf b}_\mu \cdot {\boldsymbol  \sigma}),
    \frac{\partial}{\partial {\bf x} }\check g_\mu \right\} 
- {\rm i }\sum_{\mu = \pm } [{\bf b}_\mu \cdot {\boldsymbol \sigma}, \check
g_\mu ] - {\rm i} \left[ \check \Sigma , \check g \right]
\end{equation}
\begin{equation}
\nonumber
{\bf b}({\bf p})=\alpha {\bf p}\times {\bf \hat e}_z-\omega_s {\bf \hat e}_x   \      \
 \check \Sigma=-{\rm i}\left( \frac{1}{2\tau}\langle K{\check g}\rangle
 +\frac{1}{6\tau_{sf}} \sum_l \sigma_l \langle {\check g}\rangle\sigma_l\right)
\end{equation}

\end{block}

{\small 
\begin{columns}[T]

\begin{column}{3.8cm}
\begin{alertblock}{Disorder  parameter}
\begin{equation}
\nonumber
\frac{1}{\epsilon_F\tau}\ll 1
\end{equation}
Leading terms
\end{alertblock}
\end{column}

\begin{column}{3.8cm}
\begin{alertblock}{SO  parameter}
\begin{equation}
\nonumber
\frac{\alpha p_F}{\epsilon_F}=\frac{\alpha}{v_F}\ll 1
\end{equation}
Leading terms
\end{alertblock}

\end{column}


\begin{column}{3.8cm}
\begin{alertblock}{Additional parameter}
\begin{eqnarray}
\alpha p_F \tau \ll1 &  & {\rm diffusive } \nonumber \\
\alpha p_F \tau \gg1  &  &  {\rm clean}\nonumber
\end{eqnarray}

\end{alertblock}
\end{column}
\end{columns}
}
}


\frame{
 \transsplitverticalin
\frametitle{Conclusions}

\begin{itemize}
\item Magnetic couplings yield finite SHE in a 2DEG
\item Effective Bloch's equation for spin dynamics
\item Different  time scales with long-range disorder
\item Future: inclusion of quantum corrections and extrinsic effects
\end{itemize}


}



\end{document}




