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\begin{document}
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\kalvo
\center{\normalsize Workshop on Vortices in Superfluids and
Superconductors
\\
Oulu 4.-8.1.2003}
\center{\Large INHOMOGENEOUS IMPURITY \\
\vspace{1cm}DISTRIBUTION  IN FERMI
SUPERFLUIDS}
\vspace{1cm}
\center{\href{http://boojum.hut.fi/research/theory/aerogel.html}
{http://boojum.hut.fi/research/theory/aerogel.html}}
\vspace{1cm}

{Erkki Thuneberg$^{*,\dagger}$ and Risto
H\"anninen$^{\dagger}$}
%\vspace{1cm}
\center{$^*$Department of Physical Sciences,
University of Oulu\\
 $^{\dagger}$Low Temperature Laboratory,
  Helsinki University of Technology }
\kalvoe 

\kalvo
\normalsize
\subhead{Content}

Impurities in Fermi superfluids\\
\hspace{1cm} quasiclassical theory\\
\hspace{1cm} impurity averaging\\
Homogeneous distribution of impurities\\
A model of inhomogeneous impurity distribution\\
Application to superfluid $^3$He in aerogel 
\kalvoe
 

\twocolumn
\begin{slide}
%\scriptsize\baselineskip=0.7truecm
\subhead{Impurities in metals}

 Parameters:\\
- scattering cross section $\sigma$\\
- density of impurities $n$\\
- mean free path for quasiparticles $\ell=1/n\sigma$


Scattering causes electrical resistance in normal metals\\
\includegraphics[width=0.4\linewidth]{scatter.eps}

{\bf Superconductors} (usual case)

- $T_c$ not changed (Anderson theorem)

- Ginzburg-Landau parameter increases
\kalvoe 

\kalvo
but

- magnetic impurities in usual superconductors, or

- usual impurities in unconventional superconductors 
(p, d, ... wave pairing)

$\Rightarrow$\\
\begin{center}
\includegraphics[width=0.8\linewidth]{Tchsm.eps}
\end{center}
(Abrikosov-Gorkov 1961)
\kalvoe 

\kalvo
\subhead{Quasiclassical scattering theory}
parameters:

- Fermi wave length $\lambda_F$\\
- coherence length $\xi_0=\frac{\hbar v_F}{2\pi k_BT_c}$\\
- scattering cross section $\sigma$\\
- density of impurities $n$\\
- mean free path $\ell=1/n\sigma$

Assumption: 
\begin{equation}
\lambda_F, \sqrt{\sigma} \ll  \xi_0, \ell
\end{equation}
Take leading terms, ignore terms that are smaller by factors
$\lambda_F/\xi_0$, etc. 
\kalvoe 

\kalvo
Technical tool: take an average over the locations of the impurities.

However, due to assumptions (1), this implies that all fluctuations in
the impurity density are lost.\\
$\Rightarrow$\\
Instead of true discrete impurities there is a continuous scattering
medium.\\ 
$\Rightarrow$\\ 
"Homogeneous scattering model" (HSM)
\begin{center}
\includegraphics[width=0.8\linewidth]{Tchsm.eps}
\end{center}
\kalvoe 

\onecolumn
\kalvo
\subhead{Quasiclassical equations}
\smallskip

4x4 matrix Green's function $\hat g(\hat{\bf k},{\bf r},\epsilon_m)$:
$$[i\epsilon_m\hat\tau_3-\hat\nu-\hat\Delta-\hat\rho, \hat g]+i\hbar 
{\bf v}_{\rm F}\cdot\nabla_{\bf r}\hat g=0$$
$$\hat g(\hat{\bf k},{\bf r},\epsilon_m)\hat g(\hat{\bf k},{\bf
r},\epsilon_m)=-\pi^2$$
$$\hat\rho(\hat{\bf k},{\bf r},\epsilon_m)=n({\bf r})\hat t(\hat{\bf
k},\hat{\bf k},{\bf r},\epsilon_m).$$
Equation for the scattering matrix $\hat t(\hat{\bf k},\hat{\bf k}',
\epsilon_m)$
\begin{eqnarray}
\hat t(\hat{\bf k},\hat{\bf k}',{\bf r},\epsilon_m)=\hat v(\hat{\bf
k},\hat{\bf k}')+N(0)\langle \hat v(\hat{\bf k},\hat{\bf k}'')
\hat g(\hat{\bf k}'',{\bf r},\epsilon_m)\hat t(\hat{\bf k}'',\hat{\bf
k}',{\bf r},\epsilon_m)\rangle_{\hat{\bf k}''}.\nonumber
\end{eqnarray}

Energy functional 
\begin{eqnarray}
\Omega=\int d^3r\bigg[{1\over V_{BCS}}\langle\vert\Delta(\hat{\bf
k},{\bf r})\vert^2\rangle_{\hat{\bf k}}+{1\over
2}N(0)T\sum_{\epsilon_m}\int_0^\Delta {d\Delta\over\Delta} \langle{\rm
Tr}_4[\hat g(\hat{\bf k},{\bf r},\epsilon_m)\hat\Delta(\hat{\bf k},{\bf
r})]\rangle_{\hat{\bf k}}\bigg]\nonumber
\end{eqnarray}
+ terms arising from Fermi-liquid corrections $\hat\nu$.

\kalvoe
 

\kalvo
\subhead{Need a better theory?}

- in high $T_c$ superconductors $\sqrt{\sigma}/\xi_0$ is not
negligible.\\
$\Rightarrow$\\
Fluctuations of the impurity density are important.\\
(Franz, Kallin, Berlinsky, and 
Salkola 1997)

- superfluid $^3$He in aerogel
\kalvoe 

%\kalvo \ \kalvoe 
\twocolumn
\kalvo

\subhead{Impure superfluid $^3$He}

$^3$He is a naturally pure substance

Impurity can be introduced by porous aerogel
\begin{center}
\includegraphics[width=1.0\linewidth]{aerogel_1.eps}
\end{center}
- strands of SiO$_2$\\
- typically 98\% empty\\
- small angle x-ray scattering $\Rightarrow$
homogeneous on scale above $\approx 100$ nm
\kalvoe 

\kalvo
Compare that to 
\begin{eqnarray}
\xi_0= 16\ldots 74\ {\rm nm},\nonumber
\end{eqnarray}
depending on pressure.

Experiments:\\
HSM has qualitative success, but insufficient quantitatively.

\kalvoe 

\kalvo
\subhead{Models}
\scriptsize
Use quasiclassical theory with a location dependent
impurity density
$n({\bf r})$.

%\begin{center}
\includegraphics[width=1.0\linewidth]{IISModel.eps}
%\end{center}
\kalvoe 

\kalvo
\subhead{Isotropic inhomogeneous scattering model (IISM)}
\scriptsize
%\begin{center}
\includegraphics[width=1.0\linewidth]{IISMop.eps}
%\end{center}
\kalvoe 

\onecolumn
\kalvo
\begin{center}
\subhead{Supercurrent}

\includegraphics[width=0.4\linewidth]{currecol.eps}

${\bf j}_{\rm s}$ in cylindrical coordinates $\rho,\phi,z$.
\end{center}
\kalvoe 

\twocolumn
\kalvo
\subhead{Transition temperature}
\vspace{2cm}
\scriptsize
%\begin{center}
\includegraphics[width=1.0\linewidth]{Tciism.eps}
%\end{center}
\vspace{3cm}
\kalvoe 

\kalvo
\scriptsize
Experiments (vary pressure)
\begin{center}
\includegraphics[width=1.0\linewidth]{TcComp.eps}
\end{center} 

Porto et al (1995),
Sprague et al (1995),
Matsumoto et al (1997)
\kalvoe 

\onecolumn\kalvo
\subhead{Temperature dependence}
\scriptsize
%\begin{center}
\includegraphics[width=0.8\linewidth]{gapesimcol.eps}
%\end{center}
\kalvoe 

\kalvo
\begin{center}
\subhead{Suppression of}
\includegraphics[width=0.65\linewidth]{suppcol2.eps}
\end{center}
\kalvoe 

\kalvo
\subhead{Discussion}

The IISM gives better fit to experiment than HSM.

The optimal radius of the unit cell $R\approx 140$ nm is on the same
order of magnitude than  the aerogel correlation length $\approx 84$ nm.

No perfect fit to experiments\\
- possible reason: IISM has only one length scale $R$.
\kalvoe 





\kalvo
\subhead{Conclusions}

Quasiclassical theory: inhomogeneous scattering modelled by $n({\bf
r})$.

Isotropic inhomogeneous scattering model (IISM): 

- the simplest model of
inhomogeneous scattering that reduces to homogeneous medium on a large
scale

- computationally much heavier than HSM

- $^3$He in aerogel: IISM clearly better than HSM, but still not perfect

- how to calculate vortex states?

- application to other superfluids?

\vspace{1cm}
Links: {\href{http://boojum.hut.fi/research/theory/aerogel.html}
{http://boojum.hut.fi/research/theory/aerogel.html}}



\kalvoe 





\end{document}
 