Evidence for non glass-like behavior of i-AlPdMn quasicrystals from low-temperature thermal conductivity
A. Bilu\v{s}i\'{c}$^{1,2}$
A. Smontara$^{1}$
J. Dolin\v{s}ek$^{3}$
and H. R. Ott$^{2}$
The thermal conductivities of three single-grain i-AlPdMn quasicrystals, with different compositions and grown by different growth techniques, were studied. The electronic contribution to the thermal conductivity was estimated using the law of Wiedemann and Franz. At low temperatures, below 40 K, the quasilattice thermal conductivity, obtained by subtracting the electronic from the total thermal conductivity, was analyzed by means of the Debye model. At the lowest temperatures the Casimir regime is reflected by phonon mean free paths of the order of the sample dimensions. The dominant scattering at low temperatures is not due to tunneling states, but is caused by stacking-fault-like defects. This confirms that the low-temperature lattice excitation dynamics in quasilattices is more crystal- than glass-like. Above 60 K the thermal conductivity curves deviate from the Debye model due to hopping of localized vibrational modes which results in the opening of a new heat-carrying channel.
paper1.tex
\documentclass[aps,prb,twocolumn]{revtex4}
%\usepackage{graphicx}
\usepackage{epsfig}

\begin{document}

\title{Evidence for non glass-like behavior of \emph{i}-AlPdMn quasicrystals from low-temperature thermal conductivity}

\author{A. Bilu\v{s}i\'{c}$^{1,2}$, A. Smontara$^{1}$, J. Dolin\v{s}ek$^{3}$, and H. R. Ott$^{2}$}
\affiliation{
$^{1}$ Institute of Physics, POB 304, HR-10001 Zagreb, Croatia \\
$^{2}$ Laboratorium f\"{u}r Festk\"{o}rperphysik, ETH H\"{o}nggerberg, CH-8093, Z\"{u}rich, Switzerland \\
$^{3}$ J. Stefan Institute, University of Ljubljana, Jamova 39, SI-1000 Ljubljana, Slovenia  
}

\author{I. R. Fisher}\altaffiliation[Current address: ]{Department of Applied Physics, Stanford University}
\author{P. C. Canfield}
\affiliation{Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Iowa 50011}

\date{\today}

%****************************
%*****     Abstract     *****
%****************************
\begin{abstract}
    The thermal conductivities of three single-grain \emph{i}-AlPdMn quasicrystals, with different compositions and grown by different growth techniques, were studied. The electronic contribution to the thermal conductivity was estimated using the law of Wiedemann and Franz. At low temperatures, below 40 K, the quasilattice thermal conductivity, obtained by subtracting the electronic from the total thermal conductivity, was analyzed by means of the Debye model. At the lowest temperatures the Casimir regime is reflected by phonon mean free paths of the order of the sample dimensions. The dominant scattering at low temperatures is not due to tunneling states, but is caused by stacking-fault-like defects. This confirms that the low-temperature lattice excitation dynamics in quasilattices is more crystal- than glass-like. Above 60 K the thermal conductivity curves deviate from the Debye model due to hopping of localized vibrational modes which results in the opening of a new heat-carrying channel.
\end{abstract}

%\pacs{61.14.-x, 61.44.Br, 66.70.+f}

%\keywords{icosahedral quasicrystals, thermal conductivity}

\maketitle

%********************************
%*****     Introduction     *****
%********************************
\section{Introduction}\label{sec:intro}
Quasicrystals are materials with long-range order but they exhibit symmetries which are not allowed in classical crystallographic concepts. The first report on the synthesis of these intriguing materials appeared in a paper of Schechtman \emph{et al.} in 1984 \cite{schech84}. With respect to structural symmetries, icosahedral, octa-, deca- and dodecagonal quasicrystals are encountered. The most detailed investigated quasicrystals belong to icosahedral families since it is possible to produce large enough and defect-free samples.

The first stable samples of icosahedral \emph{i}-AlPdMn quasicrystals were produced in 1990 by Tsai \emph{et al.}\cite{tsai90}. Investigations of their electrical transport properties revealed anomalous resistivities, at least in comparision with other icosahedral quasicrystals. In general, the electrical resistivity of icosahedral quasicrystals exhibits a negative temperature coefficient \cite{biggs90, klein92, berger93, guo96}, which has been ascribed to variable-range hopping \cite{janot96}. At low temperatures quantum interference effects (QIE) \cite{klein92} become dominant, in some cases resulting in maxima of the resistivity curves. For \emph{i}-AlPdMn quasicrystals, maxima in the resistivity curves appear at much higher temperatures than in other icosahedral quasicrystals \cite{klein91, lanco92, akiyama93, rodmar99-rho, escudero99}. Although it was estimated that QIE in \emph{i}-AlPdMn may not be destroyed up to 200 K \cite{chernikov93}, more recent resistivity data \cite{rodmar99, prejean02, dolinsek02} suggest that Mn magnetic moments are involved in electron scattering.

The magnetic moments of the manganese atoms do not influence the dominant thermal conductivity $\kappa_{ph}$ due to itinerant lattice excitations of \emph{i}-AlPdMn quasicrystals. In order to do that, they would have to be coupled to the quasilattice vibrational modes, which are the main carriers of heat in quasicrystals. This type of coupling is very weak in paramagnetic crystals containing Mn atoms \cite{rives68}. In \emph{i}-AlPdMn it is expected to be even weaker (if it exists at all), because only a few percent of the Mn atoms are magnetic \cite{chernikov93-magn, lasjaunias95}, and are distributed in a non-regular fashion over the lattice sites. The non-periodicity of the lattice raises the question about the nature of vibrational states in quasicrystals: are they glass- or crystal-like? Measurements of the thermal conductivity in the sub-Kelvin range may provide the necessary information. Experiments made by Chernikov and co-workers \cite{chernikov95, chernikov98} have shown that often tunneling states are the main scatterers of 'phonons'. The presence of a significant density of tunneling states in a vibrational spectrum provides strong evidence for glass-like vibrations in a material. Data of Legault \emph{et al.} \cite{legault98} gave no evidence for the presence of tunneling states. It was argued that the existence of tunneling states in quasicrystals could be conditioned by the preparation method and/or by the samples' compositions. A theoretical calculation \cite{kalugin96} predicts that a shallow maximum of the quasilattice thermal conductivity at temperatures around 10 K is an intrinsic property of quasicrystals. Experimental results show both maxima \cite{legault95, chernikov98} and plateau-type features \cite{chernikov95}. Above 100 K the thermal conductivity rises due to the activation of a variable-range hopping channel of localized vibrational states \cite{janot96}. Recent thermal conductivity data from single-grain \emph{i}-YMgZn \cite{gianno00} quasicrystals indicated that below 1 K, the lattice vibrations are not scattered by tunneling states and they revealed a pronounced peak of the quasilattice thermal conductivity at 25 K. 

This work was motivated by the fact that the question concerning the main phonon scattering mechanisms at low temperatures is still controversial (see, e.g. Ref. \cite{thompson00}). We present an investigation of the thermal conductivity of three single-grain samples of the \emph{i}-AlPdMn quasicrystal family with different compositions (Al$_{72}$Pd$_{19.5}$Mn$_{8.5}$, Al$_{70.5}$Pd$_{21.2}$Mn$_{8.3}$ and Al$_{69.6}$Pd$_{22.1}$Mn$_{8.3}$), and grown by different preparation techniques (\emph{self-flux} and Czochralsky). We show that in quasilattices with high structural perfection, tunneling states are not the main source of phonon scattering at low temperatures and hence tunneling states are definitely not an intrinsic property of quasicrystals.
%********************************
%*****      Experiment      *****
%********************************
\section{Samples and Experiment}\label{sec:exp}
Three samples of \emph{i}-AlPdMn quasicrystals with different composition and prepared by different techniques were used in our study. The first one is Al$_{72}$Pd$_{19.5}$Mn$_{8.5}$ (in the following text abbreviated as \emph{i}-AlPdMn$_{8.5}$), grown with a \emph{self-flux} technique \cite{fisher99} at Ames Laboratory. The second sample, Al$_{70.5}$Pd$_{21.2}$Mn$_{8.3}$ (abbreviated as \emph{i}-AlPdMn$_{8.3}$-a), was grown by the Czochralsky method, and 'superannealed' for 35 days at 800$^{\circ}$C in vacuum. The third specimen is Al$_{69.6}$Pd$_{22.1}$Mn$_{8.3}$ (\emph{i}-AlPdMn$_{8.3}$-b), also grown by the Czochralsky technique. The two latter samples were grown at the Forschunszentrum J\"{u}lich, Germany, by K. Urban and M. Feuerbacher. 

The thermal conductivities were measured between 0.4 K and 300 K at ETH Z\"{u}rich employing an absolute steady-state heat flow technique. Measurements on the same samples in the temperature range between 10 K and 300 K and using a comparative steady state heat-flow method, described in more detail in Ref. \cite{smontara}, were made at the Institute of Physics, Zagreb. In the range of overlap the results from both types of methods agree very well. All three samples were shaped in the form of a prism, with dimensions 0.5$\times$0.6$\times$5.9 mm$^{3}$, 0.7$\times$1.0$\times$4.4 mm$^{3}$ and 0.5$\times$0.6$\times$2.7 mm$^{3}$ for \emph{i}-AlPdMn$_{8.5}$, \emph{i}-AlPdMn$_{8.3}$-a and \emph{i}-AlPdMn$_{8.3}$-b, respectively. The thermal flux through the samples was generated by a 1 k$\Omega$ RuO$_{2}$ chip-resistor, glued to one end of the sample, while the other end was attached to a copper heat sink. Two gold wires, 50 $\mu$m in diameter, were attached to the sample and were used as thermal contacts for the resistor thermometers (see below); as a glue, EpoTek H20E was used. Between 5 K and 300 K the thermal conductivity was measured in a He$^{4}$ cryostat, using a 25 $\mu$m chromel-gold (doped with 0.07\% Fe) differential thermocouple. Between 0.4 K and 4 K, a He$^{3}$ refrigerator was used as the cooling device. For monitoring the temperature gradient of the sample in this temperature regime, two 10 k$\Omega$ RuO$_{2}$, \emph{in-situ} calibrated resistors were used. Heat losses due to radiation and conduction through contact wires can be neglected at temperatures below 300 K.  
 
%********************************
%**** Results and Discussion ****
%********************************
\section{Results and Discussion}\label{sec:results}

% ***** FIGURE *****
\begin{figure}[bt!] 
\begin{center}
    \epsfig{file=Fig1.eps,width=8.5cm,clip=}
	%\includegraphics[bb=77 94 375 563, width=8.5cm]{Fig1.eps}
    \caption{\label{fig:Ktotal}Thermal conductivity, $\kappa(T)$, for all three samples. The insets show the quasilattice thermal conductivity, renormalized to the values of the local minima near 100 K.}
\end{center}
\end{figure} 

The temperature dependence of the thermal conductivity, $\kappa(T)$, of all three samples is shown in Fig. \ref{fig:Ktotal}. Values of $\kappa(T)$ at 300 K are 4.6 W/mK, 3.2 W/mK and 5.5 W/mK for \emph{i}-AlPdMn$_{8.5}$, \emph{i}-AlPdMn$_{8.3}$-a and \emph{i}-AlPdMn$_{8.3}$-b, respectively. Such low values of the thermal conductivity are typical for quasicrystals, and are a consequence of the non-periodicity of the lattice, which hampers the existence of itinerant phonons, except those with small wave vectors and an extended wave function. The electronic contribution to the thermal conductivity $\kappa_{el}$ of quasicrystals can usually be estimated by employing the law of Wiedemann and Franz \cite{ashcroft-mermin}

% ***** EQUATION *****
\begin{equation}\label{eq:WFL}
    \kappa_{el}=L_{0} T / \rho \mathrm{,}
\end{equation}

\noindent
where $L_{0}$ is the Lorenz number, equal to $2.44 \times 10^{-8}$ W$\Omega$ K$^{-2}$, $\rho$ is the electrical resistivity and $T$ is the temperature. For an estimate of $\kappa_{el}$ we used the resistivity data obtained from corresponding measurements on the same samples \cite{bilusic99, dolinsek02, dolinsek01}. In Table \ref{tab:WFL} we list the values of $\kappa$, $\rho$ and the ratio of $\kappa_{el}/\kappa$ at 6 K. It is obvious that electrons carry less than 1\% of the heat, and it is reasonable to extrapolate $\kappa_{el}$ to 0.4 K, the lowest temperature reached in our experiments. By reducing the temperature, a big change in electrical resistivity is not expected. The variable-range hopping mechanism \cite{janot96} assumes the hopping of electrons between neighboring pseudo MacKay icosahedrons (PMIs), organized in a hierarchical way. At low temperatures only electrons from PMI's which are higher in hierarchy can be considered to be delocalized through the sample. Since the number of these electrons is limited, it may be expected that the low-temperature resistivity does not change too much compared to the value at 6 K. A question about the validity of Eq.(\ref{eq:WFL}) in the case of quasicrystals arises. Calculations of Lorenz numbers for \emph{i}-AlCuFe quasicrystals \cite{landauro01} have shown that at room temperature it is about 40\% larger than $L_{0}$ and by lowering the temperature, approaches the value of $L_{0}$. We assume  that a similar conclusion is valid in the case of \emph{i}-AlPdMn as well and the real electron contribution to the thermal conductivity is expected to not differ considerably from that given by Eq. (\ref{eq:WFL}), especially at low temperatures.

% ***** TABLE *****
\begin{table}[t!]
    \caption{\label{tab:WFL}Values of the thermal conductivity $\kappa$, the electrical resistivity $\rho$ and the ratio between the electron contribution to the thermal conductivity, estimated by the Wiedemann-Franz law given on Eq. (\ref{eq:WFL}) and the total thermal conductivity at 6 K. The values of $\rho$ are taken from the references listed below.}
    \begin{ruledtabular}
    \begin{tabular}{cccc}
        Sample & $\kappa$ [W/mK] & $\rho$ [$\mu \Omega$ m] & $\kappa_{el}/\kappa$ \\
    \hline \emph{i}-AlPdMn$_{8.5}$ & 1.9 & 13.6\footnote{from Bilu\v{s}i\'{c} \emph{et al.} \cite{bilusic99}} & $6 \times 10^{-3}$ \\
        \emph{i}-AlPdMn$_{8.3}$-a & 1.8 & 21.4\footnote{from Dolin\v{s}ek \emph{et al.} \cite{dolinsek02} } & $4 \times 10^{-3}$ \\
        \emph{i}-AlPdMn$_{8.3}$-a & 2.8 & 17.8\footnote{from Dolin\v{s}ek \emph{et al.} \cite{dolinsek01}} & $3 \times 10^{-3}$ \\
    \end{tabular}
    \end{ruledtabular}
\end{table}

The quasilattice contribution to the thermal conductivity $\kappa_{qlatt}$ is calculated by subtracting the calculated electronic part from the measured total thermal conductivity $\kappa$

% ***** EQUATION *****
\begin{equation}\label{eq:Kqlatt}
    \kappa_{qlatt}=\kappa-\kappa_{el} \mathrm{.}
\end{equation}

\noindent
The insets in Fig. \ref{fig:Ktotal} show the quasilattice thermal conductivity, renormalized to the value at the local minima $\kappa_{qlatt}(T_{min})$ at about 100 K. The ratios of $\kappa_{qlatt}/\kappa(T_{min})$ reach maximum values of 1.70, 1.52 and 1.58 for \emph{i}-AlPdMn$_{8.5}$, \emph{i}-AlPdMn$_{8.3}$-a, and \emph{i}-AlPdMn$_{8.3}$-b, respectively. These values are distinctly higher than the analogous ratios obtained by Chernikov \emph{et al.} \cite{chernikov98} (1.22) and Legault \emph{et al.} \cite{legault95} (1.27) for similar materials. A somewhat larger value for this ratio has only been established for quasicrystalline \emph{i}-YMgZn \cite{gianno00} (1.75). Following theoretical arguments \cite{kalugin96}, the maximum at temperatures of the order of 10 K is an intrinsic property of the quasilattice thermal conductivity, and the relative height provides a measure for the structural quality of the sample. It can be concluded that the \emph{i}-AlPdMn samples used in our investigation are of better structural quality than those used by Chernikov and co-workers \cite{chernikov98} or Legault and co-workers \cite{legault95}.

Since the quasicrystalline structure is not periodic, only long-wave phonons can safely be defined. Inelastic neutron (INS) \cite{janot93, boissieu93, boudard95} and X-ray (IXS) \cite{krisch02} scattering experiments on samples of the \emph{i}-AlPdMn quasicrystal family revealed the existence of isotropic acoustic phonon modes with wave vectors $q<0.3$ \AA{} and energies $E<6$ meV. Dispersionless vibrational states for quasilattice vibrations with energies higher than 12 meV were identified. The energy of 6 meV corresponds to a thermal energy of approximately 50$k_{B}$ and therefore the Debye model for analyzing thermal conductivity data can be used for temperatures less than 50 K. The relevant equation reads \cite{berman}

% ***** EQUATION *****
\begin{equation}\label{eq:Debye}
    \kappa_{Debye}=\frac{k_{B}}{2 \pi^{2} \overline{v}} \bigg( \frac{k_{B}}{\hbar} \bigg) ^{3} T^{3} \int_{0}^{\frac{\theta}{T}} \tau (x) \frac{x^{4} e^{x}}{\big( e^{x} -1 \big) ^{2}}dx \mathrm{,}
\end{equation} 

\noindent
where $k_{B}$ and $\hbar$ are Boltzmann's and the reduced Planck constant, respectively; $\overline{v}$ is the average sound velocity, $\theta$ Debye temperature, $\tau$ the phonon relaxation time and $x = \hbar \omega / k_{B} T$, where $\hbar \omega$ is phonon energy. Ultrasonic data \cite{amazit92} provided values for the transversal ($v_{T}=(3593 \pm 3$) m/s) and the longitudinal ($v_{L}=(6520 \pm 10$) m/s) sound velocities. Since  neutron and X-ray scattering experiments have shown that the excitation spectra of vibrational states are isotropic, the average sound velocity $\overline{v}$ can be obtained using the relation

% ***** EQUATION *****
\begin{equation}\label{eq:velocity}
    \frac{3}{\overline{v}^{3}}=\frac{1}{v_{L}^{3}}+\frac{2}{v_{T}^{3}}\mathrm{.}
\end{equation} 

\noindent
From this equation $\overline{v}=4004$ m/s. The Debye temperature $\theta$, was calculated by Li and Liu \cite{li01}, based on specific heat data of W\"{a}lti \emph{et al.} \cite{waelti98}. They obtained $\theta = 492$ K.

In Eq. (\ref{eq:Debye}) the different phonon scattering processes are incorporated in the relaxation time $\tau(x)$.  As usual, we assume that Matthiessen's rule is valid, and the mutual independence of the scattering processes is reflected in

% ***** EQUATION *****
\begin{equation}\label{eq:Matthiessen}
    \frac{1}{\tau}=\sum_{i} \frac{1}{\tau_{i}} \mathrm{,}
\end{equation} 

\noindent
where $\tau_{i}$ is a scattering rate related with the $i$-th scattering channel. At the lowest temperatures reached in our experiment, the Casimir scattering (i.e. scattering at sample boundaries) dominates in the total phonon relaxation rate. In that case the phonon flow between two collisions is diffusive and the corresponding scattering rate is given by the phonon mean free path $l$ and the average sound velocity $\overline{v}$ by

% ***** EQUATION *****
\begin{equation}\label{eq:Casimir}
    \tau_{Cas}^{-1}=\frac{\overline{v}}{l} \mathrm{.}
\end{equation} 

\noindent
By invoking equations (\ref{eq:Debye}) and (\ref{eq:Casimir}) it turns out that $\kappa_{ph} \propto T^{3}$ at very low temperatures.

The non-periodicity of the quasilattice implies the question whether its dynamics is similar to either glasses or crystals. At low-temperatures the particular dynamics of amorphous structures is reflected by the existence of so called tunneling states \cite{anderson72, philips72} in atom dynamics. Results of ultrasonic experiments made on \emph{i}-AlPdMn \cite{bert00, thompson00} were interpreted as to show that the dynamical properties of these material are glass-like. A similar conclusion was drawn from results of analogous experiments on \emph{i}-AlCuFe material \cite{vernier93} and it was suggested that tunneling states are an intrinsic feature of icosahedral quasicrystals in general. The presence of tunneling states may also be probed by measurements of the thermal conductivity and they are reflected in a characteristic temperature dependence of $\kappa_{ph}$ at low temperatures. The phonon scattering rate due to collisions with tunneling states is given by \cite{philips72}

% ***** EQUATION *****
\begin{equation}\label{eq:ts}
    \tau_{ts}^{-1}=\frac{\overline{P} \gamma ^{2}}{\rho \overline{v}^{2}} \pi \omega \tanh \bigg( \frac{\hbar \omega}{2 k_{B} T} \bigg) \mathrm{,}
\end{equation} 

\noindent
where $\overline{P}$ is the density of tunneling states, $\gamma$ the average phonon-tunneling states coupling constant and $\rho$ is the mass density. Inserting Eq. (\ref{eq:ts}) into Eq. (\ref{eq:Debye}), leads to a contribution to the total thermal conductivity which is proportional to $T^{2}$. Published thermal conductivity data on arc-melted \cite{chernikov95} and Czochralsky grown \cite{chernikov98} \emph{i}-AlPdMn quasicrystals show that scattering at tunneling states dominates the phonon collisions at low-temperatures. On the other hand, experiments made by Legault \emph{et al.} \cite{legault98} on samples produced by the Bridgeman method gave no evidence that phonons are scattered at tunneling states. Scattering at stacking faults had to be invoked in order to explain those data. It was argued that the presence of tunneling states in quasicrystals is sensitive to the sample composition and/or the preparation method. More recent thermal conductivity data of a single-grain \emph{i}-YMgZn quasicrystal \cite{gianno00}, grown by self-flux technique, also indicated that no tunneling states contribute to the phonon scattering at low temperatures, but scattering on stacking faults had to be considered instead. The phonon - stacking fault scattering rate is given by \cite{klemens}:

% ***** EQUATION *****
\begin{equation}\label{eq:sf}
    \tau_{sf}^{-1}=\frac{7}{10} \frac{a^{2}}{\overline{v}} \gamma ^{2} \omega ^{2} N_{s} \mathrm{,}
\end{equation} 

\noindent
where $a$ is a relevant lattice constant, $\gamma$ the Gr\"{u}neisen parameter and $N_{s}$ is the linear density of stacking faults. The phonon scattering at stacking faults leads to a contribution to $\kappa_{ph}$ which is proportional to $T$.

Theoretical calculations of the vibrational excitation spectrum of a one-dimensional quasicrystal, i.e. a Fibbonacci chain,  \cite{kalugin96}  revealed a dense distribution of energy gaps. Although not proven rigorously, the same characteristic feature is expected for three-dimensional quasilattices. In perfect periodic lattices, energy gaps appear at the Brillouin zones boundaries. They are the origin of the phonon-phonon Umklapp scattering mechanism which causes an exponential decrease of the scattering rate upon decreasing temperature. Since the quasilattice vibrational spectrum has a dense distribution of energy gaps, the phonon scattering rate due to the \emph{quasiumklapp} processes exhibits a power-law type frequency and temperature dependence with \cite{kalugin96}

% ***** EQUATION *****
\begin{equation}\label{eq:qu}
    \tau_{qu}^{-1} \propto \omega ^{2} T^{4}.
\end{equation} 

% ***** FIGURE *****
\begin{figure}[tt!]
\begin{center}
    \epsfig{file=Fig2.eps,width=8.5cm,clip=}
	%\includegraphics[bb=72 175 383 645, width=8.5cm]{Fig2.eps}
    \caption{\label{fig:Kqlatt}The main panels present $\kappa_{qlatt}/T^{3}$ below 20 K. The straight horizontal lines are the limits of $\kappa_{qlatt}/T^{3}$, indicating the Casimir limit. The insets show the low-temperature quasilattice thermal conductivity below 5 K, $\kappa_{qlatt}$, obtained by subtracting the electron contribution from the measured thermal conductivity (as explained in the text).}
\end{center}
\end{figure}

The insets in Figure \ref{fig:Kqlatt} show the quasilattice thermal conductivities, calculated using Eq. (\ref{eq:Kqlatt}). Below 1 K all three curves exhibit power law behavior with $\kappa_{ph} \propto T^{1.7}$, $T^{1.8}$ and $T^{2.1}$ for \emph{i}-AlPdMn$_{8.5}$, \emph{i}-AlPdMn$_{8.3}$-a and \emph{i}-AlPdMn$_{8.3}$-b, respectively. In the cases of \emph{i}-AlPdMn$_{8.5}$ and \emph{i}-AlPdMn$_{8.3}$-a, the power-law exponent is less than two. Since for a dominating phonon scattering at tunneling states, $\kappa_{ph}$ is expected to vary as $T^{2}$, the corresponding plots indicate that this type of scattering is weak or absent in at least two samples investigated in our work. In order to identify the dominant scattering processes, we attempted to fit the data using Eq.(\ref{eq:Debye}) with various combinations of scattering rates, given by Eqs. (\ref{eq:Casimir}), (\ref{eq:ts}), (\ref{eq:sf}), and (\ref{eq:qu}). The fitting procedure was made for temperatures below 40 K and the best fits (represented by the solid lines in Fig. \ref{fig:Kqlatt}) for all three samples were obtained with the assumption that the total relaxation rate $\tau^{-1}$ is given by the sum of Casimir, $\tau^{-1}_{Cas}$, stacking faults, $\tau^{-1}_{sf}$, and modified quasiumklapp, $\tau^{*-1}_{qu}$, scattering rates, i.e.,

% ***** EQUATION *****
\begin{equation}\label{eq:tau fit}
    \tau^{-1}=\tau^{-1}_{Cas} + \tau^{-1}_{sf} + \tau^{*-1}_{qu}.
\end{equation}

\noindent
By the modified quasiumklapp scattering rate we mean that, instead of Eq. (\ref{eq:qu}), a revised expression for $\tau_{qu}^{-1}$ had to be used, i.e., $\tau^{*-1}_{qu} \propto \omega^{2} T^{2}$. The weaker temperature dependence than that predicted by Kalugin \emph{et al.} \cite{kalugin96} may be explained by results of inelastic X-ray scattering (IXS) experiments, \cite{krisch02} which revealed the existence of low lying vibrational modes with frequencies below both the acoustic and the localized ('optical') modes. We suggest that the excitation of these vibrational modes is the reason for the modification of the quasiumklapp scattering rate. From the fit of our data to Eq. (\ref{eq:tau fit}), a phonon mean free path $l$ can be extracted. For \emph{i}-AlPdMn$_{8.5}$ it is equal to 0.82 mm, in the case of \emph{i}-AlPdMn$_{8.3}$-a, $l$ is 0.44 mm, while for \emph{i}-AlPdMn$_{8.3}$-b, it is equal to 0.26 mm. The values of the mean free paths are of the order of the sample dimensions, and the biggest one is for an \emph{i}-AlPdMn$_{8.5}$ sample. This fact, together with the highest relative maximum of $\kappa_{qlatt}$ (insets of Fig. \ref{fig:Ktotal}), suggests that this sample is the one with the best structural quality among the samples investigated in this work. Besides the phonon mean free path, the linear density of stacking faults $N_{S}$ is calculated from the fits as well. The Gr\"{u}neisen parameter $\gamma$ was calculated using the data of $\gamma$'s of the constituent elements \cite{gschneider64} with the atomic composition as the weight factor. For all three samples the same value of the Gr\"{u}neisen parameter, $\gamma = 2.1$, is valid. For the lattice constant $a$ we have used the value of 7.38 \AA{}, calculated by Elser \cite{elser96}. With this values of $\gamma$ and $a$, the density of stacking faults has been calculated; for \emph{i}-AlPdMn$_{8.5}$ and \emph{i}-AlPdMn$_{8.3}$-a, it is equal to $8.1 \times 10^{5}$ m$^{-1}$, while for \emph{i}-AlPdMn$_{8.3}$-b, $N_{S}$ is $5.6 \times 10^{5}$ m$^{-1}$. This is an order of magnitude smaller than the values obtained by Legault \emph{et al.} \cite{legault98}. Stacking-fault-like defects in quasicrystals are formed along and around the trail of dislocation-movement paths, which are characterized by a high density of phason flips \cite{beeli}. Thus, we conclude that tunneling states are not the predominant scattering mechanism at low temperatures. Our thermal conductivity data, together with some other results \cite{legault98, gianno00}, indicate that the intrinsic heat transport in quasicrystals differs distincly from that in amorphous structures.

% ***** FIGURE *****
\begin{figure}[t!]
\begin{center}
    	\epsfig{file=Fig3.eps,width=9cm,clip=}
	%\includegraphics[width=9cm]{Fig3.eps}
    \caption{\label{fig:Kqlatt-85}Quasilattice thermal conductivity of \emph{i}-AlPdMn$_{8.5}$ between 0.4 K and 300 K. The solid line represents $\kappa_{Debye}$, a fit obtained using the Debye model, as explained in the text. The difference between $\kappa_{Debye}$ and $\kappa_{qlatt}$ is shown in the inset. The solid line is a fit to the variable-range hopping model, explained in the text.}
\end{center}
\end{figure}

Figure \ref{fig:Kqlatt-85} shows the quasilattice thermal conductivity $\kappa_{qlatt}(T)$ of \emph{i}-AlPdMn$_{8.5}$ between 0.4 K and 300 K. The solid line is the fit obtained by Eqs. (\ref{eq:Debye}) and (\ref{eq:tau fit}). Above 50 K $\kappa_{qlatt}(T)$ starts to deviate from the fit. As it was mentioned before, INS \cite{janot93, boissieu93, boudard95} and IXS \cite{krisch02} experiments have shown that quasicrystals in their vibrational dispersion have localized, optic-like modes which are activated above several tens of Kelvins. This localized modes (fractons) can interact with normal, acoustical phonons, resulting in a fracton jumping through the quasilattice \cite{janot96}. With respect to heat transfer, this means an opening of a new heat-carrying channel with a thermal conductivity proportional to $T^{1.5}$. The inset in Fig. \ref{fig:Kqlatt-85} shows the difference between $\kappa_{qlatt}$ and the fit to the Debye model. Above approximately 150 K it is proportional to $T^{1.5}$, which is equal to the temperature dependence obtained by the vaiable-range hopping model. This provides further evidence that our \emph{i}-AlPdMn$_{8.5}$ sample is very close to the ideal icosahedral quasicrystal structure. We have made the same analysis for the other two samples used in our work and for the corresonding temperature dependencies above 50 K we obtained $T^{0.8}$ for \emph{i}-AlPdMn$_{8.3}$-a and $T^{1.1}$ for \emph{i}-AlPdMn$_{8.3}$-b.

%********************************
%*****      Conclusion      *****
%********************************
\section{Conclusion}\label{sec:conc}
We have measured the thermal conductivity between 0.4 K and 300 K of three single grain quasicrystals from the \emph{i}-AlPdMn family with different compositions and grown by different techniques. For the analysis of the thermal conductivity data below 40 K the Debye model was used. We have found that the dominant source of phonon scattering at low temperatures is not tunneling states, but stacking-fault-like defects, independent of sample composition and/or preparation method. At about 20 K, well developed shallow maxima were observed, which serve as indicators of the high structural perfection of the investigated samples. Above 150 K the thermal conductivity rises due to the hopping of localized quasilattice modes.

\begin{acknowledgments}
A.B. acknowledges financial support from the Swiss Federal Commission for Foreign Students. The work was supported by the Swiss National Science Foundation and the Croatian Ministry of Science and Technology. Ames Laboratory is operated for the US Department of Energy under Contract No. W-7405-Eng-82.
\end{acknowledgments}

\bibliography{QC}

\end{document} 
paper1.bbl
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\end{thebibliography}
