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\begin{document}
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\title{Transport measurements of the spin wave gap of Mn}
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\input author_list.tex
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\date{\today}
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\begin{abstract}
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Temperature dependent transport measurements on ultrathin antiferromagnetic
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Mn films reveal a heretofore unknown non-universal weak localization
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correction to the conductivity which extends to disorder strengths greater than
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100~k$\Omega$ per square. The inelastic scattering of electrons off of
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gapped antiferromagnetic spin waves gives rise to an inelastic scattering
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length which is short enough to place the system in the 3D regime. The
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extracted fitting parameters provide estimates of the energy gap ($\Delta
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\approx$~16~K) and exchange energy ($\bar{J} \approx$~320~K). %\st{which are in
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%agreement with values obtained with other techniques}.
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\end{abstract}
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\pacs{75}
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\maketitle
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Thin-film transition metal ferromagnets (Fe, Co, Ni, Gd) and
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antiferromagnets (Mn, Cr) and their alloys are not only ubiquitous in
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present day technologies but are also expected to play an important role in
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future developments~\cite{thompson_2008}. Understanding magnetism in these
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materials, especially when the films are thin enough so that disorder plays
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an important role, is complicated by the long standing controversy about the
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relative importance of itinerant and local moments~\cite%
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{slater_1936,van_vleck_1953,aharoni_2000}. For the itinerant transition
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metal magnets, a related fundamental issue centers on the question of how
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itinerancy is compromised by disorder. Clearly with sufficient disorder the
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charge carriers become localized, but questions arise as to what happens to
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the spins and associated spin waves and whether the outcome depends on the
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ferro/antiferro alignment of spins in the itinerant parent. Ferromagnets
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which have magnetization as the order parameter are fundamentally different
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than antiferromagnets which have staggered magnetization (i.e., difference
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between the magnetization on each sublattice) as the order parameter~\cite%
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{blundell_2001}. Ferromagnetism thus distinguishes itself by having soft
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modes at zero wave number whereas antiferromagnets have soft modes at finite
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wave number~\cite{belitz_2005}. Accordingly, the respective spin wave
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spectrums are radically different. These distinctions are particularly
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important when comparing quantum corrections to the conductivity near
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quantum critical points for ferromagnets~\cite{paul_2005} and
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antiferromagnets~\cite{syzranov_2012}.
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Surprisingly, although there have been systematic studies of the effect of
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disorder on the longitudinal $\sigma_{xx}$ and transverse $\sigma_{xy}$
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conductivity of ferromagnetic films~\cite%
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{bergmann_1978,bergmann_1991,mitra_2007,misra_2009,kurzweil_2009}, there
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have been few if any such studies on antiferromagnetic films. In this paper
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we remedy this situation by presenting transport data on systematically
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disordered Mn films that are sputter deposited in a custom designed vacuum
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chamber and then transferred without exposure to air into an adjacent
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cryostat for transport studies to low temperature. The experimental
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procedures are similar to those reported previously: disorder, characterized
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by the sheet resistance $R_0$ measured at $T=$~5~K, can be changed either by
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growing separate samples or by gentle annealing of a given sample through
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incremental stages of disorder~\cite{misra_2011}. Using these same procedures our results for
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antiferromagnets however are decidedly different. The data are well
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described over a large range of disorder strengths by a non-universal three
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dimensional (3d) quantum correction that applies only to spin wave gapped
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antiferromagnets. This finding implies the presence of strong inelastic
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electron scattering off of antiferromagnetic spin waves. The theory is
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validated not only by good fits to the data but also by extraction from the
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fitting parameters of a value for the spin wave gap $\Delta$ that is in
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agreement with the value expected for Mn. On the other hand, the
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exchange energy $\bar{J}$ could be sensitive to the high disorder in our
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ultra thin films, and it turns out to be much smaller compared to the known values.
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In previous work the inelastic scattering of electrons off of spin waves has
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been an essential ingredient in understanding disordered ferromagnets. For
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example, to explain the occurrence of weak-localization corrections to the
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anomalous Hall effect in polycrystalline Fe films~\cite{mitra_2007}, it was
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necessary to invoke a contribution to the inelastic phase breaking rate $%
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\tau_{\varphi}^{-1}$ due to spin-conserving inelastic scattering off
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spin-wave excitations. This phase breaking rate, anticipated by theory~\cite%
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{tatara_2004} and seen experimentally in spin polarized electron energy loss
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spectroscopy (SPEELS) measurements of ultrathin Fe films~\cite%
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{plihal_1999,zhang_2010}, is linear in temperature and significantly larger
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than the phase breaking rate due to electron-electron interactions, thus
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allowing a wide temperature range to observe weak localization corrections~%
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\cite{mitra_2007}. The effect of a high $\tau_{\varphi}^{-1}$ due to
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inelastic scattering off spin-wave excitations is also seen in Gd films
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where in addition to a localizing log($T$) quantum correction to the
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conductance, a localizing linear-in-$T$ quantum correction is present and is
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interpreted as a spin-wave mediated Altshuler-Aronov type correction to the
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conductivity~\cite{misra_2009}.
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Interestingly, this high rate of inelastic spin rate scattering becomes even
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more important for the thinnest films as shown in theoretical calculations
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on Fe and Ni which point to extremely short spin-dependent inelastic mean
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free paths~\cite{hong_2000} and in spin-polarized electron energy-loss
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spectroscopy (SPEELS) measurements on few monolayer-thick Fe/W(110) films in
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which a strong nonmonotonic enhancement of localized spin wave energies is
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found on the thinnest films~\cite{zhang_2010}.
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Inelastic spin wave scattering in highly disordered ferromagnetic films can
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be strong enough to assure that the associated $T$-dependent dephasing
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length $L_{\varphi }(T)=\sqrt{D\tau _{\varphi }}$ (with $D$ the diffusion
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constant)~\cite{lee_1985} is less than the film thickness $t$, thus putting
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thin films into the 3d limit where a metal-insulator
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transition is observed~\cite{misra_2011}. Recognizing that similarly high
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inelastic scattering rates must apply to highly disordered antiferromagnetic
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films, we first proceed with a theoretical approach that takes into account
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the scattering of antiferromagnetic spin waves on the phase relaxation rate
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and find a heretofore unrecognized non-universal 3d weak localization
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correction to the conductivity that allows an interpretation of our experimental
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results.
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We mention in passing that the 3d interaction-induced quantum correction
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found to be dominant in the case of ferromagnetic Gd
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films which undergo a metal-insulator transition\cite{misra_2011} is
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found to be much smaller in the present case and will not be considered further (for an estimate of this contribution see \cite{muttalib_unpub}.
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As discussed in detail in Ref.~[\onlinecite{wm10}], the phase relaxation
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time $\tau _{\varphi }$ limits the phase coherence in a particle-particle
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diffusion propagator $C(q,\omega )$ (Cooperon) in the form
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\begin{equation}
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C(q,\omega _{l})=\frac{1}{2\pi N_{0}\tau ^{2}}\frac{1}{Dq^{2}+|\omega
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_{l}|+1/\tau _{\varphi }}.
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\end{equation}
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where $N_{0}$ is the density of states at the Fermi level, $\tau $ is the
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elastic scattering time and $\omega _{l}=2\pi lT$ is the Matsubara
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frequency. Labeling the Cooperon propagator in the absence of interactions
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as $C_{0}$, we can write
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\begin{equation}
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\frac{1}{\tau _{\varphi }}=\frac{1}{2\pi N_{0}\tau ^{2}}[C^{-1}-C_{0}^{-1}].
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\end{equation}
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In general, $C(q,\omega )$ can be evaluated diagrammatically in the presence
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of interactions and disorder in a ladder approximation \cite{fa} that can be
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symbolically written as $C=C_{0}+C_{0}KC$ where the interaction vertex $K$
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contains self energy as well as vertex corrections due to both interactions
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and disorder. It then follows that $1/\tau _{\varphi }$ is given by
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\begin{equation}
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\frac{1}{\tau _{\varphi }}=-\frac{1}{2\pi N_{0}\tau ^{2}}K.
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\end{equation}%
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In Ref.~[\onlinecite{wm10}], the leading temperature and disorder dependence
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of the inelastic diffusion propagator was evaluated diagrammatically, in the
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presence of ferromagnetic spin-wave mediated electron-electron interactions.
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Here we consider the antiferromagnetic case. We only consider large
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spin-wave gap where the damping can be ignored. Using the antiferromagnetic
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dispersion relation $\omega _{q}=\Delta +Aq$, where $A$ is the spin
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stiffness, the inelastic lifetime is given by
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\be
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\frac{\hbar }{\tau _{\varphi }}=\frac{4}{\pi \hbar }nJ^{2}\int_{0}^{1/l}%
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\frac{q^{d-1}dq}{\sinh \beta \omega _{q}}\frac{Dq^{2}+1/\tau _{\varphi }}{%
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(Dq^{2}+1/\tau _{\varphi })^{2}+\omega _{q}^{2}}
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\ee%
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where $n=k_{F}^{3}/3\pi ^{2}$ is the 3d density, $J$ is the effective
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spin-exchange interaction and $\beta =1/k_{B}T$. Here we will consider the
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limit $\hbar /\tau _{\varphi }\ll \Delta $, relevant for our experiment on
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Mn. In this limit we can neglect the $1/\tau _{\varphi }$ terms inside the
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integral. The upper limit should be restricted to $\Delta /A$ in the limit $%
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\Delta /A<1/l$. For large disorder, we expect the parameter $x\equiv
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\hbar Dk_{F}^{2}\Delta / \bar{J}^{2}\ll 1$, where the spin-exchange energy
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is given by $\bar{J}=Ak_{F}$. In this limit, $L_{\varphi }$ can be
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simplified as
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\be
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k_{F}L_{\varphi }\approx \left( \frac{\bar{J}}{\Delta }\right) ^{3/2}\left(
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\frac{5\sinh \frac{\Delta }{T}}{12\pi }\right) ^{1/2},\;\;\;x\ll 1
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\label{L-phi-3d}
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\ee%
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which is independent of $x$, and therefore, independent of disorder.
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Given the inelastic lifetime, the weak localization correction in 3d is
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usually given by \cite{lee_1985} $\delta \sigma _{3d}=\frac{e^{2}}{\hbar \pi
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^{3}}\frac{1}{L_{\varphi }},$ where the prefactor to the inverse inelastic
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length is a universal number, independent of disorder. However, at large
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enough disorder, we show that there exists a disorder dependent correction,
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due to the scale dependent diffusion coefficient near the Anderson
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metal-insulator transition. In fact, the diffusion coefficient obeys the
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self consistent equation \cite{WV}
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\begin{equation}
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\frac{D_{0}}{D(\omega )}=1+\frac{k_{F}^{2-d}}{\pi m}\int_{0}^{1/l}dQ\frac{%
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Q^{d-1}}{-i\omega +D(\omega )Q^{2}}
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\end{equation}%
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where $D_{0}=v_{F}l/d$ is the diffusion coefficient at weak disorder. While
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the significance of the prefactor to the integral is not clear, the above
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equation remains qualitatively accurate over a wide range near the Anderson
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transition. Setting $\omega =i/\tau _{\varphi }$ and doing the $Q$-integral
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in 3d,
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\bea
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\frac{D_{0}}{D} &\approx & 1+\frac{1}{\pi mk_{F}}\int_{1/L_{\phi }}^{1/l}dQ\frac{%
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Q^{2}}{DQ^{2}}\cr
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&=& 1+\frac{D_{0}}{D}\frac{3}{\pi k_{F}^{2}l^{2}}-\delta
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\left( \frac{D_{0}}{D}\right) ,
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\label{delta}
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\eea%
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where
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\bea
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\delta \equiv \frac{D_{0}}{D}\frac{3}{\pi k_{F}^{2}l^{2}}\frac{l}{%
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L_{\varphi }}
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\eea
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is assumed to be a small correction, and Eq.~(\ref{delta})
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should not be solved self-consistently. This follows from the fact that the
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diffusion coefficient of electrons at fixed energy entering the Cooperon
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expression is that of non-interacting electrons, and is given by the limit $%
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T\rightarrow 0$, $L_{\varphi }\rightarrow \infty $ and therefore $\delta
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\rightarrow 0$. Then the correction at finite $T$ is given by
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\bea
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\frac{D}{D_{0}} &=& \frac{1}{\left( \frac{D_{0}}{D}\right) _{0}-\delta \left(
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\frac{D_{0}}{D}\right) }\cr
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&\approx & \left( \frac{D}{D_{0}}\right) _{0}+\left( \frac{D}{D_{0}}\right) _{0}
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\frac{3}{\pi k_{F}^{2}l^{2}}\frac{l}{L_{\varphi }}%
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\eea%
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where
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\be
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\lim_{T\rightarrow 0}\frac{D}{D_{0}}\equiv \left( \frac{D}{D_{0}}\right)
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_{0}.
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\ee%
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Using the relation $\sigma _{3d}=(e^{2}/\hbar )nD$ where the longitudinal
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sheet conductance $\sigma _{\square }=\sigma _{3d}t$, with $t$ being the
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film thickness, we finally get the temperature dependent weak localization
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correction term
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\bea
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\frac{\delta \sigma _{\square }}{L_{00}} &=& \left( \frac{D}{D_{0}}\right) _{0}%
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\frac{2}{\pi }\frac{t}{L_{\varphi }}\cr
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\left( \frac{D}{D_{0}}\right)_{0} &\approx &\frac{2}{1+\sqrt{1+\frac{4R_{0}^{2}}{a^{2}}}}
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\label{WL}
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\eea%
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where $R_{0}=L_{00}/\sigma _{\square }(T$=$0)$, $L_{00}=e^{2}/\pi h$, $%
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a=3\pi/2k_{F}tb_{0}$, $b_{0}$ is a number of order unity and we
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have solved the self-consistent equation for $D$ in order to express $D_{0%
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\text{ }}$in terms of $D$ and finally $R_{0}$. Thus in this case, the weak
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localization correction has a prefactor which is not universal. While this
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reduces to the well-known universal result at weak disorder $R_{0}\ll a$, it
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becomes dependent on disorder characterized by the sheet resistance $R_{0}$
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at strong disorder and at the same time substantially extends the 3d regime
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near the transition.
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Using the expression for $L_{\varphi }$ (Eq.~(\ref{L-phi-3d})) into Eq.~(\ref%
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{WL}), we finally obtain the total conductivity, including the quantum
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correction to the conductivity due to weak localization in 3d arising from
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scattering of electrons off antiferromagnetic spin waves in Mn,
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\begin{equation}
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\frac{\sigma _{\square }}{L_{00}}=A+\frac{B}{\sqrt{\sinh [\Delta /T]}},
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\label{sigmaWL}
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\end{equation}%
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\textbf{\textbf{}}where the parameter $A$ is temperature independent and the parameter
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\bea
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B &\equiv & \left( \frac{D}{D_{0}}\right) _{0}\frac{2}{\pi ^{2}}\left( \frac{%
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12\pi }{5}\right) ^{1/2}\left( \frac{\Delta }{\bar{J}}\right) ^{3/2}tk_{F}\cr%
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&=&\frac{2c}{1+\sqrt{1+\frac{4R_{0}^{2}}{a^{2}}}},
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\label{BFit}
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\eea%
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where
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\be
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c\equiv \left( \frac{\Delta }{\bar{J}}\right) ^{3/2}\left( \frac{%
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48t^{2}k_{F}^{2}}{5\pi}\right) ^{1/2}.
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\label{cFit}
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\ee
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The data presented here is for a single film prepared with an initial $R_0
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\approx$~6~k$\Omega$. Disorder was consequently increased in incremental
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stages up to 180~k$\Omega$ by annealing at approximately 280~K~\cite%
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{misra_2011}. Additional samples were grown at intermediate disorder and
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measured to check reproducibility.
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Figure~\ref{fig:cond} shows the conductivity data for two samples with
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disorder $R_{0}=$~17573~$\Omega $ and 63903~$\Omega $ with corresponding
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fittings to the expression (\ref{sigmaWL}) where $A$ and $B$ are taken as
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fitting parameters and $\Delta =$~16~K is the spin wave gap. The fits are
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sensitive to the parameters $A$ and $B$ but relatively insensitive to $%
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\Delta $. We find that $\Delta =$~16~$\pm $~4~K provides good fittings in
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the whole range of disorder (from 6 to 180~k$\Omega $).
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\begin{figure}[tbp]
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\begin{center}
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\includegraphics[width=9cm]{fig_1_16.eps}
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\end{center}
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\caption{The temperature-dependent normalized conductivity (open squares)
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for two samples with the indicated disorder strengths of $R_0 =$~17573~$%
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\Omega$ and 63903~$\Omega$ show good agreement with theory (solid lines).
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The fitting parameters $A$ and $B$ are indicated for each curve with the
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error in the least significant digit indicated in parentheses.}
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\label{fig:cond}
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\end{figure}
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Figure~\ref{fig:parb} shows the dependence of the parameter $B$ on the
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disorder strength $R_0$ (open squares) and a theoretical fit (solid line)
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using Eq.~(\ref{BFit}), where $c$ and $a$ are fitting parameters. The solid
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line for this two-paramener fit is drawn for the best-fit values $c=0.67 \pm
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0.04$ and $a= 28 \pm 3$~k$\Omega$. We note that the fit is of reasonable
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|
quality over most of the disorder range except for the film with the least
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disorder ($R_0 = 6$~k$\Omega$) where $B = 0.77$,
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somewhat below the saturated value
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$B = c = 0.67$ evaluated from Eq.~(\ref{BFit}) at $R_0 = 0$. Using higher
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values of $c$ (e.g., $c=0.8$) and lower values of $a$ (eg., $a = 22$~k$\Omega$)
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improves the fit at low disorder strengths but
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increases the discrepancy at higher disorder strengths.
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|
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%L_phi/t = 2/pi*2/(1+sqrt(1+16))/0.5, 2/pi*2/(1+sqrt(1+1))/0.25
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%http://hyperphysics.phy-astr.gsu.edu/hbase/tables/fermi.html , k_F = sqrt(2*m_e*(10.9 eV))/(hbar) = 1.7E10 1/m
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% (bar(J) / \Delta) ^ 3/2 = (48*(2e-9)^2*(2.7e9)^2/5/pi/(0.65)^2) ^0.5 = 8360 = 20 ^ 3
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%A = \bar{J} / k_F , \bar{J} = nJ
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|
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|
Substituting the Fermi energy for bulk Mn~\cite{ashcroft_1976},
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|
a thickness $t=2$~nm known to 20\% accuracy, together with the best-fit
|
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|
value for $c$ into Eq.~(\ref{cFit}), we calculate the value $\bar{J} =$~320~$%
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|
\pm$~93~K. Gao et al.~\cite{gao_2008} performed inelastic scanning tunneling
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|
spectroscopy (ISTS) on thin Mn films and reported $\Delta$ in the range from
|
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|
30 to 60~K and $\bar{J}=vk_F=$~3150~$\pm$~200~K. The agreement of energy gaps is
|
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|
good; however our significantly lower value of $\bar{J}$ is probably due to the
|
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|
high disorder in our ultra thin films.
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|
|
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|
Since the temperature-dependent correction $B/\sqrt{\sinh (\Delta /T)}$ of
|
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|
Eq.~\ref{sigmaWL} is small compared to the parameter $A$, we can write
|
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|
$\sigma_{\square} \approx 1/R_0$ so that Eq.~\ref{sigmaWL} reduces to the
|
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|
expression $A \approx 1/L_{00}R_0$. The logarithmic plot derived by taking the
|
||
|
logarithm of both sides of this approximation is shown in the inset of
|
||
|
Fig.~\ref{fig:parb}. The slope of -1 confirms the linear dependence of $A$ on
|
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|
$1/R_0$ and the intercept of 5.01 (10$^{5.01}\approx $~102~k$\Omega$) is
|
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|
within 20\% of the expected theoretical value $L_{00}=$~81~k$\Omega $,
|
||
|
for the normalization constant. Accordingly, the conductivity corrections in
|
||
|
Eq.~\ref{sigmaWL} are small compared to the zero temperature conductivity and
|
||
|
the normalization constant $L_{00}$ for the conductivity is close to the
|
||
|
expected theoretical value.
|
||
|
|
||
|
Using Eq.~(\ref{WL}) and the obtained value for $a\approx $~28~k$\Omega $ we can
|
||
|
compare the dephasing length ($L_{\varphi }$) with the thickness ($t\approx $%
|
||
|
~2~nm) at 16~K. For the sample with $R_{0}=$~63903~$\Omega $ the ratio $%
|
||
|
L_{\varphi }/t\approx $~0.5 and for the sample with $R_{0}=$~17573~$\Omega $
|
||
|
$L_{\varphi }/t\approx $~2. The latter estimate assumes no spin
|
||
|
polarization, while a full polarization would imply $L_{\varphi }/t\approx $%
|
||
|
~1. Thus $L_{\varphi }$ is smaller than or close to the thickness of the
|
||
|
film, which keeps the film in the three-dimensional regime for almost all
|
||
|
temperatures and disorder strengths considered.
|
||
|
|
||
|
\begin{figure}[tbp]
|
||
|
\begin{center}
|
||
|
\includegraphics[width=9cm]{fig_2_16.eps}
|
||
|
\end{center}
|
||
|
\caption{Dependence of the fitting parameters $B$ and $A$ (inset) on
|
||
|
disorder $R_0$ for $\Delta=$~16~K. The fitting parameters are indicated for
|
||
|
each curve with the error in the least significant digit indicated in
|
||
|
parentheses.}
|
||
|
\label{fig:parb}
|
||
|
\end{figure}
|
||
|
|
||
|
In conclusion, we have performed \textit{in situ} transport measurements on
|
||
|
ultra thin Mn films, systematically varying the disorder ($R_{0}=R_{xx}$($T=$%
|
||
|
~5~K)). The obtained data were analyzed within a weak localization theory in
|
||
|
3d generalized to strong disorder. In the temperature range considered
|
||
|
inelastic scattering off spin waves is found to be strong giving rise to a
|
||
|
dephasing length shorter than the film thickness, which places these systems
|
||
|
into the 3d regime. The obtained value for the spin wave gap was close to
|
||
|
the one measured by Gao et al.~\cite{gao_2008} using ISTS, while the
|
||
|
exchange energy was much smaller.
|
||
|
|
||
|
This work has been supported by the NSF under Grant No 1305783 (AFH).
|
||
|
PW thanks A.\ M.\ \ Finkel'stein for useful discussions and acknowledges
|
||
|
partial support through the DFG research unit "Quantum phase transitions".
|
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|
|
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|
\bibliographystyle{apsrev}
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||
|
\bibliography{bibl}
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\end{document}
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