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---
layout: deck
---
{% assign deck = site.data.deck %}
{% capture outline_content %}
<ol class="large full omega">
<li>Overview of spin injection experiment and nonlocal resistance</li>
<li>Analytic solution including effect of contacts</li>
<li>Fitting the solution to real data</li>
<li>Limitations of fitting regimes</li>
</ol>
{% endcapture %}
{% slide : Outline %}
<h3 class="full center omega">
<a href="https://evansosenko.com/spin-lifetime/">Click here for full results and the publication.</a>
</h3>
{{ outline_content }}
{% image device.svg class="two-thirds center" %}
{% endslide %}
{% slide : Outline %}
{{ outline_content }}
<h3>Familiar integral expression that ignores contact effects</h3>
<p class="large">$$R_{\text{NL}} ∝ \int_0^∞ \frac{1}{\sqrt{4 π D t}} \exp{\left[ - \frac{L^2}{4 D t} \right]} e^{-t / τ} \cos{ω t} \: dt$$</p>
{% endslide %}
{% slide#geometry : Device geometry %}
{% image device.svg %}
<ul class="equation small first">
<li>\( L \) : contact spacing</li>
<li>\( D \) : diffusion constant</li>
<li>\( τ \) : spin lifetime</li>
<li>\( λ = \sqrt{D τ} \)</li>
<li>\( ω = g μ_B B / ħ \)</li>
</ul>
<ul class="equation small last">
<li>\( μ_s = \frac{1}{2} \left( μ_↑ - μ_↓ \right) \)</li>
<li>\( J_{↑↓} = σ_{↑↓} ∇μ_{↑↓} \)</li>
<li>\( J_{↑↓}^C = Σ_{↑↓} \left( μ^N_{↑↓} - μ^F_{↑↓} \right)_c \)</li>
<li>\( J = J_↑ + J_↓ \)</li>
<li>\( J_s = J_↑ - J_↓ \)</li>
</ul>
<div class="full omega">
<p class="large">$$D ∇^2 μ_s - \frac{μ_s}{τ} + ω × μ_s = 0$$</p>
<p class="large">$$V ∝ μ_s^N(x = L)$$</p>
</div>
<p class="indent full omega">
{% reference ActaPhysicaSlovaca.57.4_5.565-907 %}<br />
{% reference PhysRevB.37.5312 %}<br />
{% reference PhysRevB.67.052409 %}<br />
{% reference PhysRevB.80.214427 %}
</p>
{% include note.html id="geometry" %}
{% endslide %}
{% slide : Nonlocal resistance %}
<p class="large half">\( R_{\text{NL}}^± = ± V / I = ± P^2 R_N f \)</p>
<p class="large half right omega">\( R_N = \frac{λ}{W} \frac{1}{σ_G} = \frac{λ}{L W} \frac{1}{σ^N} \)</p>
{% slide div.full.center.omega %}
<h3>Solution</h3>
<p class="large">$$f = \operatorname{Re}{ \left\{ \left[ 2 \left( \sqrt{1 + i ω τ} + λ / r \right) e^{\left( L / λ \right) \sqrt{1 + i ω τ}} \\ + \frac{\left( λ / r \right)^2}{\sqrt{1 + i ω τ}} \sinh{\left( L / λ \right) \sqrt{1 + i ω τ}} \right]^{-1} \right\} }$$</p>
{% endslide %}
{% slide div.full.center.omega %}
<h3>Only scales that appear in \( f \)</h3>
<ul class="equation inline">
<li>\( L / λ \)</li>
<li>\( λ / r \)</li>
<li>\( ω τ \)</li>
</ul>
{% endslide %}
{% include note.html id="nonlocal_resistance" %}
{% endslide %}
{% slide : Tunneling contacts %}
<figure class="inset-container two-thirds">
{% image plots/fig_4b_parallel.svg %}
{% image plots/fig_4b_transparent_contacts_parallel.svg class="inset" %}
<figcaption>Fit finite contact resistance case to parallel field data from Fig. 4b of W. Han, et al.
Inset: infinite contact resistance case.</figcaption>
</figure>
<div class="one-third omega">
<h3>Finite contact resistance</h3>
<ul class="equation">
<li>\( R_C = 5 × 10^5 \; \text{kΩ} \)</li>
<li>\( τ = 427 \; \text{ps} \)</li>
<li>\( D = 0.014 \; \text{m}^2 / \text{s} \)</li>
<li>\( λ = 2.5 \; \text{μm} \)</li>
</ul>
<h3>Infinite contact resistance</h3>
<ul class="equation">
<li>\( τ = 427 \; \text{ps} \)</li>
<li>\( D = 0.014 \; \text{m}^2 / \text{s} \)</li>
</ul>
</div>
<p class="indent full omega">{% reference PhysRevLett.105.167202 %}</p>
{% include note.html id="fits" %}
{% endslide %}
{% slide : Transparent contact %}
<figure class="inset-container two-thirds">
{% image plots/fig_4d_difference.svg %}
{% image plots/fig_4d_transparent_contacts_difference.svg class="inset" %}
<figcaption>Fit finite contact resistance case to difference \( |R_\text{NL}^+ - R_\text{NL}^-| \) field data from Fig. 4d of W. Han, et al.
Inset: infinite contact resistance case.</figcaption>
</figure>
<div class="one-third omega">
<h3>Finite contact resistance</h3>
<ul class="equation">
<li>\( R_C = 3 \; \text{kΩ} \)</li>
<li>\( τ = 130 \; \text{ps} \)</li>
<li>\( D = 0.021 \; \text{m}^2 / \text{s} \)</li>
<li>\( λ = 1.66 \; \text{μm} \)</li>
</ul>
<h3>Infinite contact resistance</h3>
<ul class="equation">
<li>\( τ = 78 \; \text{ps} \)</li>
<li>\( D = 0.01 \; \text{m}^2 / \text{s} \)</li>
<li>\( λ = 1.4 \; \text{μm} \)</li>
</ul>
</div>
<p class="indent full omega">{% reference PhysRevLett.105.167202 %}</p>
{% include note.html id="fits" %}
{% endslide %}
{% slide : Transparent contacts %}
<figure class="two-thirds">
{% image plots/fig_4d_normalized_difference.svg %}
<figcaption>Fit finite contact resistance case to normalized difference \( |R_\text{NL}^+ - R_\text{NL}^-| \) field data from Fig. 4d of W. Han, et al.</figcaption>
</figure>
<div class="one-third omega">
<h3>Finite contact resistance</h3>
<ul class="equation">
<li>\( R_C = 0.3 \; \text{kΩ} \)</li>
<li>\( τ = 800 × 10^4 \; \text{ps} \)</li>
<li>\( D = 0.015 \; \text{m}^2 / \text{s} \)</li>
<li>\( λ = 350 \; \text{μm} \)</li>
</ul>
</div>
<p class="indent full omega">{% reference PhysRevLett.105.167202 %}</p>
{% include note.html id="fits" %}
{% endslide %}
{% slide : Limits %}
<figure class="two-thirds">
{% image plots/PhysRevB.86.235408.fig.3.svg %}
<figcaption>Fig. 3 from T. Maassen, et al.</figcaption>
</figure>
<div class="one-third omega">
<ul class="equation">
<li>❱ \( r → ∞ \) or \( λ / r ≪ 1 \)</li>
<li>\( r \) terms are negligible.</li>
<li>Scale and zeros set by \( τ \) and \( D \).</li>
</ul>
<ul class="equation">
<li>❱ \( ω τ ≫ 1 \) & \( λ / r >1 \)</li>
<li>Zeros set by \( D \) only.</li>
<li>Normalized case scales as \( f / f_0 ∝ \frac{(λ / r)^2}{\sqrt{ω τ}} \).</li>
<li>Can fit to increased \( τ \) with moderate decrease in \( r \).</li>
</ul>
</div>
<p class="indent full omega">{% reference PhysRevB.86.235408 %}</p>
{% include note.html id="limits" %}
{% endslide %}
{% slide : Conclusion %}
<ol class="large full omega">
<li>Solve system with finite contact resistance</li>
<li>Analytic expression for \( R_{\text{NL}} \)</li>
<li>Fit to real Hanle curve data and obtain reasonable results</li>
<li>The \( r \) parameter introduces other parameter regimes and scaling freedom which can also give good fits</li>
<li>Able to explain these regimes as limits of the analytic expression</li>
</ol>
{% endslide %}