index.html

---
layout: deck
---

{% assign deck = site.data.deck %}

$${% include tex-header.tex %}$$

{% slide : Two dimensional systems %}
  <div class="outline first">
    <h3><a href="#spin-lifetime">Spin lifetime and Hanle curve fitting</a></h3>
    {% image spin-lifetime/device.svg %}
  </div>

  {% slide div.outline.last %}
    <h3><a href="#dichalcogenides">Superconducting phases of monolayer transition-metal dichalcogenides</a></h3>
    <div class="lattice-image"></div>
  {% endslide %}
{% endslide %}

{% slide#spin-lifetime : Spin lifetime %}
  <div>
    <h3>Outline</h3>
    <ol>
      <li>Motivation</li>
      <li>Model and solution</li>
      <li>Hanle curve fitting</li>
      <li>Regimes and results</li>
    </ol>
  </div>

  {% slide div %}
    <h3>Motivation</h3>
    <ol>
      <li>Theoretical lifetime predictions longer than measured values:
          \( \text{ms} \) vs. \( \text{ps} \)</li>
      <li>Finite contact resistance mismatch: a potential candidate</li>
      <li>Unified analytic solution for fitting data in all limits</li>
    </ol>
  {% endslide %}

  {% image spin-lifetime/device.svg %}

  <p>{% reference PhysRevB.89.245436 --file spintronics %}</p>
{% endslide %}

{% capture slide_fits_1 %}
{% slide.fits : Fits %}
  <div>
    <h3>Tunneling contacts</h3>
    <figure>
      {% image spin-lifetime/fig_4a_difference.svg %}
      <figcaption>Fit to parallel field data from Fig. 4a of W. Han, et al.</figcaption>
    </figure>

    <ul class="equation">
      {% include spin-lifetime-values.html id="a" %}
    </ul>
  </div>

  <div>
    <h3>Tunneling contacts</h3>
    <figure>
      {% image spin-lifetime/fig_4b_difference.svg %}
      <figcaption>Fit to parallel field data from Fig. 4b of W. Han, et al.</figcaption>
    </figure>

    <ul class="equation">
      {% include spin-lifetime-values.html id="b" %}
    </ul>
  </div>

  <p>{% reference PhysRevLett.105.167202 --file spintronics %}</p>
{% endslide %}
{% endcapture %}

{{ slide_fits_1 }}

{% slide.geometry : Device geometry %}
  <div class="device-image"></div>

  <ul class="equation 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 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>
    <p>$$D ∇^2 μ_s - \frac{μ_s}{τ} + ω × μ_s = 0$$</p>
    <p>$$V ∝ μ_s^N(x = L)$$</p>
    <p>$$R_\text{NL} = V / I$$</p>
  </div>

  <p>
    {% reference ActaPhysicaSlovaca.57.4_5.565-907 --file spintronics %}<br />
    {% reference PhysRevB.37.5312 --file spintronics %}<br />
    {% reference PhysRevB.67.052409 --file spintronics %}<br />
    {% reference PhysRevB.80.214427 --file spintronics %}
  </p>
{% endslide %}

{% slide : Motivation for solution %}
  <h3>Existing results</h3>
  <ul>
    <li>All existing analytic expressions ignore contact resistance</li>
    <li>Assume infinite resistance for convenience</li>
    <li>Integral form widely used but difficult to fit</li>
    <li>Only a numeric treatment of finite contact resistance</li>
  </ul>

  <p>$$\rNL ∝ \re{\frac{e^{- \left( L / λ \right) \sqrt{1 + i ω τ}}}{2 \sqrt{1 + i ω τ}}}$$</p>
  <p>$$\rNL ∝ \int_0^∞ \frac{1}{\sqrt{4 π D t}}
       \exp{\left[ - \frac{L^2}{4 D t} \right]} e^{-t / τ} \cos{ω t} \: dt$$</p>


  <h3>Our approach</h3>
  <ul>
    <li>Finite contact resistance</li>
    <li>Exact analytic expression</li>
    <li>Matches previous approaches in appropriate limits</li>
  </ul>

{% endslide %}

{% slide.solution : Non-local resistance %}
  <p>$$Δ \rNL = 2 P^2 R_N \left\lvert f \right\rvert$$</p>

  <ul class="equation">
    <li>\( r = \frac{R_F + R_C}{\rSQ} W \)</li>
    <li>\( \rSQ = W / σ^N \)</li>
    <li>\( R_N = \frac{λ}{W L} \frac{1}{σ^N} \)</li>
  </ul>

  <p class="long-equation">
    $$
    \begin{multline}
      f = \re \left\{ \left(
            \vphantom{
              \frac{
                \sinh{ \left[ \left( L / λ \right) \sqrt{1 + i ω τ} \right] }
              }{\sqrt{1 + i ω τ}}
            }
            2 \left[ \sqrt{1 + i ω τ} + (λ / r) \right]
            e^{\left( L / λ \right) \sqrt{1 + i ω τ}}
            \right. \right. \\ \left. \left.
            + (λ / r)^2 \frac{
                \sinh{ \left[ \left( L / λ \right) \sqrt{1 + i ω τ} \right] }
              }{\sqrt{1 + i ω τ}}
          \right)^{-1} \right\} .
    \end{multline}
    $$
  </p>

  <p>Only scales that appear in \( f \)</p>
  <ul class="equation">
    <li>\( L / λ \)</li>
    <li>\( λ / r \)</li>
    <li>\( ω τ \)</li>
  </ul>

  <p>{% reference PhysRevB.89.245436 --file spintronics %}</p>
{% endslide %}

{{ slide_fits_1 }}

{% slide.fits : Fits %}
  <div>
    <h3>Pinhole contacts</h3>
    <figure>
      {% image spin-lifetime/fig_4c_parallel.svg %}
      <figcaption>Fit to parallel field data from Fig. 4c of W. Han, et al.</figcaption>
    </figure>

    <ul class="equation">
      {% include spin-lifetime-values.html id="c" %}
    </ul>
  </div>

  <div>
    <h3>Transparent contacts</h3>
    <figure>
      {% image spin-lifetime/fig_4d_difference.svg %}
      <figcaption>Fit to parallel field data from Fig. 4d of W. Han, et al.</figcaption>
    </figure>

    <ul class="equation">
      {% include spin-lifetime-values.html id="d" %}
    </ul>
  </div>

  <p>{% reference PhysRevLett.105.167202 --file spintronics %}</p>
{% endslide %}

{% slide.regimes : Regimes %}
  <div>
    <h3>Zero field</h3>
    <p>$$Δ \rNL = \left( P_Σ^L \right)^2 R_N e^{- L / λ}$$</p>
  </div>

  <div>
    <h3>Tunneling contacts</h3>
    <p>$$f^∞ = \re{\frac{e^{- \left( L / λ \right) \sqrt{1 + i ω τ}}}{2 \sqrt{1 + i ω τ}}}$$</p>
  </div>

  <h3>Fits independent of lifetime</h3>
  <p>\( λ / r ≫ \sqrt{ω τ} ≫ 1 \)</p>
  <p>Zeros determined by
     $$L \sqrt{\frac{D}{2 ω}} + \frac{π}{4} = \frac{n π}{2}$$</p>

  <p>{% reference PhysRevB.89.245436 --file spintronics %}</p>
{% endslide %}

{% slide.fits : Fits: \( τ \) Independent Limit %}
  <div>
    <h3>Transparent contacts</h3>
    <figure>
      {% image spin-lifetime/fig_4d_difference_large_lifetime.svg %}
      <figcaption>Fit to parallel field data from Fig. 4d of W. Han, et al.</figcaption>
    </figure>

    <ul class="equation">
      {% include spin-lifetime-values.html id="d_1" %}
    </ul>
  </div>

  <div>
    <h3>Transparent contacts</h3>
    <figure>
      {% image spin-lifetime/fig_4d_difference_larger_lifetime.svg %}
      <figcaption>Fit to parallel field data from Fig. 4d of W. Han, et al.</figcaption>
    </figure>

    <ul class="equation">
      {% include spin-lifetime-values.html id="d_2" %}
    </ul>
  </div>

  <p>{% reference PhysRevLett.105.167202 --file spintronics %}</p>
{% endslide %}

{% slide#dichalcogenides : Dichalcogenides %}  <div>
  <div class="one-third">
    <h3>Outline</h3>
    <ol>
      <li>Motivation</li>
      <li>Valley physics</li>
      <li>Superconducting phase</li>
      <li>Future work</li>
    </ol>
  </div>

  {% slide div.two-thirds.omega %}
    <h3>Motivation</h3>
    <ol>
      <li>Active and emerging field</li>
      <li>Monolayer graphene-like system with new valley physics</li>
      <li>Potentially a natural spin valve material</li>
    </ol>
  {% endslide %}

  <div class="crystal-image full center"></div>
{% endslide %}

{% slide : Effective Hamiltonian %}
  <div class="lattice-image half"></div>

  <ul class="half omega">
    <li>\( \mathrm{MoS_2} \), \( \mathrm{WS_2} \), \( \mathrm{MoSe_2} \), \( \mathrm{WSe_2} \)</li>
    <li>Similar to monolayer graphene:
        two inequivalent valleys: \( \vect{K} \), \( \vect{K}' \)</li>
    <li>Strong <strong>spin-orbit coupling</strong>
        and <strong>inversion symmetry breaking</strong></li>
    <li>Leads to opposite valley Berry curvature</li>
    <li>Two state tight binding model: \( d_{z^2} \), and \( d_{xy} \), \( d_{x^2 - y^2} \)</li>
  </ul>

  {% slide div %}
    <p class="full center">
      $$
      H_0^{τ σ} \exOfK =
        a t \left(τ k_x σ_x + k_y σ_y \right) ⊗ I_2
        + \frac{Δ}{2} σ_z ⊗ I_2
        - λ τ \left(σ_z - 1 \right) ⊗ S_z
      $$
    </p>

    <p class="small full center">
      $$
      H_0^{τ σ} \exOfK =
      \left[
      \begin{matrix}
        \dfrac{Δ}{2}                     & a t \left( τ k_x - i k_y \right) \\
        a t \left( τ k_x + i k_y \right) & λ τ σ - \dfrac{Δ}{2}
      \end{matrix}
      \right]
      $$
    </p>

    <p class="small full">{% reference PhysRevLett.108.196802 %}</p>
  {% endslide %}
{% endslide %}

{% slide.bands : Energy Bands %}
  <figure class="two-thirds">
    {% image dichalcogenides/bands.svg %}
    <figcaption>The eight energy bands for \( \mathrm{MoS_2} \).</figcaption>
  </figure>

  <div class="one-third omega">
    <ul class="equation">
      <li>\( Δ \)—band splitting</li>
      <li>\( λ \)—spin splitting</li>
      <li>\( τ \)—valley index</li>
      <li>\( σ \)—spin index</li>
    </ul>

    <h3>\( \mathrm{MoS_2} \)</h3>

    <ul class="equation">
      <li>\( a t = 3.15 \: \text{Å eV} \)</li>
      <li>\( Δ = 1.66 \: \text{eV} \)</li>
      <li>\( 2 λ = 0.15 \: \text{eV} \)</li>
      <li>\( μ = -0.83 \: \text{eV} \)</li>
    </ul>
  </div>

  <p class="full omega">
    $$
    E_{τ σ}^n \exOfK =
      \frac{1}{2} \left( λ τ σ
      + n \sqrt{ (2 a t)^2 \left\lvert \vect{k} \right\rvert^2
        + \left( Δ - λ τ σ \right)^2 } \right)
    $$
  </p>
{% endslide %}

{% capture dichalcogenides_optical_transitions %}
{% slide : Optical Transitions %}
  <figure class="half">
    {% image dichalcogenides/transitions.svg %}
    <figcaption>Optical transition rates for \( H_0 \).</figcaption>
  </figure>

  <figure class="half omega">
    {% image dichalcogenides/band_transitions.svg %}
    <figcaption>Optical transitions strongly coupled to light polarization.</figcaption>
  </figure>

  <div class="half">
    <p>
      \( \vect{P}^{τ σ} \exOfK =
        \frac{m_0}{ħ}
        \left\langle u_+ \right\rvert ∇_{\vK} H_0^{τ σ} \exOfK
        \left\lvert u_- \right\rangle \)
    </p>

    <p>
      \( P_±^{τ σ} \exOfK = P_x^{τ σ} ± i P_y^{τ σ} \)
    </p>
  </div>

  <ul class="half omega">
    <li>Right circular polarization strongly couples to \( τ = + \) valley transitions</li>
    <li>Left circular polarization strongly couples to \( τ = - \) valley transitions</li>
  </ul>

  <p class="small full">{% reference PhysRevLett.108.196802 %}</p>
{% endslide %}
{% endcapture %}

{{ dichalcogenides_optical_transitions }}

{% slide : Vally Hall effect %}
    <figure class="half">
      {% image dichalcogenides/hall.svg %}
      <figcaption>Valley hall effect for electrons.</figcaption>
    </figure>

    <figure class="half omega">
      {% image dichalcogenides/band_transitions.svg %}
      <figcaption>Optical transitions strongly coupled to light polarization.</figcaption>
    </figure>

    <div class="half">
      <h3 class="">Semiclassical equations</h3>
      <p>\( \dot{\vect{r}} = ∇_{\vK} E \exOfK - \dot{\vect{k}} × \vect{Ω} \exOfK \)</p>
      <p>\( \dot{\vect{k}} = - e \vect{E} - e \dot{\vect{r}} × \vect{B} \)</p>

      <p>Anomalous velocity
         \( \vect{v} = e \vect{Ω} \exOfK × \vect{E} \)</p>
    </div>

    <div class="half omega">
      <h3 class="">Berry curvature</h3>
      <p>\(
         \vect{Ω}_{τ σ}^n \exOfK
         = ∇_{\vK} × \left\langle u_{τ σ}^n \exOfK
            \right\rvert i ∇_{\vK} \left\lvert u_{τ σ}^n \exOfK \right\rangle \)</p>

      <p>Broken inversion symmetry</p>
      <p>\( \vect{Ω}_{τ, σ}^n \exOfK = - \vect{Ω}_{-τ, σ}^n \exOfK ≠ 0 \)</p>
    </div>
{% endslide %}

{% slide : BCS Superconductivity %}
  <h3>Mean-field Hamiltonian</h3>
  <p>
    $$
    H - μ N =
      \sideset{}{}∑_{\vK σ} ξ_{\vK} c_{\vK σ}^† c_{\vK σ}
    - \sideset{}{}∑_{\vK}
        \left( \bar{Δ}_{\vK} c_{-\vK ↓} c_{\vK ↑}
          + Δ_{\vK} c_{\vK ↑}^† c_{-\vK ↓}^† \right)
    $$
  </p>

  <figure class="half">
    {% image dichalcogenides/bcs.svg %}
    <figcaption>Region of allowed parings for fixed center-of-mass momentum.</figcaption>
  </figure>

  <div class="half omega">
    <p>Number of available parings maximized when center-of-mass momentum is zero</p>

    <h3>Quasiparticle operators</h3>
    <p>\( b_{\vK σ} = σ \cos{θ_{\vK}} c_{\vK σ} + \sin{θ_{\vK}} c_{-\vK, -σ}^† \)</p>

    <h3>Diagonalized</h3>
    <p>\( \sideset{}{}∑_{\vK σ} λ_{\vK} b_{\vK σ}^† b_{\vK σ} \)</p>
  </div>
{% endslide %}

{% slide : Induced Superconductivity %}
  <h3>Intervalley pairing</h3>

  <figure class="half">
    {% image dichalcogenides/superconducting_states.svg %}
    <figcaption>BCS pairs for induced superconducting states.</figcaption>
  </figure>

  <div class="half omega">
    <ul class="full omega">
      <li>\( a^ν_{τ σ} \)—orbital operators</li>
      <li>\( b_α \)—quasiparticle operators</li>
      <li>BCS pairs in opposite valleys</li>
      <li>Reduces to standard BCS Hamiltonian
          where \( α = τ = σ \) plays the role of the spin index</li>
      <li>Not a singlet ground state: mixture of singlet and triplet states</li>
    </ul>

    <p class="small full center omega">
      $$
      \begin{equation}
        H_V = -
          \sideset{}{'}∑_{\vK} \sideset{}{}∑_{ν, τ} Δ_ν
          {a^ν_{-τ ↓}}^† \exOfMK
          {a^ν_{τ ↑}}^†  \exOfK
          + \hc
      \end{equation}
      $$
    </p>

    <p class="small full center omega">
      $$
      \begin{equation}
        H - μ N =
          \sideset{}{'}∑_{\vK} \sideset{}{}∑_α λ_{\vK}^α b_{\vK α}^† b_{\vK α}
            + \sideset{}{'}∑_{\vK} \left(ξ_{\vK ↓} + λ_{\vK}^- \right) .
      \end{equation}
      $$
    </p>
  </div>
{% endslide %}

{{ dichalcogenides_optical_transitions }}

{% slide.sc-excitations : SC Optical Excitations %}
  <figure class="half">
    {% image dichalcogenides/excitations-1.svg %}
    <figcaption>Induced superconducting optical transition rates for \( τ = + \).</figcaption>
  </figure>

  <figure class="half omega">
    {% image dichalcogenides/excitations-2.svg %}
    <figcaption>Induced superconducting optical transition rates for \( τ = - \).</figcaption>
  </figure>

  <div class="half">
    <p>
      \( \vect{P} \exOfK =
        \frac{m_0}{ħ}
        \left\langle Ω_f \right\rvert ∇_{\vK} H^{τ σ} \exOfK
        \left\lvert Ω \right\rangle \)
    </p>

    <p>
      \( P_± \exOfK = P_x ± i P_y \)
    </p>
  </div>

  <div class="half omega">
    <p>
       \( \left\lvert Ω \right\rangle
       = ∏_{\vK} b_{\vK ↑} b_{-\vK ↓} \left\lvert 0 \right\rangle \)
    </p>

    <p class="small">
       \( \left\lvert Ω_f \right\rangle =
         \begin{cases}
         {c^+_α}^† \exOfK b_{-α} \exOfMK \left\lvert Ω \right\rangle &amp; k &gt; k_μ \\
         {c^+_α}^† \exOfK b_{-α}^† \exOfMK \left\lvert Ω \right\rangle &amp; k &lt; k_μ
         \end{cases} \)
    </p>
  </div>
{% endslide %}

{% slide : SC Optical Excitations %}
  <h3>Compare to normal transitions</h3>

  <figure class="half">
    {% image dichalcogenides/excitations-closeup-1.svg %}
    <figcaption>
      Strong induced superconducting optical transition rates.
      Dashed lines are \( H_0 \) transitions.
    </figcaption>
  </figure>

  <figure class="half omega">
    {% image dichalcogenides/excitations-closeup-2.svg %}
    <figcaption>
      Weak induced superconducting optical transition rates.
      Dashed lines are \( H_0 \) transitions.
    </figcaption>
  </figure>

  <ul class="full">
    <li>Upper band excitations are now paired with lower band quasiparticle excitations</li>
    <li>Valley-polarization coupling is retained even in the superconducting case</li>
    <li>Contrast is reduced in an region around the chemical potential</li>
  </ul>
{% endslide %}

{% slide : Future Work %}
  <h3>Induced superconductivity</h3>
  <ul>
    <li>Only looked at \( \mathrm{MoS_2} \)</li>
    <li>Can tune the parameters to understand how they affect the physics</li>
    <li>Look at other properties of this state, e.g., magnetic susceptibility</li>
  </ul>

  <h3>Intrinsic superconductivity</h3>
  <ul>
    <li>Derived from density-density interactions</li>
    <li>Already have the projected interaction term</li>
    <li>Apply mean-field and an analogous analysis</li>
    <li>Both intervalley and intravalley pairing</li>
  </ul>

  {% slide div %}
    <p class="small one-third">
      $$
      H_V = - \sideset{}{'}∑_{\vK, \vK'}
      v \left( \vK' - \vK \right) A \left( \vK, \vK' \right)
      $$
    </p>

    <p class="small two-thirds omega">
      $$
      \begin{align}
        A \left( \vK, \vK' \right) =
        &amp; B^2 \left( \vK, \vK' \right) \exInteration{+ ↑}{+ ↑}
        \\ +
        &amp; B^2 \left( \vK', \vK\right) \exInteration{- ↓}{- ↓}
        \\ +
        &amp; \left\lvert B \left( \vK, \vK' \right) \right\rvert^2
        \left[
          \exInteration{+ ↑}{- ↓}
          \\ +
          \exInteration{- ↓}{+ ↑}
        \right]
      \end{align}
      $$
    </p>
  {% endslide %}
{% endslide %}