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Effect of contacts on spin lifetime measurements in graphene

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Outline

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  1. Overview of spin injection experiment and nonlocal resistance
  2. Analytic solution including effect of contacts
  3. Fitting the solution to real data
  4. Limitations of fitting regimes

Outline

  1. Overview of spin injection experiment and nonlocal resistance
  2. Analytic solution including effect of contacts
  3. Fitting the solution to real data
  4. Limitations of fitting regimes

Familiar integral expression that ignores contact effects

$$R_{\text{NL}} ∝ \int_0^∞ \frac{1}{\sqrt{4 π D t}} \exp{\left[ - \frac{L^2}{4 D t} \right]} e^{-t / τ} \cos{ω t} \: dt$$

Device geometry

  • \( L \) : contact spacing
  • \( D \) : diffusion constant
  • \( τ \) : spin lifetime
  • \( λ = \sqrt{D τ} \)
  • \( ω = g μ_B B / ħ \)
  • \( μ_s = \frac{1}{2} \left( μ_↑ - μ_↓ \right) \)
  • \( J_{↑↓} = σ_{↑↓} ∇μ_{↑↓} \)
  • \( J_{↑↓}^C = Σ_{↑↓} \left( μ^N_{↑↓} - μ^F_{↑↓} \right)_c \)
  • \( J = J_↑ + J_↓ \)
  • \( J_s = J_↑ - J_↓ \)

$$D ∇^2 μ_s - \frac{μ_s}{τ} + ω × μ_s = 0$$

$$V ∝ μ_s^N(x = L)$$

J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. Žutić, Acta Physica Slovaca 57, 565 (2007).
M. Johnson and R. H. Silsbee, Phys. Rev. B 37, 5312 (1988).
S. Takahashi and S. Maekawa, Phys. Rev. B 67, 052409 (2003).
M. Popinciuc, C. Józsa, P. J. Zomer, N. Tombros, A. Veligura, H. T. Jonkman, and B. J. van Wees, Phys. Rev. B 80, 214427 (2009).

Nonlocal resistance

\( R_{\text{NL}}^± = ± V / I = ± P^2 R_N f \)

\( R_N = \frac{λ}{W} \frac{1}{σ_G} = \frac{λ}{L W} \frac{1}{σ^N} \)

Solution

$$f = \operatorname{Re}{ \left\{ \left[ 2 \left( \sqrt{1 + i ω τ} + λ / r \right) e^{\left( L / λ \right) \sqrt{1 + i ω τ}} \\ + \frac{\left( λ / r \right)^2}{\sqrt{1 + i ω τ}} \sinh{\left( L / λ \right) \sqrt{1 + i ω τ}} \right]^{-1} \right\} }$$

Only scales that appear in \( f \)

  • \( L / λ \)
  • \( λ / r \)
  • \( ω τ \)

Tunneling contacts

Fit finite contact resistance case to parallel field data from Fig. 4b of W. Han, et al. Inset: infinite contact resistance case.

Finite contact resistance

  • \( R_C = 5 × 10^5 \; \text{kΩ} \)
  • \( τ = 427 \; \text{ps} \)
  • \( D = 0.014 \; \text{m}^2 / \text{s} \)
  • \( λ = 2.5 \; \text{μm} \)

Infinite contact resistance

  • \( τ = 427 \; \text{ps} \)
  • \( D = 0.014 \; \text{m}^2 / \text{s} \)

W. Han, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong, A. G. Swartz, and R. K. Kawakami, Phys. Rev. Lett. 105, 167202 (2010).

Transparent contact

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.

Finite contact resistance

  • \( R_C = 3 \; \text{kΩ} \)
  • \( τ = 130 \; \text{ps} \)
  • \( D = 0.021 \; \text{m}^2 / \text{s} \)
  • \( λ = 1.66 \; \text{μm} \)

Infinite contact resistance

  • \( τ = 78 \; \text{ps} \)
  • \( D = 0.01 \; \text{m}^2 / \text{s} \)
  • \( λ = 1.4 \; \text{μm} \)

W. Han, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong, A. G. Swartz, and R. K. Kawakami, Phys. Rev. Lett. 105, 167202 (2010).

Transparent contacts

Fit finite contact resistance case to normalized difference \( |R_\text{NL}^+ - R_\text{NL}^-| \) field data from Fig. 4d of W. Han, et al.

Finite contact resistance

  • \( R_C = 0.3 \; \text{kΩ} \)
  • \( τ = 800 × 10^4 \; \text{ps} \)
  • \( D = 0.015 \; \text{m}^2 / \text{s} \)
  • \( λ = 350 \; \text{μm} \)

W. Han, K. Pi, K. M. McCreary, Y. Li, J. J. I. Wong, A. G. Swartz, and R. K. Kawakami, Phys. Rev. Lett. 105, 167202 (2010).

Limits

Fig. 3 from T. Maassen, et al.
  • ❱ \( r → ∞ \) or \( λ / r ≪ 1 \)
  • \( r \) terms are negligible.
  • Scale and zeros set by \( τ \) and \( D \).
  • ❱ \( ω τ ≫ 1 \) & \( λ / r >1 \)
  • Zeros set by \( D \) only.
  • Normalized case scales as \( f / f_0 ∝ \frac{(λ / r)^2}{\sqrt{ω τ}} \).
  • Can fit to increased \( τ \) with moderate decrease in \( r \).

T. Maassen, I. J. Vera-Marun, M. H. D. Guimarães, and B. J. van Wees, Phys. Rev. B 86, 235408 (2012).

Conclusion

  1. Solve system with finite contact resistance
  2. Analytic expression for \( R_{\text{NL}} \)
  3. Fit to real Hanle curve data and obtain reasonable results
  4. The \( r \) parameter introduces other parameter regimes and scaling freedom which can also give good fits
  5. Able to explain these regimes as limits of the analytic expression