# Effect of contacts on spin lifetime measurements in graphene

- Evan Sosenko
- with
- Vivek Aji

## Outline

### Click here for full results and the publication.

- Overview of spin injection experiment and nonlocal resistance
- Analytic solution including effect of contacts
- Fitting the solution to real data
- Limitations of fitting regimes

## Outline

- Overview of spin injection experiment and nonlocal resistance
- Analytic solution including effect of contacts
- Fitting the solution to real data
- 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

### 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

### 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

## Limits

- ❱ \( r → ∞ \) 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

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