# Effect of contacts on spin lifetime measurements in graphene

## 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

## 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

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

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

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

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
• Spin injection experimental geometry.
• Ferromagnetic contacts (F) on semiconductor (N) (graphene).
• Current flows on the left only from the injector (I) at $$x = 0$$ into (N).
• Spin current diffuses to the detector (D) at $$x = L$$.
• Measured voltage is proportional to the spin chemical potential at (D).
• Modeled by diffusion equation.
• Include effects on contacts with $$J_{↑↓}^C$$.
• Solve by assuming appropriate continuity conditions on current and spin current at (I) and (D).
• Applied transverse magnetic field causes spins to precess as they diffuse.
• Nonlocal resistance is the voltage normalized by the injected current.
• Boundary conditions give linear system, solve to obtain $$μ_s^N$$.
• Sign will change when alignment of the field (as compared to the ferromagnet) changes from parallel (P) to antiparallel (AP).
• $$P$$ is a polarization factor, $$P ≤ 1$$. Depends on resistances and conductivities.
• $$R_N$$ is effectively the resistance of (N) over the spin diffusion length $$λ$$.
• $$r$$ is a new length scale introduced by the contact resistance and effectively proportional to it.
• Expression reduces to the typical case where contacts are ignored in the limit $$r → ∞$$.
• Fit done with Python using least squares.
• Tunneling contacts with finite $$r$$ give same $$τ$$.
• This is to be expected since tunneling contacts have high effective resistance.
• Transparent contacts with finite $$r$$ only changed $$τ$$ by a factor of 2.
• All fits have Chi-squared much less than unity.
• There are a few key limits we considered.
• Tunneling contacts when $$r ≫ λ$$, essentially $$R_C → ∞$$, and all the terms with $$λ / r$$ become small. Then, $$f$$ reduces to the previously considered case that ignores the contact resistance.