Spin-Orbit Based
Spin Ratchets

Manuel Strehl

Outline

A Short Introduction

Ratchets

Classical ratchets have two general properties:


H. Linke et.al.,
Science 286, 2314

Aim: Extract useful work out of unbiased fluctuations

Ratchet condition to get directed current: I(+U) \ne -I(-U)

P. Reimann, Phys. Rep. 361, 57 (2002)

Spin Ratchets

A. Pfund, D. Bercioux, K. Richter cond-mat/0601118

Spin Currents

Rashba, J. Supercond. 18, 137 (2005)

We are interested in \left<I_S\right> = \frac{1}{2} [I_S^{R/L}(+U_0) + I_S^{R/L}(-U_0)]

Problem: Leads must be selected, because spin current is not conserved → 2 possible definitions

Two definitions

Spin current measured in the lead with lower chemical potential:

\left<\mathcal{I}_S(U_0)\right> = \frac{1}{2}\left[I_S^R(+U_0) + I_S^L(-U_0)\right]
= \frac{1}{8\pi} \int_{E_C}^\infty \Delta f(E,U_0)\times[T_S(E,U_0)-T'_S(E,-U_0)]d\epsilon

with
T_S(E,U_0) = \sum_{\sigma=\pm1\in L}\left[T_{+,\sigma}(E,U_0)-T_{-,\sigma}(E,U_0)\right]
T'_S(E,U_0) = \sum_{\sigma=\pm1\in R}\left[T'_{+,\sigma}(E,U_0)-T'_{-,\sigma}(E,U_0)\right]

Spin current measured in the right lead for the two rocking situations:

\left<I_S^R(U_0)\right> = \frac{1}{2}\left[I_S^R(+U_0) + I_S^R(-U_0)\right]
= \frac{1}{8\pi} \int_{E_C}^\infty \Delta f(E,U_0)\times[T_S(E,U_0)-T_S(E,-U_0)]d\epsilon

with
T_S(E,U_0) = \sum_{\sigma=\pm1\in L}\left[T_{+,\sigma}(E,U_0)-T_{-,\sigma}(E,U_0)\right]

Matthias Scheid, Master's thesis

Comparison in Linear Response

In linear response (that is, for \pm U_0 \to 0):

\lim_{U_0\to 0} \left<I_S(U_0)\right> = \frac{1}{8\pi} \int_{E_C}^\infty \Delta f(E,0)\times[T_S(E,0)-T_S(E,0)]d\epsilon
= 0

\lim_{U_0\to 0} \left<\mathcal{I}_S(U_0)\right> = \frac{1}{8\pi} \int_{E_C}^\infty \Delta f(E,0)\times[T_S(E,0)-T'_S(E,0)]d\epsilon
can be different from 0

Meaning: Spin current ⇔ polarization differs in the two directions.

We concentrate mostly on \left<I_S(U_0)\right> (measuring on one side):

Spin Orbit Ratchets

System

Mireles and Kirczenow, Phys. Rev B 64, 024426 (2001)

Ingredients

We look at the system for \pm U_0:

\left<I_{C/S}\right> = \frac{1}{2} [I(+U_0) + I(-U_0)]

Hamiltonian

\mathcal{H}_c = \frac{\hat{p}^2}{2m^\star} + V_c(z) + \frac{\hbar k_{SO}^R}{m^\star}(\hat{\sigma}_x\hat{p}_z-\hat{\sigma}_z\hat{p}_x) + V_b(x)

Rashba interaction leads to a spin splitting in the dispersion relation:

Additional term for the rocking potential:

\mathcal{H}' = -eU(t)g(x,z;V)

Numerical Results


... to be explained


Series of potentials


T_{++} in one rocking situation


T_{--} in one rocking situation


T_{++} + T_{+-} - T_{--} - T_{-+} = T_S(+U_0) in one rocking situation


Resulting \Delta T_S = T_S(+U_0) - T_S(-U_0)


Full picture

T_S - T'_S for a scatterer with 5 serial adiabatic potentials, 3 open modes in linear response:

The Ratchet Mechanism

Landau-Zener Approach


Examine a particle moving to the right


So we get for the spin transmission:

T_{m,\sigma',n,\sigma} (1+) (1-) (2+) (2-)
(1+) 1 0 0 0
(1-) 0 (1-P)(1-P) (1-P)P 0
(2+) 0 P(1-P) PP 0
(2-) 0 0 0 0

and finally T_{-+} = T_{+-}, but T_{++} \ne T_{--}

Consequence: The more adiabatic U(x) is, the better is the ratchet.

Deficiancies of the Model

Role of Dresselhaus Coupling

\mathcal{H}_D = \frac{\hbar k_{SO}^D}{m^\star}(\hat{\sigma}_x\hat{p}_x-\hat{\sigma}_z\hat{p}_z)

depends on the direction of the crystallographic axes.

⇒ additional terms to the tight-binding Hamiltonian \propto sin(2\phi) and \propto cos(2\phi)

⇒ dependence of the spin current from the orientation of the scatterer

Ganichev et. al., Phys. Rev. Lett. 92, 25 (2004)


T_{++} for one rocking situation


Spin polarization for one rocking situation


Resulting \Delta T_S


Note the vanishing \Delta T_S for k_\alpha = k_\beta (i.e., for identical Rashba and Dresselhaus coupling)

Conclusion

Outlook

Acknowledgements