Spin-Polarized Current

Intuition

In an ordinary metal (Cu, Al), every conduction electron is just as likely to be spin-up as spin-down, and the current is unpolarized: $J_{\uparrow }=J_{\downarrow }$. Pass that same current through a ferromagnet, however, and the situation changes radically.

Inside a ferromagnet, the Stoner exchange splitting rigidly shifts the two spin sub-bands relative to each other. As a consequence, the densities of states at the Fermi level for the two spin channels are different:

$$g_{\uparrow }({\epsilon }_F)\ne g_{\downarrow }({\epsilon }_F).$$

The current that emerges from the ferromagnet is then biased towards the majority spin. The ferromagnet acts as a spin filter — it both magnetizes ($M\ne 0$) and polarizes its current. This dual role is the cornerstone of spintronics.

Formal definition

Define the spin polarization of a current $\vec{J}$ as

$$P=\frac{J_{\uparrow }-J_{\downarrow }}{J_{\uparrow }+J_{\downarrow }},|P|\le 1.$$

$P=0$ is an unpolarized current; $P=\pm 1$ is a fully polarized current carried by a single spin channel. In a two-current model (Mott, 1936) the two spin channels conduct in parallel with different conductivities ${\sigma }_{\uparrow }$, ${\sigma }_{\downarrow }$, so

$$P\approx \frac{{\sigma }_{\uparrow }-{\sigma }_{\downarrow }}{{\sigma }_{\uparrow }+{\sigma }_{\downarrow }}\sim \frac{g_{\uparrow }({\epsilon }_F)-g_{\downarrow }({\epsilon }_F)}{g_{\uparrow }({\epsilon }_F)+g_{\downarrow }({\epsilon }_F)},$$

with corrections coming from the spin-dependent Fermi velocity and scattering rate.

Key results

1. Typical polarizations

For the elemental 3d ferromagnets and a few engineered materials:

Material	$P$ (%)	Notes
Fe	40–45
Co	35–45
Ni	25–35
Permalloy (Ni${}_{80}$Fe${}_{20}$)	35–45	the workhorse soft FM in spintronics
CrO${}_2$	$\to 100$	half-metal, only one spin channel at ${\epsilon }_F$
Heusler alloys (Co${}_2$MnSi, …)	$\to 100$	designer half-metals

A half-metal is the holy grail of spintronics: $P=1$ would give infinite GMR and TMR ratios. Practical half-metals work only at low temperature and lose polarization at room temperature through spin-flip scattering.

2. Spin accumulation and the spin diffusion length

When a spin-polarized current crosses an interface into a non-magnetic metal, the imbalance does not disappear instantly. The two spin channels relax back to equilibrium over the spin-diffusion length ${\lambda }_{\text{sf}}$:

$${\lambda }_{\text{sf}}=\sqrt{D{\tau }_{\text{sf}}},$$

where $D$ is the diffusion constant and ${\tau }_{\text{sf}}$ the spin-flip relaxation time. Typical values: Cu $\sim 500$ nm at 4 K, Al $\sim 1$ µm, Pt $\sim 5$ nm (large spin-orbit, fast spin-flip). Devices must be smaller than ${\lambda }_{\text{sf}}$ for spin to survive transit; this is the reason GMR/TMR stacks have layer thicknesses of only a few nanometers.

3. Measuring P

$P$ is not directly observable but inferred from:

• Point-contact Andreev reflection (PCAR) — a superconducting tip on the FM forces every electron to find a spin-flipped partner; the fraction that fails to do so reveals $P$.
• Tunneling spectroscopy (Meservey–Tedrow) — splitting of a superconducting tip’s DOS in a magnetic field.
• Indirectly, from the GMR or TMR ratio via the Jullière model.

Applications {#applications}

A spin-polarized current is the carrier of every spintronic device:

• It is read in GMR and TMR stacks — different relative orientations of the two FM layers give different resistances.
• It is written by spin-transfer torque — a polarized current dumps its angular momentum into a free layer and flips its magnetization.
• It is the active ingredient of every commercial MRAM cell, every modern hard-disk read head, and every proposed neuromorphic spintronic oscillator.

The fact that a single phenomenon — asymmetric DOS at ${\epsilon }_F$ — underlies all of these is the unifying message of the lecture.

Summary

Inside a ferromagnet the two spin sub-bands have different DOS at ${\epsilon }_F$. The current that leaves the ferromagnet inherits this imbalance and carries a spin polarization $P=(J_{\uparrow }-J_{\downarrow })/(J_{\uparrow }+J_{\downarrow })$. Typical 3d ferromagnets give $P\sim 30$–$45\%$; engineered half-metals approach $P=100\%$. Spin survives in a non-magnetic conductor only over the spin-diffusion length ${\lambda }_{\text{sf}}$, which is why useful spintronic devices are nanometer-thin.

Connections

• stoner-model — origin of the asymmetric DOS
• spin-valve — how P is read out via GMR
• magnetic-tunnel-junction — how P is read out via TMR
• spin-transfer-torque — how P is used to write a magnetic state
• magnetic-materials — where ferromagnets sit among the five families

References

• N. F. Mott, Proc. Roy. Soc. A 153, 699 (1936) — two-current model.
• I. Žutić, J. Fabian & S. Das Sarma, Spintronics: Fundamentals and Applications, Rev. Mod. Phys. 76, 323 (2004).
• R. Meservey & P. M. Tedrow, Phys. Rep. 238, 173 (1994).