Representable functor

Magnetic tunnel junction (schematic)

Tunnel magnetoresistance (TMR) is a magnetoresistive effect that occurs in a magnetic tunnel junction (MTJ), which is a component consisting of two ferromagnets separated by a thin insulator. If the insulating layer is thin enough (typically a few nanometers), electrons can tunnel from one ferromagnet into the other. Since this process is forbidden in classical physics, the tunnel magnetoresistance is a strictly quantum mechanical phenomenon.

Magnetic tunnel junctions are manufactured in thin film technology. On an industrial scale the film deposition is done by magnetron sputter deposition; on a laboratory scale molecular beam epitaxy, pulsed laser deposition and electron beam physical vapor deposition are also utilized. The junctions are prepared by photolithography.

Phenomenological description

The direction of the two magnetizations of the ferromagnetic films can be switched individually by an external magnetic field. If the magnetizations are in a parallel orientation it is more likely that electrons will tunnel through the insulating film than if they are in the oppositional (antiparallel) orientation. Consequently, such a junction can be switched between two states of electrical resistance, one with low and one with very high resistance.

History

The effect was originally discovered in 1975 by M. Jullière (University of Rennes, France) in Fe/Ge-O/Co-junctions at 4.2 K. The relative change of resistance was around 14%, and did not attract much attention. In 1991 Terunobu Miyazaki (Tohoku University, Japan) found an effect of 2.7% at room temperature. Later, in 1994, Miyazaki found 18% in junctions of iron separated by an amorphous aluminum oxide insulator and Jagadeesh Moodera found 11.8% in junctions with electrodes of CoFe and Co. The highest effects observed to date with aluminum oxide insulators are around 70% at room temperature.

Since the year 2000, tunnel barriers of crystalline magnesium oxide (MgO) have been under development. In 2001 Butler and Mathon independently made the theoretical prediction that using iron as the ferromagnet and MgO as the insulator, the tunnel magnetoresistance can reach several thousand percent. The same year, Bowen et al. were the first to report experiments showing a significant TMR in a MgO based magnetic tunnel junction [Fe/MgO/FeCo(001)]. In 2004, Parkin and Yuasa were able to make Fe/MgO/Fe junctions that reach over 200% TMR at room temperature. In 2009, effects of up to 600% at room temperature and more than 1100% at 4.2 K were observed in junctions of CoFeB/MgO/CoFeB.

Applications

The read-heads of modern hard disk drives work on the basis of magnetic tunnel junctions. TMR, or more specifically the magnetic tunnel junction, is also the basis of MRAM, a new type of non-volatile memory. The 1st generation technologies relied on creating cross-point magnetic fields on each bit to write the data on it, although this approach has a scaling limit at around 90–130 nm. There are two 2nd generation techniques currently being developed: Thermal Assisted Switching (TAS) and Spin Torque Transfer (STT). Magnetic tunnel junctions are also used for sensing applications.

Physical explanation

Two-current model for parallel and anti-parallel alignment of the magnetizations

The relative resistance change—or effect amplitude—is defined as

$\mathrm {TMR} :={\frac {R_{\mathrm {ap} }-R_{\mathrm {p} }}{R_{\mathrm {p} }}}$

where $R_{\mathrm {ap} }$ is the electrical resistance in the anti-parallel state, whereas $R_{\mathrm {p} }$ is the resistance in the parallel state.

The TMR effect was explained by Jullière with the spin polarizations of the ferromagnetic electrodes. The spin polarization P is calculated from the spin dependent density of states (DOS) ${\mathcal {D}}$ at the Fermi energy:

$P={\frac {{\mathcal {D}}_{\uparrow }(E_{\mathrm {F} })-{\mathcal {D}}_{\downarrow }(E_{\mathrm {F} })}{{\mathcal {D}}_{\uparrow }(E_{\mathrm {F} })+{\mathcal {D}}_{\downarrow }(E_{\mathrm {F} })}}$

The spin-up electrons are those with spin orientation parallel to the external magnetic field, whereas the spin-down electrons have anti-parallel alignment with the external field. The relative resistance change is now given by the spin polarizations of the two ferromagnets, P₁ and P₂:

$\mathrm {TMR} ={\frac {2P_{1}P_{2}}{1-P_{1}P_{2}}}$

If no voltage is applied to the junction, electrons tunnel in both directions with equal rates. With a bias voltage U, electrons tunnel preferentially to the positive electrode. With the assumption that spin is conserved during tunneling, the current can be described in a two-current model. The total current is split in two partial currents, one for the spin-up electrons and another for the spin-down electrons. These vary depending on the magnetic state of the junctions.

There are two possibilities to obtain a defined anti-parallel state. First, one can use ferromagnets with different coercivities (by using different materials or different film thicknesses). And second, one of the ferromagnets can be coupled with an antiferromagnet (exchange bias). In this case the magnetization of the uncoupled electrode remains "free".

The TMR decreases with both increasing temperature and increasing bias voltage. Both can be understood in principle by magnon excitations and interactions with magnons.

It is obvious that the TMR becomes infinite if P₁ and P₂ equal 1, i.e. if both electrodes have 100% spin polarization. In this case the magnetic tunnel junction becomes a switch, that switches magnetically between low resistance and infinite resistance. Materials that come into consideration for this are called ferromagnetic half-metals. Their conduction electrons are fully spin polarized. This property is theoretically predicted for a number of materials (e.g. CrO₂, various Heusler alloys) but has not been experimentally confirmed to date.

Spin-filtering in Tunnel Barriers

Prior to the introduction of epitaxial magnesium oxide (MgO), amorphous aluminum oxide was used as the tunnel barrier of the MTJ and typical room temperature TMR was in the range of tens of percent. MgO barriers increased TMR to hundreds of percent due to the ability to filter spin, which is complementary to the electrode spin polarization effect described above. The physical origin of this spin filtering is actually symmetry filtering because the electron wavefunctions of opposite spin originate from different bands at the Fermi level. These bands correspond to different orbitals for majority and minority spin and thus have different symmetries. The MgO conduction and valence bands have the same symmetry as majority spin electrons, so they experience a lower barrier height than minority spin electrons. This exponentially increases the tunneling probability so parallel configuration current exceeds anti-parallel current by a much larger amount.

Spin-transfer torque in Magnetic Tunnel Junctions (MTJs)

The effect of spin-transfer torque (STT) has been studied in MTJs where there is an tunnelling barrier sandwiched between a set of 2 ferromagnetic electrodes such that there is (free) magnetization of the right electrode, while assuming that the left electrode (with fixed magnetization) acts as spin-polarizer. This would then be pinned to some selecting transistor in an MRAM device.

The STT vector, driven by the linear response voltage, can be computed from the expectation value of the torque operator:

$\mathbf {T} =\mathrm {Tr} [{\hat {\mathbf {T} }}{\hat {\rho }}_{\mathrm {neq} }]$

where ${\hat {\rho }}_{\mathrm {neq} }$ is the gauge-invariant nonequilibrium density matrix for the steady-state transport, in the zero-temperature limit, in the linear-response regime, and the torque operator ${\hat {\mathbf {T} }}$ is obtained from the time derivative of the spin operator:

${\hat {\mathbf {T} }}={\frac {d{\hat {\mathbf {S} }}}{dt}}=-{\frac {i}{\hbar }}\left[{\frac {\hbar }{2}}{\boldsymbol {\sigma }},{\hat {H}}\right]$

Using the general form of a 1D tight-binding Hamiltonian:

${\hat {H}}={\hat {H}}_{0}-\Delta ({\boldsymbol {\sigma }}\cdot \mathbf {m} )/2$

where total magnetization (as macrospin) is along the unit vector $\mathbf {m}$ and the Pauli matrices properties involving arbitrary classical vectors $\mathbf {p} ,\mathbf {q}$, given by

$({\boldsymbol {\sigma }}\cdot \mathbf {p} )({\boldsymbol {\sigma }}\cdot \mathbf {q} )=\mathbf {p} \cdot \mathbf {q} +i(\mathbf {p} \times \mathbf {q} )\cdot {\boldsymbol {\sigma }}$

$({\boldsymbol {\sigma }}\cdot \mathbf {p} ){\boldsymbol {\sigma }}=\mathbf {p} +i{\boldsymbol {\sigma }}\times \mathbf {p}$

${\boldsymbol {\sigma }}({\boldsymbol {\sigma }}\cdot \mathbf {q} )=\mathbf {q} +i\mathbf {q} \times {\boldsymbol {\sigma }}$

it is then possible to first obtain an analytical expression for ${\hat {\mathbf {T} }}$ (which can be expressed in compact form using $\Delta ,\mathbf {m}$, and the vector of Pauli spin matrices ${\boldsymbol {\sigma }}=(\sigma _{x},\sigma _{y},\sigma _{z})$).

The STT vector in general MTJs has two components: a parallel and perpendicular component:

A parallel component: $T_{\parallel }={\sqrt {T_{x}^{2}+T_{z}^{2}}}$

And a perpendicular component: $T_{\perp }=T_{y}$

While in symmetric MTJs (made of electrodes with the same geometry and exchange splitting), the STT vector has only one active component, as the perpendicular component disappears:

$T_{\perp }\equiv 0$.

Therefore, only $T_{\parallel }$ vs. $\theta$ needs to be plotted at the site of the right electrode to characterise tunnelling in symmetric MTJs, making them appealing for production and characterisation at an industrial scale.

Note: In these calculations the active region (for which it is necessary to calculate the retarded Green's function) should consist of the tunnel barrier + the right ferromagnetic layer of finite thickness (as in realistic devices). The active region is attached to the left ferromagnetic electrode (modeled as semi-infinite tight-binding chain with non-zero Zeeman splitting) and the right N electrode (semi-infinite tight-binding chain without any Zeeman splitting), as encoded by the corresponding self-energy terms.

References

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See also

• Quantum tunnelling
• Magnetoresistance
• Giant Magnetoresistance
• Spin-transfer torque