Theory of the tunneling magnetoresistance

Theory of Tunneling Magneto-Resistance

In this section, the theory of tunneling magneto-resistance in Spintronics is discussed, building upon the concept of magneto-resistance in multi-layered devices from the previous video.

Tunneling Magneto-Resistance Theory

• The advancement in Spintronics has led to significant improvements in the tunneling magneto-resistance ratio and magnetic tunnel junctions. Devices consist of two magnetic layers separated by an insulating spacer.

• Two concise results are derived: current through the tunneling structure depends on the relative orientation of magnetizations, and in the limit of an opaque barrier, tunneling magneto-resistance (TMR) is related to spin polarizations of ferromagnetic contacts.

• Magnetic tunnel junctions (MTJs) can be configured based on alignment between magnetizations: anti-parallel configuration leads to high resistance (RAP), while parallel configuration results in low resistance (RP). TMR quantifies resistance difference between these configurations.

Spin Dependent Tunneling

• Electron transport across a tunneling barrier involves treating spin up and spin down electrons as separate transport problems when magnetizations are collinear. This leads to spin-dependent tunneling due to different transmission probabilities for spin up and spin down electrons.

• The band structure of ferromagnetic metals like BCC iron shows energy splitting for opposite spin states due to exchange interaction, resulting in magnetism. Density of states reveals that spin up states are energetically shifted compared to spin down states, highlighting the basis for spin-dependent tunneling phenomenon in MTJs.

Spinner Wave Function and Magnetization Rotation

• A simplified model with parabolic bands capturing exchange split features demonstrates spin-dependent tunneling. Spinner wave functions differentiate between wave functions for spin up and down electrons relative to a z-axis quantization axis. Rotation operators apply when considering spins along different axes such as z prime direction defined by theta and phi angles relative to z-axis.

The Effect of Magnetization Rotation on Band Structure

This section discusses how the color-coded bands change based on the expectation value of the spin operator as the magnetization direction is rotated.

Bands Color Coding and Spin States

• The bands are color coded based on the z component of the spin operator's expectation value.

• When magnetization rotates, band structure remains, but spin states' directions change.

• Bands turn green when there is no z component in individual bands.

Magnetization Rotation Impact

• Changing magnetization direction alters spin states colors due to new quantization axis.

• Band structure remains invariant with magnetization rotation unless considering spin-orbit interaction.

Tunneling Across a Barrier in MTJ Device

This part introduces tunneling of spin-polarized states across a barrier in a magnetic tunnel junction (MTJ) device.

Device Structure and Model Assumptions

• Describes an iron-magnesium oxide-iron multilayer MTJ device with an insulating spacer.

• Focuses on a simplified toy model to capture fundamental features before discussing realistic MTJ details.

Tunnel Junction Model Assumptions

• Assumes free-spin electron model for ferromagnetic contacts and insulating spacer.

• Considers non-magnetic insulator with zero spin splitting and misaligned magnetizations for ferromagnetic contacts.

Scattering Process and Wave Functions

• Discusses tunneling rates through scattering of electron states and transmission probabilities.

• Details spinner wave functions in different regions: electrode, barrier, transmitted states.

Boundary Conditions and Transmission Amplitudes Calculation

Explains how boundary conditions are used to determine transmission amplitudes for electrons passing through an MTJ device.

Boundary Conditions Application

• Electrons partially transmit or reflect at interfaces; transmission probability relates to transmitted wave function amplitude.

• Applying boundary conditions ensures continuity for wave functions at interfaces for both spin up and down states.

Transmission Amplitudes Determination

• s Obtaining coefficients like see up and see down helps calculate transmission amplitudes for both spin channels.

New Section

In this section, the speaker discusses the derivation of coefficients by Sloan Zuski in a 1989 paper and their application in computing the total tunneling current across a magnetic tunnel junction.

Derivation of Coefficients and Tunneling Current Computation

• The coefficients crucial for calculating tunneling current were first derived by Sloan Zuski in 1989.

• The total tunneling current across the magnetic tunnel junction can be computed using these coefficients.

• The final result is explicitly shown, with the current density calculated from a textbook expression that emphasizes its dependence on the transmitted wave function.

• Coefficients g1 and g3 are influenced by electron states' wave vectors in ferromagnetic contacts and the complex wave vector of electrons in the tunneling barrier.

New Section

This part delves into the angular dependence of mTj resistance, emphasizing the significance of tunneling spin polarizations in describing this phenomenon.

Angular Dependence and Spin Polarizations

• Angular dependence of mTj resistance is primarily determined by tunneling spin polarizations p1 and p3.

• These polarizations are essential for understanding differences in currents between parallel and anti-parallel configurations.

• When polarization values are zero, there is no distinction between currents in different configurations, resulting in a zero tunneling magneto-resistance ratio.

New Section

Exploring high barrier limits and their implications on system properties related to isolated contacts versus junction properties.

High Barrier Limit Analysis

• In high barrier limits, system properties relate solely to isolated contacts rather than junction characteristics.

• The high barrier approximation occurs when kappa (inverse of wave function penetration length) surpasses any wave vector magnitude within the problem.

• Tunneling spin polarization can be expressed based on density of states for spin up and down channels, particularly evident in three-dimensional free electron models.

New Section

Discusses how tunneling spin polarization accounts for coupling between electron wave functions from both contacts, highlighting its importance as a property specific to the junction.

Tunneling Spin Polarization as Junction Property

• Tunneling spin polarization reflects coupling between electron wave functions from both contacts, making it a property intrinsic to the junction itself.

• This property distinguishes itself from being an attribute of isolated contacts, justifying its characterization as "tunneling spin polarization."

New Section

Examining explicit expressions for tunneling magneto-resistance ratio (TMR) concerning thick barrier approximations and its dependency on tunneling spin polarizations.

Tunneling Magneto-resistance Ratio Analysis

• TMR explicitly depends on tunneling spin polarizations p1 and p3 under thick barrier approximations.

• Jolier's work in 1975 first introduced TMR concepts regarding mTjs.

• Misalignment angles impact charge current flow through mTjs significantly; minimum occurs at 180 degrees (anti-parallel configuration), while maximum arises at 0 degrees (parallel configuration).

New Section

Analyzing predictions regarding optimal conditions for achieving significant differences between parallel and anti-parallel configurations based on tunneling spin polarization products.

Optimal Conditions for Maximum TMR

• Maximum difference between parallel and anti-parallel configurations occurs when product of tunneling spin polarizations approaches unity.