Magnetoresistance

Magnetoresistance Jeff FitzgeraldPhysics 211A

The Basic Idea: • What happens when B fields are applied to metals? • 2. What happens when those metals are magnetic?

Free Electron Model • Metals are gases of electrons (MB/FD), ion lattice doesn’t move • Each electron sees a constant potential • Scattering only from ion lattice • Relaxation time not a function of position or velocity • Calculated later using quantum mechanics

DC Resistivity Use Ohm’s law for a wire carrying current density to get resistivity in terms of the relaxation time:

Hall Effect Hall Resistivity:

For small fields For very high fields Ordinary Magnetoresistance Experimental observations: Always an increase in resistance

Ordinary Magnetoresistance Kohler’s Rule Why increase resistance? Electrons orbit around B field until scattered, thus longer relaxation time = larger change in resistance What does this mean physically? # times electron goes around orbit is proportional to mean free path divided by cyclotron radius

Ordinary Magnetoresistance lowest order More on Kohler’s Rule Deflecting charges left or right gives an increase in resistance, thus Taylor expand (weak field)

Anomalous Hall Effect In a ferromagnet M couples to j due to spin-orbit interaction. Equations: B H (Ordinary) B M (Anomalous)

Hall Resistivity: Anomalous Hall Effect Two types of scattering processes: 1. Skew or s-d scattering (M) 2. Side-jump scattering (impurities) High-field slope due to OHE High-field extrapolation to zero is AH resistivity

Anisotropic Magnetoresistance Experimental Observations

AMR Easy axis  no resistivity change Hard axis  resistivity changes depending on angle between j and M Insight into domain structure

AMR Using Ohm’s Law, we can get results that match observation in many systems: Then rotate field to y direction to measure the resistivity, and subtract:

Since resistivity is measured along current direction Ohm’s Law Plug in E Define: Then This result matches observations in many systems

What’s actually going on in there…? Things to consider: How are the different spins behaving in the FM? What is the main scattering mechanism? How does the magnetization affect the current?

Ferromagnetic Band Structures Mott’s Two-Current Model majority minority (Mathiessen’s Rule)

Scattering in a FM Assumptions: • current carried mostly by s electrons (d electrons have much higher effective masses) • s-d scattering more likely than s-s scattering because

and in our case Transition Rates Relaxation time: remember From quantum mechanics: So for s-d scattering, But we’re missing something important…

Spin-Orbit Interaction The d electrons’ (responsible for the magnetization) spin couples to their angular motion. The d states experience an energy shift according to the Hamiltonian Use first order perturbation theory SOI allows spin-mixing and creates d band holes so that majority spin electrons can scatter into empty d states

Recipe for resistivity • Pick unperturbed d electron wavefunctions • Choose the magnetization • Perturb the d states (using SOI) • Calculate scattering rate • Use rate to get relaxation time • Sum over all atoms • Repeat for other scattering mechanisms (if you want) • Use Mathiessen’s rule to get total resistivity Potter calculated this in 1974 with mediocre results

Other types of MR GMR TMR

OUTLINE • Free Electron Model • Resistance changes in metals • Ordinary Hall Effect • Ordinary MR • Resistance changes in ferromagnets • Anomalous Hall Effect • Anisotropic MR Experiment • AMR Theory • Other types of MR

Potter’s Calculation Potter (1974) calculated the resistivity by • Using tight binding model to get • Assuming the d-bands are nearly full • Assuming (Yukawa potential)

where and Spin mixing takes therefore

Magnetoresistance