Magnetoresistance Jeff Fitzgerald Physics 211 A The Basic
Magnetoresistance Jeff Fitzgerald Physics 211 A
The Basic Idea: 1. What happens when B fields are applied to metals? 2. 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:
Ordinary Magnetoresistance Experimental observations: For small fields For very high fields Always an increase in resistance
Ordinary Magnetoresistance Why increase resistance? Electrons orbit around B field until scattered, thus longer relaxation time = larger change in resistance Kohler’s Rule What does this mean physically? # times electron goes around orbit is proportional to mean free path divided by cyclotron radius
Ordinary Magnetoresistance More on Kohler’s Rule Deflecting charges left or right gives an increase in resistance, thus lowest order 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)
Anomalous Hall Effect Hall Resistivity: 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:
Ohm’s Law Since resistivity is measured along current direction 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
Transition Rates Relaxation time: remember From quantum mechanics: and in our case 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 1. 2. 3. 4. 5. 6. 7. 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) 8. Use Mathiessen’s rule to get total resistivity Potter calculated this in 1974 with mediocre results
Other types of MR GMR TMR
OUTLINE 1. Free Electron Model 2. Resistance changes in metals 1. Ordinary Hall Effect 2. Ordinary MR 3. Resistance changes in ferromagnets 1. Anomalous Hall Effect 2. Anisotropic MR Experiment 3. AMR Theory 4. 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)
Spin mixing where therefore and takes
- Slides: 24