Gavin W Morley Department of Physics University of Warwick Diamond Science & Technology Centre for Doctoral Training, MSc course Module 2 – Properties and Characterization of Materials Module 2 – (PX904) Lectures 15 & 16 – Magnetic properties and characterization

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 2 Overview Lecture 15 & 16Magnetic properties of materials - Paramagnetism - Diamagnetism - Ferromagnetism Magnetic characterization - SQUID magnetometry - Neutron scattering - Magnetic resonance - Electron paramagnetic resonance - Nuclear magnetic resonance

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 3 Magnetism Current, I Area, A Current loop has magnetic moment, µ = I A Maxwell 4:  × B = μ 0 J Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 4 Magnetism Current, I Area, A Current loop has magnetic moment, µ = I A NSNS Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 5 What happens if we try to cut a magnet in half? Current, I Area, A NSNS Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 6 What happens if we try to cut a magnet in half? NSNS Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 NSNSNSNS a) b)

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 7 Magnetism NSNSNSNS Magnetic Monopoles Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 8 Angular momentum and Magnetism Current, I Area, A Current loop has magnetic moment, µ = I A NSNS µ = γ L (L is angular momentum, γ is gyromagnetic ratio) Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 9 What happens if you put a magnetic moment into a uniform magnetic field? NSNS Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 a)It moves b)It lines up c)It precesses d)Hey, I thought you were supposed to be teaching me

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 10 What happens if you put a magnetic moment into a uniform magnetic field? NSNS Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 a)It moves b)It lines up c)It precesses d)Hey, I thought you were supposed to be teaching me

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 11 Precession of a magnetic moment Current, I Area, A Energy of the magnetic moment in a magnetic field, B : E = - µ B Larmor precession frequency = γ B Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 Joseph Larmor (1857 – 1942)

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 12 Magnitude of magnetic moment Electron has charge, e and mass, m, so Current: I = -e/t as speed, v = 2 π r/t for radius, r. Magnetic moment, μ = I A = I π r 2 = - e ℏ/ 2m (as electron angular momentum = mvr = ℏ ) Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 v e- r ≡ -μ B

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 13 Magnetization, M: Magnetic moment per unit volume in a solid Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 In vacuum: B = µ 0 H permeability of free space, µ 0 = 4π × Hm -1 In a magnetic solid: B = µ 0 (H + M) For a linear material, M = χ H for susceptibility, χ So then B = µ 0 (1+ χ )H = µ 0 µ r H For relative permeability, µ r

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 14 Magnetization, M: Magnetic moment per unit volume in a solid Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 In vacuum: B = µ 0 H permeability of free space, µ 0 = 4π × Hm -1 In a magnetic solid: B = µ 0 (H + M) For a linear material, M = χ H for susceptibility, χ So then B = µ 0 (1+ χ )H = µ 0 µ r H For relative permeability, µ r

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 15 Magnetization, M: Magnetic moment per unit volume in a solid Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 In vacuum: B = µ 0 H permeability of free space, µ 0 = 4π × Hm -1 In a magnetic solid: B = µ 0 (H + M) For a linear material, M = χ H for susceptibility, χ So then B = µ 0 (1+ χ )H

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 16 Relative permeability, µ r Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 In vacuum: B = µ 0 H permeability of free space, µ 0 = 4π × Hm -1 In a magnetic solid: B = µ 0 (H + M) For a linear material, M = χ H for susceptibility, χ So then B = µ 0 (1+ χ )H = µ 0 µ r H µ r = 1+ χ

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 17 Magnetization, M: Magnetic moment per unit volume in a solid Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 In vacuum: B = µ 0 H permeability of free space, µ 0 = 4π × Hm -1 In a magnetic solid: B = µ 0 (H + M) For a linear material, M = χ H for susceptibility, χ So then B = µ 0 (1+ χ )H

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 18 Susceptibility, χ Chapter 1, Blundell, Magnetism in Condensed Matter, OUP 2001 In vacuum: B = µ 0 H permeability of free space, µ 0 = 4π × Hm -1 In a magnetic solid: B = µ 0 (H + M) For a linear material, M = χ H for susceptibility, χ So then B = µ 0 (1+ χ )H = µ 0 µ r H For relative permeability, µ r Table from Kaye & Laby

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 19 Magnetism of an atom 1.From the electrons 1.Spin angular momentum 2.Orbital angular momentum 3.An applied magnetic field can change their orbital angular momentum 2.From the nuclei 1.Spin angular momentum

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 20 A beam of atoms hits a screen Classical prediction

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 21 See also: File%3AQuantum_spin_and_the_Stern- Gerlach_experiment.ogv SNSN Classical prediction Stern-Gerlach experiment

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 22 Magnetism of an atom 1.From the electrons 1.Spin angular momentum 2.Orbital angular momentum 3.An applied magnetic field can change their orbital angular momentum 2.From the nuclei 1.Spin angular momentum

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 23 Page 240, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985 → Discrete energy levels Erwin Schrödinger (1887 – 1961) Solve Schrödinger’s equation for an electron in a box:

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 24 Solve Schrödinger’s equation for electron in Coulomb potential and include spin n 123 l mlml 00-1,0,+10 -2,-1,0,+1,+2 msms +½,-½ Number of degenerate eignenfunctions for each l Subshell name1s2s2p3s3p3d Page 241, Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, Wiley 1985

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 25 1s 2s 2p x 2p y 2p z Atomic orbitals

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 26 Magnetism of an atom 1.From the electrons 1.Spin angular momentum 2.Orbital angular momentum 3.An applied magnetic field can change their orbital angular momentum 2.From the nuclei 1.Spin angular momentum

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 27 Magnetic Field Sample magnetization Paramagnetic 1.From the electrons 1.Spin 2.Orbital For spin ½, Magnetization is M = M s tanh(µ B B/k B T)

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 28 Magnetic Field Sample magnetization Paramagnetic 1.From the electrons 1.Spin 2.Orbital Paramagnetic susceptibility follows the Curie Law: χ = C Curie /T

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 29 Page 20, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 30 1s 2s 2p x 2p y 2p z Atomic orbitals

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 31 Page 20, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 32 Fermi-Dirac distribution function, Page 9, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001 Conduction electrons have “Pauli paramagnetism” (Chapter 7 of Blundell’s book)

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 33 Magnetism of an atom 1.From the electrons 1.Spin angular momentum 2.Orbital angular momentum 3.An applied magnetic field can change their orbital angular momentum 2.From the nuclei 1.Spin angular momentum

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 34 Sample magnetization Paramagnetic - From spin and orbital angular momentum From change in orbital angular momentum- Diamagnetic Magnetic Field

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 35 Sample magnetization Paramagnetic - From spin and orbital angular momentum From change in orbital angular momentum - Diamagnetic Magnetic Field

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 36 Page 20, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 37 Magnetism of an atom 1.From the electrons 1.Spin angular momentum 2.Orbital angular momentum 3.An applied magnetic field can change their orbital angular momentum 2.From the nuclei 1.Spin angular momentum

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 38 1.From the electrons 1.Spin angular momentum 2.Orbital angular momentum 3.An applied magnetic field can change their orbital angular momentum 2.From the nuclei 1.Spin angular momentum Interactions → Ferromagnetism Ferromagnet in zero applied magnet field ( J > 0 ):

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 39 Sample magnetization Ferromagnetic Paramagnetic Diamagnetic Magnetic Field

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 40 Sample magnetization Ferromagnetic Magnetic Field Saturation magnetization Remanent magnetization Coercive field

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 41 Ferromagnetic domains Page 131, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 42 1.From the electrons 1.Spin angular momentum 2.Orbital angular momentum 3.An applied magnetic field can change their orbital angular momentum 2.From the nuclei 1.Spin angular momentum Interactions → Antiferromagnetism Antiferromagnet in zero applied magnet field ( J < 0 ):

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 43 Page 202, Singleton, Band Theory and Electronic Properties of Solids, OUP 2001 Diamond Superconductivity → perfect diamagnetism In vacuum: B = µ 0 H permeability of free space, µ 0 = 4π × Hm -1 In a magnetic solid: B = µ 0 (H + M) For a linear material, M = χ H for susceptibility, χ So then B = µ 0 (1+ χ )H = µ 0 µ r H

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Boron-doped Diamond: Superconductivity 44 E Bustarret et al, Dependence of the Superconducting Transition Temperature on the Doping Level in Single-Crystalline Diamond Films, Physical Review Letters, 93, (2004)

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 45 Diamond Magnetic characterization In a magnetic solid: B = µ 0 (H + M) For a linear material, M = χ H Measure magnetization, M which could be a function of temperature, magnetic field, orientation etc.

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 46 Diamond Magnetic characterization In a magnetic solid: B = µ 0 (H + M) For a linear material, M = χ H Measure magnetization, M which could be a function of temperature, magnetic field, orientation etc. Extraction magnetometer: V V = 0 V > 0

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 47 Diamond Magnetic characterization V V = 0 V ac > 0 Vibrating sample magnetometer (VSM):

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 48 SQUID magnetometer V V = 0 V ac > 0 Vibrating sample magnetometer (VSM) with SQUID detection: SQUID = superconducting quantum interference device Bias current

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 49 V V = 0 V ac > 0 Vibrating sample magnetometer (VSM) with SQUID detection in an applied magnetic field → susceptibility Bias current M = χ H for susceptibility χ SQUID magnetometer

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 50 Analogous to X-ray diffraction with neutrons instead of X-rays. Neutrons have no charge but spin ½ Neutron Scattering Page 104, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization 51 Analogous to X-ray diffraction with neutrons instead of X-rays. Neutrons have no charge but spin ½ Neutron Scattering Page 106, Blundell, Magnetism in Condensed Matter, OUP 2001

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Energy of a spin 0 Magnetic field, B Magnetic resonance Energy of the magnetic moment in a magnetic field, B : E = - µ B

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Energy of a spin 0 Photon energy = h f Magnetic field, B Energy of the magnetic moment in a magnetic field, B : E = - µ B Magnetic resonance

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Electron paramagnetic resonance …NMR for electrons The crucial difference is that the electron magnetic moment is 660 times larger than that of a proton

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Electron paramagnetic resonance S Bridge source detector Microwave resonator Modulation coils Circulator Main magnet

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Pulsed magnetic resonance Felix Bloch ( ) Photo courtesy Stanford News Service

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Rotating frame

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Rotating frame

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Spin echo In rotating frame

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Spin echo In rotating frame

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Nuclear magnetic resonance S RF coil provides a horizontal magnetic field which is oscillating Main magnet with a vertical field Probe nuclear paramagnetism

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Nuclear Magnetic Resonance NMR periodic table from Philip Grandinetti

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Diamond NMR 99% of carbon is 12 C with zero nuclear spin. 1% is 13 C with nuclear spin I = ½ L. H. Merwin, C. E. Johnson and W. A. Weimer, 13 C NMR investigation of CVD diamond: Correlation of NMR and Raman spectral linewidths, Journal of Materials Research 9, 631 (1994).

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Pure 13 C Diamond NMR K. Lefmann et al., NMR spectra of pure 13 C diamond, Physical Review B 50, (1994).

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Lecture Summary Lecture 15 & 16Magnetic properties of materials - Paramagnetism - Diamagnetism - Ferromagnetism Magnetic characterization - SQUID magnetometry - Neutron scattering - Magnetic resonance - Electron paramagnetic resonance - Nuclear magnetic resonance

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization Module Summary Module 2 – Properties and Characterization of Materials - Summary

Module 2 – Properties and Characterization of Materials - Lectures 15 & 16 – Magnetic properties & characterization LecturesLecturer 1-3Philip MartineauCrystallography 4-6Gavin MorleyElectronic properties 7-8Stephen LynchOptical 9Gavin MorleyElectronic characterization 10Richard BeanlandElectron microscopy 11-12Claire DancerMechanical 13-14Martin KuballThermal 15-16Gavin MorleyMagnetic Module 2 – Properties and Characterization of Materials - Summary