module: magnetism on the nanoscale, WS 2019/2020
chapter 2: magnetism in metals – part II (Landau diamagnetism)
chapter 3: from microscopic to macroscopic
chapter 4: spectroscopic techniques
Dr. Sabine Wurmehl
Dresden, January 6th, 2020
reminder….
2.0 magnetism in metals
example: metallic Fe, Co, Ni, Gd
important: NON-integer number!
2.1 Free electron model
assumptions:
1) electrons are free
atom ions and e- do not interact (but atom ions needed for setting boundary conditions)
2) electrons are independent
e- do not interact
3) no lattice contribution
Bloch's theorem:
• unbound electron moves in a periodic potential as a free electron in vacuum
• electron mass may be modified by band structure and interactions effective mass m*
4) Pauli exclusion principle
each quantum state is occupied by a single electron
Fermi–Dirac statistics
Description similar as particle in a box
free electron gas in magnetic field
Landau diamagnetism Pauli paramagnetism
2.2 Pauli paramagnetism
origin of Pauli paramagnetism
Zeemann splitting in magnetic field in a metal
g (E) /2 g (E) /2
E
2mBB
B
E = EF
if conductions electrons are weakly interacting and delocalized (Fermi gas)
magnetic response originates in interaction of spin with magnetic field
replace integral by EF
temperature independent
2.3 Landau diamagnetism
2.3 Landau diamagnetism
weak counteracting field that forms when the electrons' trajectories are curved due to the Lorentz force
harmonic oscillator plane wave
…some mathematics….
energy Eigenvalues for harmonic oscillator
plane waves in
x, y direction
quantized states
along B
Landau levels (tubes)
http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf
with magnetic field:
k-vectors condense on tubes paralell to field
no magnetic field:
discrete states
Landau susceptibility of conduction electrons
application of magnetic field quantized Landau levels changes energetic state
thermodynamics: magnetic field induced change of energy magnetization
with tentative assumption: all metals are paramagnets as c Pauli >> c Landau
disclaimer: bandstructure effects may matter since g(EF) ~ m*/me
for most metals m* ~ me most metals are paramagnets
occupation of Landau levels
B1< B2 < B3
De Haas-van Alphen effect
http://lampx.tugraz.at/~hadley/ss2/problems/fermisurf/s.pdf
specific heat
quantum oscillations in metals
2.0 magnetism in metals
example: Metallic Fe, Co, Ni, Gd
Important: NON-Integer number!
spin resolved DOS
http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf
2.4 band ferromagnetism
Stoner criterion, s-d model (see lectures by J. Dufoleur)
3 from microscopic to macroscopic
lessons learned on microscopic level:
localized electrons diamagnetism of paired e- ; paramagnetism of unpaired electrons
itinerant electrons Landau diamagnetism & Pauli paramagnetism of conduction electrons
3 from microscopic to macroscopic
macroscopic behaviour of magnetization results from minimization of contributions of 4 interactions
• Zeemann interaction, viz. interaction with an external magnetic field (Fex):
minimization of energy by alignment of magnetic moments along field
• dipolar interaction (Fdip):
minimization of energy by avoiding formation of magnetic poles
weak but long-ranged
• exchange interaction (FH):
minimization of energy by uniform magnetization
very strong but short-ranged
• magnetic anisotropy (Fan) :
minimization of energy by orienting magnetic moments along preferred directions
for a homogeneous ferromagnetic material, minimization of free energy F:
F = Fex + Fdip + FH + Fan
3.1 magnetic anisotropy
anisotropy: when a physical property of a material is a function of direction
types of magnetic anisotropies:
• 3.1.1 magnetocrystalline anisotropy (spin-orbit-coupling, crystal structure)
• shape anisotropy (demagnetization field)
• 3.1.2 magnetoelastic anisotropy (stress)
• 3.1.2 induced anisotropy (processing, treatment, annealing)
3.1.1 magnetocrystalline anisotropy
most important contribution: orbital motion of the electrons couple to crystal electric field
energy is minimzed if magnetic moments are aligned along specific preferred directions easy axis
different orientations of spins correspond to different orientations of atomic orbitals relative to crystal structure
demagnetization field
http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-8.pdf
at sample edges: magnetization diverges
costs energy by formation of stray fields with demagnetization field HD (demagnetization energy, dipolar energy)
also see Maxwell equations
with Nij the demagnetization factor
(shape anisotropy)
magnetic domains
http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-8.pdf
formation of stray fields costs dipolar energy energy costs minimized formation of magnetic domains
dipolar energy is minimized if as many domains as possible are formed
BUT: formation of domains costs energy
closure domain structure
costs for formation of domains (details: lecture T. Mühl)
M
if ferromagnetic material forms domains:
no divergence of magnetization at sample edge
within domain, all spin moments are aligned
not all domains are aligned along preferred easy axis
between domains, spin moments need to rotate
dipolar fields minimized
exchange energy J minimized
costs anisotropy energy
costs exchange energy
balance between costs determines width of domain wall
types of domain walls
P. Li-Cong et al. Chinese Physics B 27, 066802 (2018)
magnetization rotates in plane parallel
to plane of domain wall
magnetization rotates in plane perpendicular
to plane of domain wall
no stray fields on sample surface thin films
magnetic hysteresis loop
http://hydrogen.physik.uni-wuppertal.de/hyperphysics/hyperphysics/hbase/solids/hyst.html
reversible wall
displacements
irreversible wall
displacements
coherent rotation of domains
https://en.wikipedia.org/wiki/Magnetocrystalline_anisotropy#/media/File:Easy_axes.jpg
http://www.ifmpan.poznan.pl/~urbaniak/Wyklady2012/urbifmpan2012lect5_03.pdf
http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf
different crystallographic structure different magnetic anisotropy different hysteresis curves
hard and soft magnetic materials
hard soft
H. D. Young, University Physics, 8th Ed., Addison-Wesley, 1992
hard magnetic materialsmagnetic anisotropy in Nd-Fe-B
D. Goll and H. Kronmüller, Naturwissenschaften 87, 423 (2000).
3.1.2 microstructure and it‘s impact on magnetic hysteresis
D. Goll and H. Kronmüller, Naturwissenschaften 87, 423 (2000).
magnetic domains as seen by Kerr microscopy
grain
magnetic
domains
http://en.wikipedia.org/wiki/Magnetic_domain
AlNiCo annealed with and without magnetic field
X. Han et al. J. Alloys Cmpds. 785, 715 (2019)
typical grain size < 3mm
typical grain size > 10 mm
irregular morphology
& inhomogeneous distribution
very regular morphology
& homogeneous distribution
3.1.2 magnetization in response to processing
X. Han et al. J. Alloys Cmpds. 785, 715 (2019)
shopping list for hard magnetic materials (simplified)
• highly anisotropic crystallographic structure
• highly anisotropic atomic orbitals
• high magnetic moment
• high Curie temperature
• many pinning centers
SOC
high magnetocrystalline
anisotropy
mainly determines
high remanence
microstructure,
stress, strain
intrinsic
extrinsic
mainly determines
high coercive field
soft magnetic materials
Wurmehl et al.
Appl. Phys. Lett. 88 (2006) 032503
Phys. Rev. B 72 (2005) 184434
shopping list for soft magnetic materials (simplified)
• isotropic crystallographic structure, fcc or bcc
• as less pinning centers as possible
intrinsic
extrinsic
4 spectroscopic techniques
local spectroscopic methods
• Nuclear magnetic resonance spectroscopy (NMR)
• Mößbauer spectroscopy (Mößbauer)
Method I:
Nuclear magnetic resonance (NMR)
• nucleus has nuclear magnetic moment mN with mN= ħI
I is nuclear spin qn (I≠0 → nucleus NMR active)
• nuclear magnetic moment precesses
around steady magnetic field B0
• frequency of precession
→ Larmor frequency with L= B0
• energy of nuclear precession quantized E=-mNħB0
nucleus
nuclear Zeeman splitting
(2I+1) sub-levels
Population described by
Boltzman statistics
Nuclear
Zeemann
splitting
dipolar transitions
Selection rule for transition:
m=1
E=(h/2p)L= gmN B0
resonance frequency depends on local (magnetic and electronic)
environment of nucleus
Nuclear Magnetic Resonance (NMR)
Resonance frequency / hyperfine field
L= B0
resonance
• dipolar transition observed if resonance condition is fulfilled:
L= B0
• dipolar transition induced by radio frequency pulses
• rf pulses applied by coil wrapped around sample
• signal inductively measured
pulsed NMR
• superposition static field B0 and rf field
• rf pulses are time dependent external fields
“corkscrew scenario”
description quite complicated
simplification rotating frame formalism
• frame rotates with around B0
• transformation of coordinates
• rotating frame formalism: rf pulses rotate precessing spins around one of the axis of rotating frame
rotating frame formalism
relaxation
• two types of relaxation
→ longitudinal (paralell to B0) components of mN T1
→ transverse (perpendicular to B0) components of mN T2
spin lattice relaxation
• after rf pulse spins repopulate initial energy levels(back to thermal equilibrium)
• relaxation time T1
))/exp(1()( 10 TtMtM z
spin-spin relaxation time
• spins exchange polarization (dipole-dipole interaction, loss of phase coherence)
• relaxation time T2
))/(exp()( 20 TtMtM
MATCOR summer school, Rathen bei Dresden 2008
spin Echo NMR
Knight shift K
• metals: small polarization of unpaired conduction electrons due to applied field
(compare Pauli spin susceptibility)
→ small frequency shiftcompared to dia- or paramagnetic materials
Korringa relation:
B
B
kTTK
2
2
21
p
m
field at nucleus
• condensed matter:
static field B0≠ Bapplied magnetic field
electronic magnetization yields additional field at nucleus
nuclei experience “effective field”
hyperfine field (NMR and Mößbauer)
results from all electron spin and orbital moments within ion radius
hyperfine interactionInteraction of nuclear magnetic moments with magnetic fields due to spin and orbital currents of the surrounding electrons
courtesy H.-J. Grafe
hyperfine interactions
courtesy A.U.B. Wolter
physics: a typical 59Co NMR spectrum
courtesy of H. Wieldraaijer, TU Eindhoven
different local environments have different hyperfine field
NMR active nuclei
I≠0 → nucleus NMR active
Method II:
Mößbauer spectroscopy
resonant absorption of -quants
resonant absorption of -quant, BUT…
excited state
ground state
Nnucleus emission resonant absorption
Z,N
Z,N
source absorber
Z,N
Z,N
EgEg
Ee Ee
recoil!
solid state matter:recoil passed to crystal lattice
-quant
Z,N
nucleus
E=E0-Er
Er
recoil
conservation of momentum recoil of nucleus
in gases and molecules no resonant absorption
22 2/ mcEEr
radiation for e.g. 57Fe Mößbauer
http://pecbip2.univ-lemans.fr/~moss/webibame/
Mößbauer effect
how to make use of the Mößbauer effect?
Up to now:
Ideal, model solid state system
• no recoil
• maximum resonant absorption due to exactly matching nuclear energy levels
• „no chemistry“
how to make use of the Mößbauer effect?
real solid state system:
• no recoil
• nuclear energy levels shifted due to interactions
Excited state
Ground state
Nucleus Emission
Z,N
Z,NSourceAbsorber
Z,N
Z,N
Eg
Eg
Ee
Ee
??
E0(absorber) ≠ E0(source)
Doppler effect
http://de.wikipedia.org/w/index.php?title=Datei:Dopplerfrequenz.gif&filetimestamp=2007012718204
policecar is not moving
observer/absorber
policeman and observer “hear siren” with same frequency
observer/absorber
policeman and observer “hear siren” with different frequency
)/( cvEE
experimental setup
P. Gütlich, CHIUZ 4, 133 (1970)
Source DetectorAbsorber/sample
thin foils or powder samples (thickness <50mm)
resonance line doppler effect
v=0
v>0
v<0
P. Gütlich, CHIUZ 4, 133 (1970)
Mößbauer spectrum
http://chemwiki.ucdavis.edu/@api/deki/pages/1813/pdf
100%
0%
what affects the hyperfine interaction?
• monople-monopole interaction isomer shift (chemical shift)
• quadrupole interaction quadrupole splitting
• hyperfine interaction magnetic splitting
http://iacgu32.chemie.uni-mainz.de/moessbauer.php?ln=d
isomer shift
variation of electron density at nucleus
quadrupole splitting
inhomogenous electrical field interacts with quadrupole moment at nucleus
nuclear Zeemann splitting
magnetic splitting (e.g. 57Fe with I=3/2)
Selection rule for dipolar transition:
I= 1 ; m=0,1
Em magnetic field Beff at nucleus
Nucleus with magnetic dipole m(I>0):57Fe
http://chemwiki.ucdavis.edu/@api/deki/pages/1813/pdf
Mößbauer active nuclei
50% of all Mößbauer experiments
57Fe Mößbauer spectrum
http://en.wikipedia.org/wiki/File:Mossbauer_51Fe.png
literature
http://www.cis.rit.edu/htbooks/nmr/inside.htm
http://alexandria.tue.nl/extra2/200610857.pdf
http://alexandria.tue.nl/extra3/proefschrift/boeken/9903019.pdf
Wurmehl S, Kohlhepp JT, Topical review in J. Phys. D. Appl. Phys. 41
(2007) 173002
Panissod P, 1986 Nuclear Magnetic Resonance, Topics in Current
Physics: Microscopics Models in Physics
de Jonge W, de Gronckel HAM and Kopinga K, 1994 Nuclear magnetic
resonance in thin magnetic films and multilayers
Ultrathin Magnetic Structures II
Gütlich P, CHIUZ 4 (1970) 133
2.1 Free electron model – 3 dimensions, N fermions
N particles in box (fermions with spin ½)
Eigenvalues for energy
with plane waves
occupied states
2 spins (Pauli)volume of every state in k-space
distance between each dot 2p/L
volume of k-space
http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf
plane waves in k-space
2.1 Free electron model – density of states
N particles in box (fermions with spin ½)
2 spins per state
(Pauli) volume of every state in k-space
distance between each dot 2p/L
volume of k-space
increasing the density of states (DOS)
2.1 Free electron model – finite T
𝐻Ψ = 𝑝2
2𝑚Ψ = − ℏ
2𝑚⋁2Ψ= 𝐸Ψ 𝑤𝑖𝑡ℎ Ψ 𝑟 = 1
𝑉𝑒−𝑖𝑘𝑟
N particles in box (fermions with spin ½)
Filling up of energy levels up to n = N/2
Temperature dependence Fermi function f (E,T) = (1
𝑒𝐸−𝜇
𝑘𝐵𝑇
+1)
T = 0 K corresponding Fermi wave vector kF (Fermi level);
well-defined border between occupied and unoccupied states (f(E,T) is step function)
T>> 0 K Fermi function f (E,T) = (1
𝑒𝐸−𝜇
𝑘𝐵𝑇
+1) with m the chemical potential
http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf