chapter 2: magnetism in metals part ii (landau ... · 2.1 free electron model assumptions: 1)...

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