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Home , Density of states, Diamagnetism, Fermi gas, Free electron model

module: on the nanoscale, WS 2019/2020

chapter 2: magnetism in – part II (Landau ) 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 model

assumptions:

1) are free 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 • electron mass may be modified by band structure and interactions effective mass m*

4) Pauli exclusion principle each is occupied by a single electron  Fermi–Dirac statistics

Description similar as free electron in

Landau diamagnetism Pauli 2.2 Pauli paramagnetism origin of Pauli paramagnetism if conductions electrons are weakly interacting and delocalized ()  magnetic response originates in interaction of with magnetic field

Zeemann splitting in magnetic field in a

E

E = EF

B replace integral by EF

2mBB g (E) /2 g (E) /2

independent 2.3 Landau diamagnetism 2.3 Landau diamagnetism weak counteracting field that forms when the electrons' trajectories are curved due to the

…some mathematics….

plane waves in quantized states x, y direction along B harmonic oscillator plane wave

energy Eigenvalues for harmonic oscillator Landau levels (tubes)

no magnetic field: with magnetic field: discrete states k-vectors condense on tubes paralell to field

http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf Landau susceptibility of conduction electrons

application of magnetic field quantized Landau levels changes energetic state

thermodynamics: magnetic field induced change of energy

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 quantum oscillations in metals

specific heat De Haas-van Alphen effect

http://lampx.tugraz.at/~hadley/ss2/problems/fermisurf/s.pdf 2.0 magnetism in metals spin resolved DOS

example: Metallic Fe, Co, Ni, Gd

Important: NON-Integer number!

http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf 2.4 band

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 (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 :

• 3.1.1 magnetocrystalline anisotropy (spin-orbit-coupling, )

• 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

different orientations of spins correspond to different orientations of atomic orbitals relative to crystal structure

energy is minimzed if magnetic moments are aligned along specific preferred directions easy axis demagnetization field

at sample edges: magnetization diverges

 costs energy by formation of stray fields with demagnetization field HD (demagnetization energy, dipolar energy)

with Nij the demagnetization factor (shape anisotropy)

also see Maxwell equations http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-8.pdf magnetic domains

formation of stray fields costs dipolar energy energy costs minimized  formation of magnetic domains

closure domain structure

dipolar energy is minimized if as many domains as possible are formed

 BUT: formation of domains costs energy http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-8.pdf costs for formation of domains (details: lecture T. Mühl)

if ferromagnetic material forms domains: no divergence of magnetization at sample edge dipolar fields minimized within domain, all spin moments are aligned exchange energy J minimized not all domains are alignedM along preferred easy axis costs anisotropy energy between domains, spin moments need to rotate costs exchange energy

balance between costs determines width of domain wall types of domain walls

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

P. Li-Cong et al. Chinese B 27, 066802 (2018)

magnetic hysteresis loop

coherent rotation of domains

irreversible wall displacements reversible wall displacements

http://hydrogen.physik.uni-wuppertal.de/hyperphysics/hyperphysics/hbase//hyst.html different crystallographic structure different magnetic anisotropy  different hysteresis curves

http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf http://www.ifmpan.poznan.pl/~urbaniak/Wyklady2012/urbifmpan2012lect5_03.pdf https://en.wikipedia.org/wiki/Magnetocrystalline_anisotropy#/media/File:Easy_axes.jpg hard and soft magnetic materials

hard soft

H. D. Young, University Physics, 8th Ed., Addison-Wesley, 1992 hard magnetic materials magnetic 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 typical grain size < 3mm

irregular morphology & inhomogeneous distribution

very regular morphology & homogeneous distribution

typical grain size > 10 mm

X. Han et al. J. Alloys Cmpds. 785, 715 (2019) 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 SOC mainly determines  high magnetocrystalline • highly anisotropic atomic orbitals anisotropy high remanence • high

• high mainly determines microstructure, • many pinning centers high coercive field stress, strain

intrinsic

extrinsic 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 (NMR)

• Mößbauer spectroscopy (Mößbauer) Method I: Nuclear magnetic resonance (NMR) nucleus

• 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=-mNħB0 nuclear Zeeman splitting

Nuclear Zeemann (2I+1) sub-levels splitting Population described by Boltzman statistics dipolar transitions

Selection rule for transition: m=1

E=(h/2p)L= gmN B0 Nuclear Magnetic Resonance (NMR)

L=  B0

Resonance frequency / hyperfine field

resonance frequency depends on local (magnetic and electronic) environment of nucleus 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 rotating frame formalism

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 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

M z (t)  M 0 (1 exp(t /T1 )) spin-spin relaxation time

• spins exchange polarization (dipole-dipole interaction, loss of coherence)

• relaxation time T2

M(t)  M0 (exp(t /T2 )) spin Echo NMR

MATCOR summer school, Rathen bei Dresden 2008 Knight shift K

• metals: small polarization of unpaired conduction electrons due to applied field (compare Pauli spin susceptibility)

→ small frequency shift compared to dia- or paramagnetic materials

Korringa relation:

2 2 mB K T1T  p 2kB 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 radius

hyperfine interaction Interaction 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

different local environments have different hyperfine field courtesy of H. Wieldraaijer, TU Eindhoven NMR active nuclei

I≠0 → nucleus NMR active Method II: Mößbauer spectroscopy resonant absorption of -quants

Nnucleus emission resonant absorption

excited state Ee Z,N Z,N Ee

Z,N Eg Z,N Eg source absorber

resonant absorption of -quant, BUT… recoil! conservation of  recoil of nucleus

nucleus recoil -quant Z,N

Er

2 2 E=E0-Er Er  E / 2mc

in and molecules no resonant absorption

state matter: recoil passed to crystal lattice  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?

Nucleus Emission Z,N Ee Excited state Ee Z,N

??

Z,N Ground state Eg Source Z,N Eg Absorber

E0(absorber) ≠ E0(source)

real solid state system: • no recoil • nuclear energy levels shifted due to interactions Doppler effect

policecar is not moving

policeman and observer “hear siren” with same frequency

observer/absorber

E  E (v / c)

policeman and observer “hear siren” with different frequency

observer/absorber http://de.wikipedia.org/w/index.php?title=Datei:Dopplerfrequenz.gif&filetimestamp=2007012718204 experimental setup

Source Absorber/sample Detector

thin foils or powder samples (thickness <50mm)

P. Gütlich, CHIUZ 4, 133 (1970) resonance line doppler effect

v=0

v>0

v<0

P. Gütlich, CHIUZ 4, 133 (1970) Mößbauer spectrum

100%

0%

http://chemwiki.ucdavis.edu/@api/deki/pages/1813/pdf 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)

Nucleus with m(I>0): 57Fe

Selection rule for dipolar transition: I= 1 ; m=0,1

Em  magnetic field Beff at nucleus 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 plane waves in k-space 2.1 – 3 dimensions, N

with plane waves

Eigenvalues for energy occupied states

N particles in box (fermions with spin ½) volume of k-space

2 spins (Pauli) volume of every state in k-space distance between each dot 2p/L

http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf 2.1 Free electron model –

N particles in box (fermions with spin ½) volume of k-space

2 spins per state (Pauli) volume of every state in k-space distance between each dot 2p/L

increasing the density of states (DOS) 2.1 Free electron model – finite T

푝2 ℏ 1 −푖푘푟 ⋁2Ψ = 퐸Ψ 푤𝑖푡ℎ Ψ 푟 = 푒 퐻Ψ = 2푚 Ψ = −2푚 푉

N particles in box (fermions with spin ½)

Filling up of energy levels up to n = N/2

1 Temperature dependence Fermi function f (E,T) = ( 퐸−휇+1) 푘 푇 푒 퐵

T = 0 K  corresponding Fermi kF ();  well-defined border between occupied and unoccupied states (f(E,T) is step function)

1 T>> 0 K  Fermi function f (E,T) = ( 퐸−휇+1) with m the 푘 푇 푒 퐵

http://www.wmi.badw.de/teaching/Lecturenotes/magnetismus/Kapitel-5.pdf