Magnetism and Spin-Orbit Interaction: Some basics and examples. Dieter Weiss Experimentelle und Angewandte Physik

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1 Magnetism and Spin-Orbit Interaction: Some basics and examples Dieter Weiss Experimentelle und Angewandte Physik

2 (Our initial) Motivation Transport in a two-dimensional electron gas with superimposed periodic magnetic field (l e >> a) ferromagnet (Ni) a 2DEG Magnetization pattern of ferromagnet determines stray field

3 Transport in a periodic magnetic field Commensurability (Weiss) Oscillations : ( ) 2R = n ± 1 a SdH oscillations C 4 ρ xx (Ω) B (T) see, e.g., P.D. Ye PRL 74, 3013 (1995)

4 A little bit of history SciencephotoLIBRARY Hans Christian Ørsted 1820: Electric current generates magnetic field Albert Einstein 1905: On the electrodynamics of moving bodies Magnetic field stems from Lorentz contraction of moving charges

5 Magnetism and Spin-Orbit Interaction: Magnetism Basic concepts magnetic moments, susceptibility, paramagnetism, ferromagnetism exchange interaction, domains, magnetic anisotropy, Examples: detection of (nanoscale) magnetization structure using Hall-magnetometry, Lorentz microscopy and MFM Spin-Orbit interaction Some basics Rashba- and Dresselhaus contribution, SO-interaction and effective magnetic field Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces

6 Magnetism and Spin-Orbit Interaction: Magnetism Basic concepts magnetic moments, susceptibility, paramagnetism, ferromagnetism exchange interaction, domains, magnetic anisotropy, Examples: detection of (nanoscale) magnetization structure using Hall-magnetometry, Lorentz microscopy and MFM Spin-Orbit interaction Some basics Rashba- and Dresselhaus contribution, SO-interaction and effective magnetic field Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces

7 Magnetic moment and angular momentum Current in a loop generates magnetic moment µ, constitutes magnetic dipole area S S μ = IS I As mass is transported around the loop µ is connected to angular momentum μ 2 = γl [Am ] γ = gyromagnetic ratio Size of atomic magnetic moment e ev 1 1 e L I = = µ = evr = L = µ B T 2πr 2 2 m ħ µ B eħ μ = L with µ B = = Bohr Magneton ħ 2m L µ - -e v Magnetization M: magnetic moments / volume [A/m]

8 Angular momentum: quantum mechanical picture Total angular momentum of an atom is composed of angular momentum of all occupied electron states L and of all corresponding spins S Total angular momentum (L-S coupling): μ J = L + S Connection between µ and J with Landé factor gµ B = ħ ( + ) + ( + ) ( + ) 2J( J + 1) J J 1 S S 1 L L 1 g = 1+ J

9 Susceptibility Connection between field H (e.g. generated by coil) and magnetization M: H Linear materials: diamagnetic if χ < 0 M = χh paramagnetic if χ > 0 0 ( H M) ( ) magnetic susceptibility Connection between B and H field : B = µ + (for M << H) B = µ 0 H + χ H = µ 0 (1 + χ ) H Not the whole story! permeability of free space ( ) B = µ H + H + M 0 d demagnetizing field or stray field (important for ferromagnets)

10 Demagnetizing field H B d [ H M ] M = µ 0 ( d + ) = 0 H d = M Any divergence of M creates a magnetic field can be seen as sort of (bound) magnetic charges Easy expression for strayfield (demagnetizing field) H d only for ellipsoids: H d Nxx 0 0 = 0 Nyy 0 M 0 0 Nzz with N + N + N = 1 xx yy zz

11 Some examples: Nxx = 0, Nyy = N zz = 1 2 N = N = N = 1 xx yy zz 3 long cylindrical rod Nxx = Nyy = 0, Nzz = 1 z y x sphere H flat plate M Hd = M B = µ H + + M = µ ( H ) 0 d 0 H

12 Paramagnetism (of isolated magnetic moments) Requisite: Atoms with non-vanishing angular momentum J Energy of magnetic moment E = μ B Classically: arbitrary values and orientation of µ U/µ B UmB πp p 3 πp 2 p 0 π 2π 2 2 fφ Quantum mechanically: discrete values and orientations of μ, e.g. J = 5 / 2 J ħ ħ ħ ħ ħ ħ gµ µ = = µ ħ B z Jz mjg B J z = mħ J mj = J,J 1,... (J 1), J 2J+1 discrete values (and hence energy levels) Associated energy levels Em = m J Jgµ BB get thermally occupied µ z

13 Magnetization (thermal average) Magnetization = µ ( ) M NgJ B x J: total angular momentum B J B Brillouin function B ( x) = coth x coth ; with x = Saturation: for x >> 1: B 1 and M NgJµ = M J N : Density of magnetic atoms B J S 2J + 1 2J x gjµ B 2J 2J 2J 2J k T B J (x) B 1 3 J =,1,, "High T": 1 x 0 x << 1 coth( x ) +... x 3 J + 1x B x = gjµ BB / kt J J g J(J + 1) µ B g J(J + 1) µ B M g J(J + 1) µ Bµ 0 1 M = NgJµ BBJ ( x) N B = N µ 0H χ = N 3k T 3k T H 3k T B B B

14 Paramagnetism (of free electrons) E ( ) B B Magnetization: M = N N µ = N µ E F 1 3 N N = D( E ) 2gµ B = gµ B 2 2 E F B B F 3 N D( E F ) = in 3D 2 E F 2gµ B B D(E) Resulting magnetization: 3 N 3 N M = gµ B = gµ µ H 2 E 2 k T F 2 2 B B 0 B F χ Taking into account diamagnetism of free electrons (M = M ) N 2 χ = g µ Bµ 0 Independent of temperature! k T B F dia 1 3 para

15 Exchange Interaction (a la Heisenberg) Exchange due to Pauli exclusion principle and Coulomb repulsion An electron pushes a second electron with the same spin orientation out of a region with diameter λ exchange hole forms λ Consequences of spin arrangement for total energy Exchange hole For parallel orientation, e.g., potential energy gets reduced compared to anti-parallel arrangement, e.g., since Coulomb-repulsion is reduced due to larger average interparticle distance. On the other hand kinetic energy is larger for since Fermi-energy increases due to fixed electron number. delicate interplay!

16 Increase of kinetic energy (example) E E δe E F E F D (E) D (E) D (E) D (E) Fermi energy E F of a free electron gas increases if, for fixed electron density only one spin species is permitted

17 Exchange Interaction (a la Heisenberg) r 1 R1 r 2 R2 Two electron wave function of hydrogen molecule like system: 2 ˆ ħ ( 2 2 Hψ = ) ψ + V( r1, r2 ) ψ = Eψ 2m Fermi-Dirac statistics requires that ψ ( r, r ) = ψ( r, r ) exchange of electrons Construction of Ψ with Heitler-London approximation: Spin of both electrons anti-parallel (Singulett, S=0) [ + ϕ ϕ ] ψ ( r, r ) = ϕ ( r ) ϕ ( r ) ( r ) ( r ) χ s S symmetric anti-symmetric under particle exchange Spin of both electrons parallel (Triplett, S=1) [ ϕ ϕ ] ψ ( r, r ) = ϕ ( r ) ϕ ( r ) ( r ) ( r ) χ T T anti-symmetric symmetric under particle exchange

18 Spin wave function χ r 1 S m s χ S1 S2 R1 r 2 R Triplett χ T symmetric Singulett χ S anti-symmetric see, e.g. Stephen Blundell Magnetism in Condensed Matter Oxford University Press

19 ψ Exchange splitting ( r, r ) and ψ ( r, r ) have different energy E and E S 1 2 T 1 2 S T ψs Hˆ ψs ψ ˆ T H ψt ES E T =... ψ ψ ψ ψ S S T exclusively due to the exchange of the two electrons T Ashcroft, Mermin e e e e = 2 d 1d 2 1( 1) 2( 2) r r 1( 1) 2( 2 ) ϕ r ϕ r + ϕ r ϕ r r 1 r R 2 1 R2 r 1 R1 r2 R 2 S energy 1 J/4 0-3J/4 J m s exchange integral J sign of J determines spin ground state J > 0 E S > ET spins parallel J < 0 E < E spins anti-parallel S T B

20 Heisenberg Hamiltonian Expressing spin Hamiltonian with Eigenvalues E S and E T in terms of S 1 and S 2 : + 1 S = Hˆ = (E ˆ ˆ 4 S + 3E T ) (ES E T ) S1 S ˆ ˆ 2 S1 S2 = 3 S = 0 Hˆ has Eigenvalue E for triplett ( or ) and T E for singulett ( or ) S Shift of origin results in usual form of Heisenberg Hamiltonian Hˆ = (E E ) Sˆ Sˆ = JSˆ Sˆ S T Generalisation for many spins: H ˆ = J ˆ ij Si S ˆ j i,j 4 ˆ 2 using that S = S(S + 1) Exchange integral between spin i and spin j

21 Forms of exchange interaction Direct exchange interaction from direct overlap Superexchange mediated by paramagnetic atoms or ions RKKY Interaction (After M.A. Rudermann, C. Kittel, T. Kasuya, K. Yosida) coupling mediated by mobile charge carriers

22 Example: p-d exchage interaction in a 2DHG Mn (S=5/2) in two-dimensional hole gas Nature Physics 6, 955 (2010) Wurstbauer et al. see: WP-20 Anomalous Hall effect in Mn doped,p-type InAs quantum wells C. Wensauer, D. Vogel et al.

23 Weiss molecular field Pierre Weiss 1907 Idea: Replace interaction of a given spin with all other spins by interaction with an effective field (molecular field) Sum of all S j act on one particular spin S i magnetic moment μ of spin S i i g i E = 2 µ J, = S = μħ g µ B i ex Si ijsj μi Si j ħ E 2 J J 2 μiħ 2ħ ex = ijsj = μi μ 2 i ij = i M g µ B μ B j (g µ B ) j = λm Weiss molecular field B M S replaced by constant moment: μ = g µ S / ħ S = ħ μ / gµ j j B j j j B B Magnetic moments get aligned in an effective field B eff = B 0 +B M >>B 0

24 Ferromagnetism within the molecular field picture Molecular field B m =λm allows to map ferromagnetism onto paramagnetism 2J + 1 2J 1 1 x gjj B(B ) M NgJJ B coth + x coth µ + λm = µ with x = 2J 2J 2J 2J kt M on both sides of equation: graphical or numerical solution T C = g Jµ B(J + 1) λm 3k B S M C S 3 Für T ~ 10 K und B = λm ~ 1500T J = 1/ 2

25 Band magnetism and Stoner criterion Heisenberg Model unable to describe all aspects of ferromagnetism e.g. moment of iron is 2.2 µ B. Stoner picture: E Redistribution of spins: Energy cost E kin 1 2 Ekin = D(E F ) δe 2 E F δe Resulting magnetization ( ) B F B M = n n µ = D(E ) µ δe Energy of electron s magnetization dm in effective (Weiss) field B eff of other electrons D (E) D (E) M 2 M 1 2 depot = dm Beff Epot = dm λ M = λ = BD(E F ) E 2 2 λ µ δ Energy gain if Ekin + Epot = D(E F ) δ E [1- λµ BD(E F )] < B 0 F 2 2 Ferromagnetism occurs if λµ D(E ) 1 Stoner criterion (holds for Fe,Ni,Co)

26 Ferromagnet: Density of States D(E) Schematic Nickel majority spins majority spins s-electrons minority spins minority spins d-electrons

27 Magnetic Domains 1 1 Energy of stray field H : E = µ H dv = µ H M dv 2 d d 0 d 0 d 2 2 space sample Formation of domains reduces stray field energy!

28 Width of a Bloch wall Classical Heisenberg Hamiltonian Eex = 2JS1 S2 = 2JS2 cos θ θ = 0 Eex = 2JS2 Energy to rotate spin by angle θ θ 0 Eex = JS2θ2 for θ << 1 2 π 2 π Spin rotation by π over N sites θ = energy cos t per line : Eex = JS N N Energy per unit area εex = Eex 2 a a: lattice constant 2 = JS π2 Na2 best N?

29 Magnetocrystalline anisotropy Ferromagnetic crystals have easy and hard axes. Along some crystallographic directions it is easy to magnetize the crysal, along others harder. Reflects the crystal symmetry. Extra energy associated with magnetocrystalline anisotrpy: uniaxial anisotropy (e.g. Co) cubic anisotropy (e.g. Fe) = 2 θ E K sin ( ) K sin ( θ ) +... E = K1 ( sin ( θ)sin (2 ϕ ) + cos ( θ) ) sin ( θ ) ϕ θ M

30 Example: GaMnAs core-shell nanowires system with very strong uniaxial anisotropy see: WP-22, C. Butschkow et al.

31 Width of a Bloch wall, continued assume simple form of anisotropy energy Eani = K sin2 ( θ); K > 0 a energy cost of rotation out of easy direction N π N NK 2 K sin θi π K sin θdθ = 2 i=1 0 2 energy per unit area= NKa 2 exchange energy + anisotropy energy εbw From dεbw / dn = 0 N = πs 2 NKa 2 π = + JS 2 Na2 2J 2J = π S Na Ka Ka3 domain wall width

32 Reversal of magnetization: Hysteresis macroscopic bulk material consisting of many domains M S M r M -H C H S H Processes 1. Shift of domain walls (reversible and irreversible) 2. Domain rotation towards nearest easy axis 3. Coherent rotation at high fields

33 Magnetization of a cube with uni-axial anisotrpy micromagnetic simulation sample size vortex-state multi-domain-state two-domain-state single-domain-state normalized anisotropy constant K / K W. Rave, K. Fabian, A. Hubert, JMMM 190, 332 (1998) u d anisotropy

34 Magnetism and Spin-Orbit Interaction: Magnetism Basic concepts magnetic moments, susceptibility, paramagnetism, ferromagnetism exchange interaction, domains, magnetic anisotropy, Examples: detection of (nanoscale) magnetization structure using Hall-magnetometry, Lorentz microscopy and MFM Spin-Orbit interaction Some basics Rashba- and Dresselhaus contribution, SO-interaction and effective magnetic field Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces

35 Measurement of domains: magnetic force microscopy cantilever with magnetic tip lift height 1. topography scan 2. MFM scan (lift mode) MFM measures magnetic charges magnetization M div M sim. MFM picture real MFM picture

36 Domains, measured by MFM Perpendicular magnetization in-plane magnetization 4Å Fe / 4Å Gd (75 layers) 50 Mbyte hard drive

37 Between multidomain and single-domain state: vortex structure 4 ground state configurations in-plane and out-of plane magnetization component can be switched independently! Picture: Ref 3) 1) J. Raabe, R. Pulwey et al., J. Appl. Phys. 88, 4437 (2000) 2) T. Shinjo et al., Science 289, 930 (2000) 3) A. Wachowiak et al., Science 298, 577 (2002)

38 300 nm Detection of in-plane vortex Lorentz-microscopy probes in-plane magnetization J. Raabe, R. Pulwey et al., J. Appl. Phys. 88, 4437 (2000)

39 Detection of vortex singularity J. Raabe, R. Pulwey et al., J. Appl. Phys. 88, 4437 (2000)

40 Another technique: Micro Hall magnetometry allows to measure hysteresis traces M M Measures directly the stray field, averaged over the central area of the Hall junction U H = I en s B

41 Hysteresis trace of a vortex state M. Rahm et al., J. Appl. Phys. 93, 7429 (2003)

42 Vortex-Golf 1.8 mm

43 Magnetism and Spin-Orbit Interaction: Magnetism Basic concepts magnetic moments, susceptibility, paramagnetism, ferromagnetism exchange interaction, domains, magnetic anisotropy, Examples: detection of (nanoscale) magnetization structure using Hall-magnetometry, Lorentz microscopy and MFM Spin-Orbit interaction Some basics Rashba- and Dresselhaus contribution, SO-interaction and effective magnetic field Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces

44 Origin of spin-orbit (SO) interaction Spin-orbit interation E = µ B B eff due to orbital motion magnetic moment of electron nucleus + - electron electron s rest frame - + B eff Hˆ E p = µ σˆ 2mc SO B 2 H ˆ = µ σˆ B Zeeman B vector of Pauli spin matrices B = E p 2mc eff 2

45 Electric field E in solids Bulk inversion asymmetry (BIA) Lack of inversion symmetry in III-V semiconductors "Dresselhaus contribution γ" B BIA kx γ ky k y B BIA k x ψ(z) 2 V(z) Structure inversion asymmetry (SIA) due to macroscopic confining potential: "Rashba contribution α". Tunable by external electric field! k y k x B SIA k y α k x B SIA

46 SO interaction in 2DEG: Rashba & Dresselhaus terms 2 2 ˆ ħ k H = + Hˆ SO with 2m tunable by gate voltage Pauli spin matrix Hˆ = α( σ k σ k ) + γ( σ k σ k ) SO x y y x x x y y Rashba Dresselhaus

47 Calculation of Eigenvalue Rashba Hˆ = α( σ k σ k ) SO x y y x σˆ = 0 1 x ; σˆ = 0 i y ; σˆ z = i Rashba Hˆ = α( σ k σ k ) = SO x y y x Pauli Spin matrices 0 αky 0 iαkx 0 α (ky + ik x ) = = α α ky 0 iαkx 0 (ky ik x ) 0 Eigenvalues of the matrix : ±α k + k = ±αk 2 2 x y

48 Description of zero-field spin splitting by B eff Ĥ = α( σ k σ k ) + γ( σ k σ k ) ~ σˆ B ; SO x y y x x x y y eff σˆ B eff = σ B + σ B + σ B x x y y z z Rashba Dresselhaus Comparison of coefficients. E.g. only Rashba contribution: B eff ky α k x B eff E.A. de Andrada e Silva PRB 46, 1921 (1992)

49 Presence of Rashba & Dresselhaus contributions Rashba or Dresselhaus Rashba and Dresselhaus 2 k 2 ħ = ± α + γ E ( ) k 2m ] 2 k 2 ħ = ± α γ E ( ) k 2m

50 SO-interaction in a InGaAs quantum well B V x ρ xx Beating of SdH-oscillations I V g Nitta et al., Phys. Rev. Lett 78, 1335 (1997)

51 Quantum oscillations (SdH) reflect k-space area k y Periodicity of Shubnikov-de Haas (SdH) oscillations k F A k x 1 2πe = B ħa 2 2 F Note that A = π k = 2π n s Origin of beating: two periodicities due to two k-space areas A 1 and A 2 ρ xx A 1 A 2 Nitta et al., Phys. Rev. Lett 78, 1335 (1997) B

52 Tuning of Rashba coefficient a by gate voltage V g Corresponds to tuning of spin orbit field, i.e., B eff ky = α k x Nitta et al., Phys. Rev. Lett 78, 1335 (1997)

53 Magnetism and Spin-Orbit Interaction: Magnetism Basic concepts magnetic moments, susceptibility, paramagnetism, ferromagnetism exchange interaction, domains, magnetic anisotropy, Examples: detection of (nanoscale) magnetization structure using Hall-magnetometry, Lorentz microscopy and MFM Spin-Orbit interaction Some basics Rashba- and Dresselhaus contribution, SO-interaction and effective magnetic field Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces

54 Spin-orbit interaction plays also role in tunneling Epitaxial Fe-GaAs interface [110] Fe As Ga [110] a GaAs =5.653 Å a Fe =2.867 Å

55 Device fabrication Substrate Au Fe GaAs 8 nm As-Cap Fe GaAs Substrate Substrate J. Moser et al. Appl. Phys. Lett. 89, (2006)

56 Fe GaAs AlGaAs TEM-micrograph: C.-H. Lai, Tsing Hua University, Taiwan

57 TAMR: Tunneling Anisotropic Magnetoresistance Are always two ferromagnetic layers necessary to see a magnetization dependent resistance? Our model system: Fe/GaAs/Au with epitaxial Fe/GaAs interface M M Fe R 1 GaAs R 2 Au R R 1 2 See also: Gould et al. PRL 93, (2004) (Ga,Mn)As/Al 2 O 3 /Au

58 Tunneling magnetoresistance: B dependence Spin-valve like signal -88 mv Double step/single step switching due to magnetic anisotropy Fe on GaAs: Cubic + uniaxial anisotropy [110 easy [110 φ B φ = 0 B J. Moser et al., Phys. Rev. Lett. 99, , 2007

59 Angular dependence of TAMR: negative bias Co/Fe(epi)/GaAs(8nm)/Au measured at T = 4.2 K and -90 mv B=0.5T R M Fe GaAs Au J. Moser et al., Phys. Rev. Lett. 99, , 2007

60 Angular dependence of TAMR: positive bias Co/Fe(epi)/GaAs(8nm)/Au measured at T = 4.2 K and +90 mv B=0.5T R M Fe GaAs Au J. Moser et al., Phys. Rev. Lett. 99, , 2007

61 H = H0 + Hz + HBR + HD H 1 H = α ( σ p σ p ) δ (z z ) H 0 z Modelling 2 ħ 1 = + V(z) 2 m(z) BR i x y y x i ħ i= l, r (z) = n σ 2 A. Matos-Abiague & J. Fabian, PRB 79, (2009) 1 H D = ( σxp x σyp y ) (z) ħ z γ z 2 T 3 σ(e,k ħ σ= 1,1 Fe e I = ded k )[f l(e) f r(e)] (2 π) α particle transmissivity l γ GaAs αr Au f b z l z r z E F

62 Spin-orbit field w(k) ky ky α = 0 γ = 0 kx [100] kx [100] ky w( k ) = αky - γk -α kx + γk 0 x y ky w(k ) [100] α > γ > 0 α < 0 γ > 0 kx kx [100]

63 Origin of anisotropic resistance: superposition of Rashba & Dresselhaus SO-contribution αl γ φ b E F Au GaAs Fe x n z Fe z z l z r Anisotropy det ermined by 2 Au T( k ) f([ n w( k )] ) R / R 1~ αγ(cos2φ 1) [110] Anisotropy vanishes for αγ 0 w(k ) w( k ) = y αky - γk -α kx + γk 0 x y J. Moser et al., Phys. Rev. Lett. 99, , 2007

64 Angular dependence of TAMR: bias dependence TAMR = R( φ) R R [110] [110] 2 α l (eva ) γ in GaAs: 24 eva 3 J. Moser et al., Phys. Rev. Lett. 99, , 2007 See also:t. Uemura, Appl. Phys. Lett. 98, (2011)

65 Fe on (Ga,Mn)As Au Fe (Ga,Mn)As si-gaas (001) F. Maccherozzi et al. PRL 101, (2008) M. Sperl et al., Phys. Rev B 81, (2010)

66 XMCD: antiferromagnetic coupling RT US pat 61/133,344 Thin ferromagnetic iron film induces antiferromagnetic coupling in (Ga,Mn)As at room temperature. Minimum thickness of the coupled (Ga,Mn)As layer is at least 1 nm F. Maccherozzi et al. PRL 101, (2008) M. Sperl et al., Phys. Rev B 81, (2010)

67 Spin injection: Proximity effect enhances T C Proximity effect: exchange coupling between Fe and Mn at Fe/(Ga,Mn)As interface FP-34 C. Song et al. WP-30 M. Ciorga et al. WP-105 J. Shiogai et al.

68 Magnetism and Spin-Orbit Interaction: In retrospect Magnetism Basic concepts magnetic moments, susceptibility, paramagnetism, ferromagnetism exchange interaction, domains, magnetic anisotropy, Examples: detection of (nanoscale) magnetization structure using Hall-magnetometry, Lorentz microscopy and MFM Spin-Orbit interaction Some basics Rashba- and Dresselhaus contribution, SO-interaction and effective magnetic field Example: Ferromagnet-Semiconductor Hybrids: TAMR involving epitaxial Fe/GaAs interfaces The End