Linear excitations and domain walls

Transcription

1 and domain walls Alessandro Vindigni Laboratorium für Festkörperphysik, ETH Zürich ETH November 26, 2012

2 Lecture plan Real systems Lecture plan 1. Atomic magnetism (Pescia) 2. Magnetism in solids (Pescia) 3. Magnetic order at finite temperature (A. V.) Mean-filed theory and Landau approach Classical spin models 4. Magnetic domains and domain walls (A. V.) Magnetic anisotropy and domain walls Dipolar interaction and magnetic domains

3 Lecture plan Real systems Susceptibility and correlation length 2d-3d systems T T C 1d-system Paramagnet χ(t, 0) χ Γ ± T T C γ χ ξ T χ = C T In 1d systems, the universality class is determined by the temperature dependence of the correlation length MFA ξ 1 T T C 1/2 Ising ξ e /k BT Heisenberg and XY ξ 1 T

4 Lecture plan Real systems 1d Ising model: molecular spin chains S z i S z i+r = 1 4 e r/ξ with ξ = 1 [ ( )] e J/2k BT ln tanh βj 4

5 Harmonic Oscillators Landau theory of correlations Digression: chain of Harmonic Oscillators H = 1 2 K e (u k+1 u k ) m k k In the canonical ensemble, the partition function is given by [ ] Z = T r e βh(u k, u k ) 1 = (2π ) N Π k du k d u k e βh(u k, u k ) H = 1 [ ] Z T r He βh(u k, u k ) = 1 ( ) Z β Z = (ln Z) β u 2 k

6 Harmonic Oscillators Landau theory of correlations Digression: chain of Harmonic Oscillators, Kinetic energy Case K e = 0 Πk du k = V N so that Z = ( ) V N 2π Π k d u k e 1 2 βm k u2 k each Gaussian integral gives d u k e 1 2 βm u2 k and consequently H = (ln Z) = β β = N β ( ) V N = Π k 2π 2π = βm ( ) N V 2π ln 2π βm [ 12 ] ln β + const. = 1 2 N 1 β = 1 2 Nk BT d u k e 1 2 βm u2 k

7 Harmonic Oscillators Landau theory of correlations Digression: chain of Harmonic Oscillators, Potential energy Case K e 0 ( ) V N Z = 2π ( ) V N = 2π Π k du k e 1 2 βke k (u k+1 u k ) 2 Π k du k e 1 2 βke(u k+1 u k ) 2 After the transformation u k = 1 N q ũqe iqk one has [K e (1 cos q) ũ q ] m ũ q 2 Coupled H = q Z = 1 (2π ) N Π k dũ q d ũ q e βh(ũq, ũ q) { K e (1 cos q) ũ q 2 = 1 2 k BT 1 2 m ũ q 2 = 1 2 k BT

8 Harmonic Oscillators Landau theory of correlations Landau theory of correlations L GL [φ] = [ 1 2 J ( φ)2 + b 2 φ2 + λ ] 4 φ4 d d x = f GL [φ]d d x perturbations of the mean-field solution: φ = φ + δφ the integrand of th GL functional reads f GL [ φ + δφ] = 1 2 J ( δφ)2 + b 2 ( φ + δφ) 2 + λ 4 ( φ + δφ) 4 = = 1 2 J ( δφ)2 b δφ 2 + f GL [ φ] + (... ) The partition function Z (B, T ) = e βl GL[ φ] + D[δφ] e βl fl[δφ]

9 Harmonic Oscillators Landau theory of correlations Fluctuation quadratic Hamiltonian Fourier space L fl [δφ] = L fl [δφ] = 1 (2π) d Equipartition theorem [ ] 1 2 J ( δφ)2 b (δφ) 2 d d x. 1 ( Jq 2 2b ) δφ(q) 2 d d q. 2 1 ( Jq 2 2b ) δφ(q) 2 fl = k BT δφ(q) 2 fl = k BT Jq 2 2b

10 Harmonic Oscillators Landau theory of correlations Landau theory of correlations 1 ( Jq 2 2b ) δφ(q) 2 fl = k BT δφ(q) 2 fl = correlation function in the continuum formalism k BT Jq 2 2b G(x, x ) = σ(x)σ(x ) σ(x) σ(x ) = δφ(x)δφ(x ) fl Ornstein-Zernicke form G(q) = k BT J 1 q 2 + ξ 2 with ξ 1 τ 1/2 G(x, x ) is actually the term that we neglected in the MFA

11 Magnetic Anisotropy Applications Uniaxial magnetic anisotropy DS 2 0 2

12 Magnetic Anisotropy Applications Magnetic anisotropy away from the bulk

13 Magnetic Anisotropy Applications Heisenberg model with anisotropy H = 1 2 J Ŝ(n) Ŝ(n ) + gµ B B Ŝ(n) D n n n =1 n (Ŝ z (n)) 2. When D becomes large with respect to J D J + Ising model D J XY / planar model.

14 Magnetic Anisotropy Applications in experiments (a) Pescia group I II (b) IBM Rüschlikon

15 Magnetic Anisotropy Applications and Racetrack memory

16 Magnetic Anisotropy Applications Domain-wall energy N x [ E H = J J cos (θi+1 θ i ) + D sin 2 ] θ i. i= Broad DW 1.0 Sharp DW