Alessandro Vindigni Laboratorium für Festkörperphysik, ETH Zürich ETH October 29, 2012
Lecture plan N-body problem 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
Lecture plan N-body problem Characteristic energy scales intra-atomic exchange interaction (Hund s rules) 2 3 ev 10 4 K gµ B Sh: Zeeman splitting for B-field=1 T 0.1 mev 1 K J: inter-atomic exchange interaction (RKKY) 10 50 mev 100 500 K T c [K] C[K] µ[µ B ] J[eV] J[k] M(0, 0)[Gauss] Fe 1043 2.22 2.22 0.012 139 1746 Co 1395 2.24 1.71 0.015 174 1446 Ni 629 0.588 0.605 0.013 151 0.510 Gd 289 7.1 0.00025 2.9 2.060 Dy 87 2920
Lecture plan N-body problem N-body problem Magnetic moment of an isolated paramagnetic atom [ ] m = gµ B Ŝ z gµb S B = gµ B S B S k B T where B S is the Brillouin function Heisenberg Hamiltonian for N-coupled spins H = 1 2 J Ŝ(n) Ŝ(n ) + gµ B B n n =1 n Ŝ z (n) Hilbert space (2S + 1) N
Lecture plan N-body problem State-of-the-art of numerical methods Actual number of spin 1/2 10 7 4D 3D 2D 1D 10 6 10 5 10 4 10 3 10 2 10 1 Quantum Classical Monte Carlo Monte Carlo 10 0 ED 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 ED: exact diagonalization Linear size of the system
Lecture plan N-body problem Possible ways to simplify the problem 1. Reduce the many-body problem to a s.p. problem (MFA) 2. Simplify the problem replacing the quantum-spin operators by classical vectors 3. Use symmetry arguments 4. Consider only a selected family of excitations: localized: domain walls delocalized: spin waves
Critical exponets MFA: single-particle Hamiltonian H sp (n) = gµ B (B + B W ) Ŝ z (n) with B W = zj Ŝ z (n) Hsp /gµ B Weiss field Many-body problem Single-particle problem MFA effective field
Critical exponets By analogy with an isolated atom Ŝ z (n) Hsp = SB S (α) with α = gµ B S B t k B T = gµ B S B zjs Ŝ z (n) Hsp k B T. Setting m = gµ B Ŝ z (n) Hsp, one gets [ gµb S B m = gµ B S B S k B T + zj S m ] gµ B k B T
Critical exponets For graphic visualization { σ = m gµ B S = B S(α) σ = k BT α gµ BB zjs 2 zjs When B = 0 and α 0 B S (α) S + 1 3 S α +... so that a spontaneous magnetization (σ 0) only arises for T < T C with T C = S + 1 zjs 2 3 S k B
Critical exponets Curie-Weiss law For T > T C and α 0 { σ = S+1 by using the definition of T C σ or equivalently (using T C ) m = gµ B Sσ = (gµ B) 2 zj 3 S α = k BT α gµ BB zjs 2 zjs (1 T T C )σ = gµ B zjs B T C B = (gµ B) 2 S(S + 1) 1 B T T C 3k B T T C The pre-factor of h on the right-hand side is right the susceptibility C χ = with C = (gµ B) 2 S(S + 1) T T C 3k B Curie-Weiss law.
Critical exponets Critical exponents in general Being τ := T T C 1 the reduced temperature, α, β, γ and δ critical exponents are defined as follows: C(τ, B = 0) τ α M(τ, B = 0) ( τ) β, τ < 0 χ(b = 0, τ) τ γ M(τ = 0, B) B 1/δ. MFA 3d-Heisenberg α 0 (Jump) 0.11 ± 0.006 β 0.5 0.365 ± 0.002 γ 1.0 1.386 ± 0.004 δ 3.0 4.46
Critical exponets Experiments on a Fe film on W(110)