in the de Haas van Alphen effect Magnetic domains of non-spin origine related to orbital quantization Jörg Hinderer, Roman Kramer, Walter Joss Grenoble High Magnetic Field laboratory Ferromagnetic domains Permalloy Ni-Fe, 10 x 20 µm Valery Egorov Kurchatov Institute, Moscow Louis Jansen Département de Recherche Fondamentale sur la Matière Condensée, CEA-Grenoble 1
de Haas van Alphen effect oscillatory susceptibility in Bi de Haas and van Alphen, Leiden Communication, 1930 periodic in 1/H prediction of oscillatory magnetization Landau, Z. Phys., 1930 Electrons on Landau cylinders occupied within the Fermi sphere 2
Landau quantization k x A extr k z k y B Electrons in a magnetic field: Landau levels: E n = ħω c (n+1/2) + ħ 2 k z 2 cyclotron frequency: ω c = eb /m cross section of the n-th cylinder: A n = 2π (n+1/2) eb/ħ Fermi-surface topology period in inverse field (1/B) = 2πe/ħ A extr When B is increased the Landau cylinders pass through the Fermi surface, leading to oscillations in the density of states at the Fermi level. 3
dhva method Powerful method: electronic structure in metals Fermi surface topology effective band mass electronic scattering (Dingle) spin g-factor Ag Bi 971 T 942 T heavy-fermion compounds, low-dimensional organic metals, MgB 2, borocarbides YNi 2 B 2 C,.. 4
Lifshitz - Kosevic formula (1958) Fermi-surface topology from the dhva frequency F: Effective band mass m * from temperature dependence reduction factor R T : Scattering time τ from magnetic field dependence Dingle reduction factor R D : Spin-slitting g-factor from damping factor dephasing by Zeeman splitting E ± gµ B B Monograph D. Shoenberg, Magnetic oscillations in metals, 1984 5
even more in the dhva effect dhva in the superconducting state (down to 0.5 B c2 ) NbSe 2, V 3 Si, Nb 3 Sn, YNi 2 B 2 C, Ba(K)BiO 3 Very recently in underdoped YBaCuO no change frequency at B c2 no change effective mass dhva signal for YNi 2 B 2 C reduced amplitude in superconducting state B c2 6
even more in the dhva effect 2D systems (quasi-2d) Field-induced Fermi-surface changes: phase transitions Magnetic break-down.. Magnetic interaction effects 7
Magnetic interaction or Shoenberg effect, M versus H Electrons feel the external field but also the field from all electrons around B = µ 0 (H + M) M(B) = M(µ 0 H + µ 0 M) = A sin [k (µ 0 h + M)] Instability for a µ 0 dm/db = ka 1 jump in magnetization at critical field from M Q to M P with gap in the induction values Shoenberg observed deformation of line-shape in Au (1962) 8
Magnetic interaction, B versus H Small magnetic interaction a < 1 High magnetic interaction a > 1 H a long rod-like sample ignore demagnetization window of B-values does not occur similarity with p-v diagram of liquid-gas transition (van der Waals) 9
Energy arguments for the thermodynamic instability E magnetization = (1/2µ 0 ) (B µ 0 H) 2 E osc (dhva effect) 10
Finite samples, demagnetization field H a B = µ 0 H + µ 0 (1-n)M (for plate-like sample n=1 and B = µ 0 H) slope ~1/n J.H. Condon, Phys.Rev. 145, 526 (1966) 11
Domain size thickness of domain wall w Contribution of surface energy (per vol) ~ wh a M 2 /p Contribution of magnetostatic energy (per vol) ~ H a M 2 p/t minimum of total free energy at equilibrium for p ~ (wt) 1/2 [ w ~ r c ] typical domain period of about 30 µm for cyclotron radius of 1 µm (Bi at 2.3 T, Ag at 9.0 T) samples thickness 1 mm 12
Condon domain phase diagram condition a=1 (magnetization amplitude equal field period) determines presence of (using Lifshitz-Kosevic theory) Ag sub-structure in each dhva period 13
example of interaction effect in dhva signal Be needle-like sample distorted dhva signal indication of Condon domain state for dm/dh a > 0 at lowest temperatures 14
NMR experiment First evidence of magnetic domains in pure Ag dhva period J.H. Condon and R.H. Walstedt, Phys. Rev. Lett. 21, 612 (1968) 15
µsr experiment two resonance lines from two domains Be crystal 3 T, 800 mk G. Solt, C. Baines, V. S. Egorov, et al. Phys. Rev. Lett. 76, 2575 (1996) 16
Condon domain studies --- Direct visualization of the domains using local Hall probes --- Condon domain phase diagram determination hysteresis effect at the transition 17
Hall probes Local probe of the induction on top of single crystals Be single cristal single crystals Be, Ag chemically polished Chip carrier with Hall probe 18
Direct observation of Using Hall probes, 10 x 10 µm 2 Linear array of Hall probes No. 1 2 3 4 5 Longitudinal array Transverse array distance d = 40 µm dhva period Ag Difference in B between neighbouring Hall probes L-array R.B.G. Kramer, et al., PRL 95, 267209 (2005) 19
Comparison T- and L-arrays T-array Arbitrary order between successive probes L-array Order in successive probes, But inversed for successive dhva periods Domains walls move along long side of sample 20
Tilted Ag sample (15 around long sample axis) L-array Field perpendicular inversed pattern Field rotated by 15 regular pattern By tilting, regular domain pattern parrallel to sample boundaries 21
Analogy with domains in type-1 superconductors Intermediate state domains in Sn domain size ~20 µm perpendicular field field tilted 15 22
Domains observed using an array of Hall probes preferentially aligned domains (reinforced for tilted magnetic field) domain parameters deduced for Ag domain period at least 150 µm (expected 30 µm) domain width around 20 µm 23
Condon domain phase diagram --- Hysteresis observed at Condon domain transition --- Condon domain phase diagram determination Techniques Hall probes pick-up coils 24
Hysteresis at Condon domain transition Hall probe signal Be 1.3 K Easier to observe via AC techniques R.B.G. Kramer, et al., PRL 95, 187204 (2005) 25
AC response at the hysteresis transition Be for H // (0001) AC DC Non-linear response of AC detection for modulation amplitude ~ hysteresis width 26
Non-linear AC response out-of-phase signal 3rd harmonic signal Be for H // (0001) Determination of Condon phase diagram boundary 27
Non-linear pick-up signal for Be Pick-up signal Be for H // (0001) 28
Condon phase diagram for Be Be for H // (0001) Sub-structure in phase diagram due to beating pattern. Comparison with LK formula not ideal. 29
Condon phase diagram for Ag Ag for H // (100) phase boundary calculated from LK formula 30
Origin of hysteresis -- irreversible domain wall motion -- rearrangement of domain volume fractions Energy barrier between domain states in ferromagnets: - magnetic anisotropy - wall pinning at defects - inhomogeneities - geometry Rayleigh model (N. Logoboy) Experimental determination hysteresis width Energy barrier (LK formula) Be 31
in the dhva effect Domains observed on the sample surface of Be and Ag Hysteresis allows to determine the phase diagram scanning the Hall probe! 32