Dynamics of polar nanodomains and critical behavior of the uniaxial relaxor SBN W. Kleemann, Th. Braun, Angewandte Physik, Univ. Duisburg, Germany J. Banys, Physics Department, University of Vilnius, Lithuania Th. Granzow, Institut für Mineralogie, Universität zu Köln, Germany Z. Kutnjak, Jozef Stefan Institute, Ljubljana, Slovenia R. Pankrath, Fachbereich Physik, Universität Osnabrück, Germany Ce Charge disordered congruently melting tungsten bronze SBN61 Sr 0.61-x Ba 0.39 Nb 2 O 6 :(RE,TM) x RE=Ce,La,...; TM=Co,Cr,... 0 x 0.02 Random electric fields W. Kleemann et al., Fund. Phys. Ferroelectr. (Aspen) ed. R.E. Cohen, AIP Conf. Proc. 535 (2000) 26
Features of the 3D Random-Field Ising Model phase transition expected at T c ( robust against RFs ) spatial fluctuations of quenched random fields E i compete with thermal fluctuations of the order parameter modified critical behavior: new critical exponents 3d Ising 3d RFIM α (specific heat) 0.11-0.01 Middleton, Fisher 2001 β (order parameter) 0.31 0.02 Middleton, Fisher 2001 γ (dc susceptibility) 1.31 1.89 Newman, Barkema 1996 ν (correlation length) 0.63 1.07 Middleton, Fisher 2001 enhanced critical slowing-down: activated dynamic scaling τ = τ 0 exp[{t 0 /(T m - T c )} Θν ] instead of τ = τ 0 [(T m -T c )/T c ] -zν Fisher 1986 100 nm PFM image smearing of phase transition & formation of metastable domains Imry, Ma 1975; Villain 1985
Random-field induced polar nanodomains in a 2D model system kt/j=10 <N> = 2 RF distribution kt/j=0.6 <N>= 97 Random-field 4-state Potts model in d = 2 dimensions ( order ( local ( σ, σ ) H = J δ l lm σ l, m= ± Px, ± Py m parameter components) field components) + h γlσl, l 1 σl = σm δ ( σl, σm) = if 0 σl σm hγ l=± Ex, ± Ey kt/j= 1 <N> = 8 kt/j=0.8 <N> 25 kt/j=0.3 <N>=205 +P x -P x +P y -P y kt/j=0.2 <N>=206 H. Qian, L. Bursill, Int. J. Mod. Phys. B 10 (1996) 2027 coarse graining @ fluctuations of RFs
Fingerprint experiments of RFIM behavior of SBN61:Ce cluster formation optical second harmonic precursor @ T>T c linear birefringence precursor critical slowing-down polydispersivelinear linear susceptibility activated dynamic scaling Θν = 1.1 new critical behavior linear susceptibility γ =1.0 1.39 RBIM 1.89 RFIM 97 Nb - NMR β = 0.14 linear birefringence β = 0.12 second harmonic generation (SHG) β = 0.14 pyroelectric current β = 0.14 0.30 specific heat α 0 nanodomains piezoresponse scanning force microscopy (PFM) domain walls repoling, aging and memory effects creep and non-debye relaxation dynamic light scattering.
Dielectric permittivity at elevated frequencies 10 5 T = 320 K 340 K 360 K 380 K 400 K ε ' 10 4 domain walls & ionic conductivity ε" 10 3 10 5 10 4 10 3 two dispersion steps T = 320 K 340 K 360 K 380 K 400 K nanoregions 10 2 ν min 10 1 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 ν [Hz]
Distribution of relaxation times (unpoled SBN) g(τ) 0.8 0.4 Tikhonov ε ( ω) = ε regularization of g( τ ) d lnτ + ε 0 1 + iωτ thermal shift of largest relaxation time τ max (T c +10K) T = 300 K 305 310 315 320 325 330 340 345 350 355 360 365 370 375 380 385 390 0.0 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 τ [s]
Divergence of largest relaxation time 10-2 10-3 10 log(τ/s) -2-4 -6 dynamic scaling of average relaxation time T c = 329.2±0.8 K -8 Θν = 1.08±0.04 τ = τ 0 exp [{T 0 /(T m -T c )} Θν ] τ 0 = 2.4x10-12 s, T 0 = 399.3 K T c = 320.5 K, Θν = 1.13 f 0 =6.6x10 10 Hz activated dynamic scaling 10-4 <τ> [s] 10-5 10-6 10-7 Dec et al., EPL (2001) 0.004 0.01 0.04 T m /T c -1 350 360 370 380 390 T [K]
Order Parameter Criticality measured via NMR 93 Nb (I = 9/2) Spin-Spin relaxation time T 2 P/P 0 1.0 0.5 0.0 1.0 0.5 0.2 10-4 10-2 1-T/T c 200 300 400 Temperature (K) Blinc, Zalar, Lushnikov, WK et al. Phys. Rev. B 64 (2001) 134109 B = 9 T, ν L = 99,92 MHz 90 x - τ - 90 y Echo 1 T 2Q P = (1 P 0 P 2 ) 3 / 2 (1 T / Tc) Experiment (333< T < 349K): β = 0.14±0.03 Theory: β = 0.02±0.01 3-dim. Random-Field Ising Model (Middleton, Fisher 2001) ß
10 2. n ac - n th ac 6 4 2 long-range order n (T c -T) 2β (a) (b) (c) T c - Linear Birefringence of SBN:Ce (revisited) (a) x = 0 β = 0.13 ± 0.01 (b) x = 0.007 β = 0.12 ± 0.01 (c) x = 0.011 β = 0.11 ± 0.01 T c + precursor nanodomains n <δp 2 > 1-tan -1 (ξq m )/(ξq m ) Ornstein-Zernike function Lehnen et al., EPJB (2000) 0 100 200 300 400 500 600 700 Temperature [K]
Hyper Rayleigh scattering (SHG), revisited 1.2 SBN scattering geometry x(zz)x n = a(t c -T) 2β T c =332.22±0.03 Intensity I/I 0 0.8 0.4 I/I 0 1 a=0.443±0.005 β=0.143±0.003 precursor nanodomains I ξ 2-η P. Lehnen (2001) 2β=0.288 0.0 data: Th. Braun (2004) 10-3 10-2 10-1 1-T/T c 300 320 340 Temperature [K]
Pyroelectric response and criticality of polarization polarization P (µc/cm 2 ) 0.4 0.3 0.2 0.1 1.5% poled 100% poled 0.0 290 300 310 320 330 340 350 360 0 temperature T (K) 25 20 15 10 5 2 polarization P (µc/cm ) p Granzow, Woike, WK et al., PRL 72 (2004) 065701 = pyroelectric response P β, where P = P0 (1 T / Tc ) T β(100% poled) = 0.13 RFIM β(1.5% poled) = 0.30 3d-Ising normalized polarization P/P(293 K) 1 0.9 0.8 0.7 0.6 0.5 SBN:Cr 1.13 mol% 100% a.) 16% b.) 1.5% c.) 1E-5 1E-4 1E-3 0.01 0.1 reduced temperature (1 - T/T ) C critical exponent β 0.30 0.25 0.20 0.15 0.10 0 5 10 15 20 25 2 polarization P (20 C) (µc/cm )
Pyroelectric response and criticality of polarization Compensation of random fields by charges on head-tohead and tail-to-tail domains critical exponent β 0.30 0.25 0.20 0.15 0.10 0 5 10 15 20 25 polarization P (20 C) (µc/cm 2 ) Crossover Analogy Dilute uniaxial antiferromagnets in a uniform magnetic field (DAFF Fishman, Aharony 1979) β =0.30±0.02 β = 0.14±0.02 (Fe 0.88 Zn 0.12 F 2 Belanger et al. 2002)
Specific heat anomaly of SBN61 ac calorimetry Z. Kutnjak et al. (submitted) non-ergodicity upon FC and upon ZFH/ZFC
Specific heat anomaly of SBN61 Z. Kutnjak, WK, et al., submitted C p = t -α± + Bt + C T= T/T c -1 Experiment: α ± = - 0.02 Simulations: α = - 0.01 DAFF systems: α 0 Theory: α = -1.0... 0.5
Scaling relations of the critical exponents α = -0.02 ± 0.02 (c p ) β = 0.14 ± 0.02 (NMR, LB, SHG, pyroelectric current) γ = 1.78 ± 0.05 (susceptibility, linear birefringence) Rushbrooke relation α+2β+γ = 2.04±0.05 Critical isotherm: δ =1+γ/β = 13.7 ± 0.05? how to overcome hysteresis? f=4x10-4 Hz T c T. Granzow (PhD Thesis 2003
Conclusion Ce dynamics of polar nanoregions and critical behavior qualify the uniaxial relaxor SBN Sr 0.61-x Ba 0.39 Nb 2 O 6 :(RE,TM) x to belong to the 3D Random Field Ising Model universality class (first ever realized ferroic RFIM!) open questions dynamics in the Terahertz region? (birth of the polar nanoregions @ soft mode frequencies) critical behavior @ T c? (critical exponent δ )