Dipole glass

A dipole glass is an analog of a glass where the dipoles are frozen below a given freezing temperature T_f introducing randomness thus resulting in a lack of long-range ferroelectric order.<ref name=":0">Vugmeister, B. E.; Glinchuk, M. D. (1990-10-01). "Dipole glass and ferroelectricity in random-site electric dipole systems". Reviews of Modern Physics. 62 (4): 993–1026. Bibcode:1990RvMP...62..993V. doi:10.1103/RevModPhys.62.993.</ref> A dipole glass is very similar to the concept of a spin glass where the atomic spins don't all align in the same direction (like in a ferromagnetic material) and thus result in a net-zero magnetization. The randomness of dipoles in a dipole glass creates local fields resulting in short-range order but no long-range order.

The dipole glass like state was first observed in Alkali halide crystal-type dielectrics containing dipole impurities.<ref>Känzig, W.; Hart, H. R.; Roberts, S. (1964-11-02). "Paraelectricity and Ferroelectricity Due to Hydroxyl Ions in Alkali Halides; Paraelectric Cooling". Physical Review Letters. 13 (18): 543–545. Bibcode:1964PhRvL..13..543K. doi:10.1103/PhysRevLett.13.543.</ref><ref>Peressini, P. P.; Harrison, J. P.; Pohl, R. O. (1969-04-15). "Thermal Conductivity of KCl: Li. Isotope and Electric Field Effects". Physical Review. 180 (3): 926–930. Bibcode:1969PhRv..180..926P. doi:10.1103/PhysRev.180.926.</ref><ref>Fiory, A. T. (1971-07-15). "Electric Dipole Interactions Among Polar Defects in Alkali Halides". Physical Review B. 4 (2): 614–627. Bibcode:1971PhRvB...4..614F. doi:10.1103/PhysRevB.4.614.</ref> The dipole impurities in these materials result in off-center ions which results in anomalies in certain properties like specific heat, thermal conductivity as well as some spectroscopic properties.<ref name=":0" /> Other materials which show a dipolar glass phase include Rb_(1-x)(NH₄)_xH₂PO₄ (RADP) and Rb_(1-x)(ND₄)_xD₂PO₄ (DRADP).<ref>Kim, Bog-Gi; Kim, Jong-Jean; Jang, Hyun M. (1999-09-01). "Correlated domain model of deuterated dipole glass". Physical Review B. 60 (10): 7170–7177. Bibcode:1999PhRvB..60.7170K. doi:10.1103/PhysRevB.60.7170.</ref><ref>Yuzyuk, Yu I; Gregora, I; Vorlicek, V; Pokorny, J; Petzelt, J (1995-01-16). "Raman spectra of DRADP-50 dipolar glass". Journal of Physics: Condensed Matter. 7 (3): 683–695. Bibcode:1995JPCM....7..683Y. doi:10.1088/0953-8984/7/3/022. ISSN 0953-8984. S2CID 250808735.</ref> In materials like DRADP the dipole moment is introduced due to the deuteron O-D--O bond. Dipole glass like behavior is also observed in materials like ceramics,<ref>He, Ju; Lu, Xiaomei; Ti, Ruixia; Zhu, Weili; Huang, Fengzhen; Zhou, Min; Jin, Yaming; Zhu, Jinsong (2017-07-01). "Dipole glass behavior of Fe-doped SrTiO3 ceramics". Journal of Materials Science: Materials in Electronics. 28 (14): 10700–10706. doi:10.1007/s10854-017-6845-2. ISSN 1573-482X. S2CID 136008779.</ref> 3D water framework<ref>Huang, Rui-Kang; Wang, Sha-Sha; Liu, De-Xuan; Li, Xin; Song, Jian-Ming; Xia, Yuan-Hua; Zhou, Dong-Dong; Huang, Jin; Zhang, Wei-Xiong; Chen, Xiao-Ming (2019-04-10). "Supercooling Behavior and Dipole-Glass-like Relaxation in a Three-Dimensional Water Framework". Journal of the American Chemical Society. 141 (14): 5645–5649. doi:10.1021/jacs.9b01866. ISSN 0002-7863. PMID 30908017. S2CID 85516311.</ref> and perovskites.<ref>Svirskas, Šarūnas; Adamchuk, Dzmitry; Grigalaitis, Robertas; Jablonskas, Džiugas; Macutkevič, Jan; Canu, Giovanna; Buscaglia, Maria Teresa; Buscaglia, Vincenzo; Curecheriu, Lavinia; Mitoseriu, Liliana; Banys, Jūras (2021-02-15). "Dipolar glass state in BaCe0.3Ti0.7O3 perovskite solid solutions". Journal of Alloys and Compounds. 854: 155755. doi:10.1016/j.jallcom.2020.155755. ISSN 0925-8388. S2CID 219916164.</ref><ref>Maiti, Tanmoy; Saxena, Mandvi; Roy, Pinku (2019-01-01). "Double perovskite (Sr2B′B"O6) oxides for high-temperature thermoelectric power generation—A review". Journal of Materials Research. 34 (1): 107–125. doi:10.1557/jmr.2018.376. ISSN 2044-5326. S2CID 139136260.</ref>

Random-bond-random-field Ising model (RBRF)

The model describing the pseudo-spins (dipole moments) is given by the Hamiltonian as:

<math>\mathcal{H}=-\frac{1}{2}\sum_{ij}{J_{ij}}{S_i}^{z}{S_j}^{z}-\sum_{i}{f_i}{S_i}^{z}-E\sum_{i}{S_i}^{z}</math>,

where <math>{S_i}^{z}</math> is the Ising dipole moments. The <math>{J_{ij}}</math> refers to the random bond interactions which are described by a gaussian probability distribution with mean <math>{J_{0}}</math> and variance <math>{J^{2}/N}</math>. The second term provides a description of the interactions of the pseudo-spins in presence of random local fields where <math>{f_{i}}</math> are represented by an independent gaussian distribution with zero mean and variance <math>\Delta</math>. The final term denotes the interaction in presence of an external electric field <math>{E}</math>.

The replica method is used to obtain the glass order parameter:

<math>q=\int Dz{\tanh}^{2}[J(q+\frac{\Delta}{J^{2}})^{1/2}\frac{z}{T}]</math>.

where <math>\int Dz</math> is the gaussian measure and under the assumption that <math>{E}=0</math> the free energy is given by:<ref name=":1">Blinc, Robert (2011-08-25). Advanced Ferroelectricity. Oxford University Press. doi:10.1093/acprof:oso/9780199570942.001.0001. ISBN 978-0-19-957094-2.</ref><ref name=":2">Kleemann, Wolfgang (September 2002). "Random fields in dipolar glasses and relaxors". Journal of Non-Crystalline Solids. 307–310: 66–72. Bibcode:2002JNCS..307...66K. doi:10.1016/S0022-3093(02)01441-2.</ref>

<math>\beta{f}=-(\beta{J}/2)^{2}[(1-q_{1})^{2}-m(q_{1}^{2}-q_{0}^{2})]-m^{-1}\int Dz \ln\int Dy Z^{m}(y,z)</math>.

where <math>\beta = 1/T</math> and <math>Z^{m}(y,z) = 2\cosh[\beta{h(y,z)}]</math> with <math>h(x,y)= J[(q_{1}-q_{0})^{1/2}y+(q_{0}+\Delta/J^{2})^{1/2}z]</math>.

The <math>{f_{i}}</math> term is zero in case of magnetic spin glasses and with no presence of an external electric field this model reduces to the Edwards–Anderson model which is used to describe spin glasses. This model has been used to give quantitative description of DRADP type systems.<ref name=":1" /><ref name=":2" />