阻挫性材料是凝態物理中目前很受關注的課題，在低溫下會因阻挫性引起一系列有趣的磁特性。自旋冰(spin ice) 材料為一高阻挫性材料。自旋冰材料的磁矩排列就像氫離子與氧離子在一般冰塊中的排列。目前為止實驗上仍沒有找到自旋冰系統長而有序(long range order) 的排列。自旋冰二鈦酸鏑(Dy2Ti2O7，簡稱DTO) 和二鈦酸鈥(Ho2Ti2O7，簡稱HTO) 的結構是燒綠石，在晶格內磁性稀土離子是以頂點相接的正四面體單元。由於晶體場和幾何結構所造成的異向性(anisotropy)，在低溫時是由冰法則(ice rule) 的微觀太限制。此研究探討DTO 和 HTO 的混合化合物HoDyTi2O7 (簡稱，HDTO) 的低溫磁性。本文將呈現HDTO 在直流磁化率、交流磁化率、比熱和中子散射實驗上的結果。藉由直流磁化率得
到單晶及粉末樣本的飽和磁矩各別為11.5 uB 和11.6 uB 大於原預估的10 uB。交流磁化率則發現在15 K 時虛部有能量的吸收，與鏑離子在晶體場下的能階有關。比熱量測從45 K 至50 mK 發現在1 K 及0.6 K 有峰值。1 K 的峰對應冰法則局限之磁矩自凍結態‘融化’至動態所釋放的熵，0.6K 的峰是目前未知來源的新現象。低溫中子繞射數據顯示在低散射向量有漫散射的訊號，是典型的短程關聯的象徵，為一般自旋冰在中子繞射下會看到的訊號。在非彈性中子散射實驗，低溫低能量下發現有兩個非彈性散射的訊號，它們各別為E = 0.26 meV 和E = 0.45meV。我們推測這兩個能量可能各別是鏑和鈥離子基態雙重狀態的分裂。這兩個訊號在原HTO 和DTO 下並沒有發生。先前對HTO 和DTO 參雜的實驗結果都顯示自旋冰態並非容易被破壞，且樣品保留原物理特性。在此研究我們發現當這兩個具代表性的自旋冰材料合成為一之後觀察到明顯的改變，尤其是較大的飽和磁矩，低溫比熱發現新的相以及非彈性散射觀察到的雙重基態能階的分裂。

Frustrated spin systems give rise to a host of interesting magnetic property. Among them lies the spin ice compound whereby spins arrange in a manner similar to ice-rule configuration found in water ice. In this thesis we present a study on the mixture compound of two canonical spin ice candidates, HoDyTi2O7 (HDTO). The parent compounds Ho2Ti2O7 (HTO) and Dy2Ti2O7 (DTO) are similar in so many ways such as having large moment on the magnetic sites Ho and Dy respectively, and due to crystal electric field properties, the spins are essentially Ising. However a few publications have highlighted their subtle differences, which include the slightly different spin freezing temperature and dynamic properties. In this thesis, the proposed HDTO is grown in both polycrystalline and single crystal form and measured for various low temperature physical properties and compared against their parent counterpart. We present results from magnetization, heat capacity measurements on polycrystalline samples and heat capacity measurements on single crystal samples respectively as well as neutron scattering experiments.
Temperature dependent magnetization reveals no transition down to 2 K. High temperature paramagnetic phase was fitted using Curie-Weiss law and an effective moment of 11.5 uB. High field magnetization yield a saturation moment of 11.2 uB for powder and 11.6 uB for single crystal along [111] direction. The higher saturation moment compared to parent compound points to a softening in the Ising anisotropy constraint or slight tilt in spin easy axis. AC susceptibility at high field present a frequency dependent peak related to single ion property of Dy. Fitting with Arrhenius law obtained a energy barrier 183 k < EB < 231 K. Specific heat measurement on single crystal samples down to 50 mK showed a spin freezing peak at 1.2 K as well as a novel peak at 0.6 K. The novel peak may point towards a new ordering in our system, however its origin is currently unknown.
Under applied field along [111] direction both peaks experience a shift which suggest that it is magnetic. Aside from above mentioned peaks, a field induced peak was also seen at 0.4 K, which may be related to the transition between kagome ice state and fully saturated state. Interestingly, nuclear hyperfine signal measured in our system is much smaller than typical Ho containing compound. Powdered diffraction measurement yield a diffuse scattering profile at the low Q region at temperature below 5 K, similar to other dipolar spin ice materials. Inelastic neutron scattering measurement on HDTO powder showed two inelastic peaks at low temperature. The energy transfer are at 0.26 meV and 0.5 meV respectively. These inelastic signal hints at a splitting of ground state doublet of Holmium ion and Dysprosium ion respectively.
More work to study the microscopic behavior in our system need to be carried out to explain the changes brought in by mixing two spin ice compounds.

[1] Claudine Lacroix, Philippe Mendels, and Frédéric Mila. Introduction to Frustrated
Magnetism: Materials, Experiments, Theory. Vol. 164. Springer Science
& Business Media, 2011.
[2] Samuel Frederick Edwards and Phil W Anderson. “Theory of spin glasses”.
Journal of Physics F: Metal Physics 5.5 (1975), p. 965.
[3] GH Wannier. “Antiferromagnetism. the triangular ising net”. Physical Review
79.2 (1950), p. 357.
[4] PHILIP W Anderson. “Ordering and antiferromagnetism in ferrites”. Physical
Review 102.4 (1956), p. 1008.
[5] Jacques Villain. “Insulating spin glasses”. Zeitschrift für Physik B Condensed
Matter 33.1 (1979), pp. 31–42.
[6] Jason S Gardner, Michel JP Gingras, and John E Greedan. “Magnetic pyrochlore
oxides”. Reviews of modern Physics 82.1 (2010), p. 53.
[7] MJ Harris et al. “Liquid-gas critical behavior in a frustrated pyrochlore ferromagnet”.
Physical review letters 81.20 (1998), p. 4496.
[8] J. D. Bernal and R. H. Fowler. “A Theory of Water and Ionic Solution, with
Particular Reference to Hydrogen and Hydroxyl Ions”. Journal of Chemical
Physics 1.8 (1933), p. 515.
[9] Water ice arrangement. url: https://en.wikipedia.org/wiki/Spin_ice.
[10] Linus Pauling. “The structure and entropy of ice and of other crystals with
some randomness of atomic arrangement”. Journal of the American Chemical
Society 57.12 (1935), pp. 2680–2684.
[11] MJ Harris et al. “Geometrical frustration in the ferromagnetic pyrochlore Ho2
Ti2 O7”. Physical Review Letters 79.13 (1997), p. 2554.
[12] Ludovic DC Jaubert. “Topological constraints and defects in spin ice”. PhD
thesis. Ecole normale supérieure de lyon-ENS LYON, 2009.
[13] Arthur P Ramirez et al. “Zero-point entropy in‘spin ice’”. Nature 399.6734
(1999), pp. 333–335.
[14] ST Bramwell et al. “Spin correlations in Ho2Ti2O7: a dipolar spin ice system”.
Physical Review Letters 87.4 (2001), p. 047205.
[15] D Pomaranski et al. “Absence of Pauling/’s residual entropy in thermally
equilibrated Dy2Ti2O7”. Nature Physics 9.6 (2013), pp. 353–356.
[16] H Fukazawa et al. “Magnetic anisotropy of the spin-ice compound Dy2Ti2O7”.
Physical Review B 65.5 (2002), p. 054410.
[17] Byron C den Hertog and Michel JP Gingras. “Dipolar interactions and origin
of spin ice in Ising pyrochlore magnets”. Physical review letters 84.15 (2000),
p. 3430.
[18] T Sakakibara et al. “Observation of a Liquid-Gas-Type Transition in the Pyrochlore
Spin Ice Compound Dy2Ti2O7 in a Magnetic Field”. Physical review
letters 90.20 (2003), p. 207205.
[19] Y Tabata et al. “Kagome ice state in the dipolar spin ice Dy2Ti2O7”. Physical
review letters 97.25 (2006), p. 257205.
[20] Zenji Hiroi et al. “Specific heat of kagome ice in the pyrochlore oxide Dy2Ti2O7”.
Journal of the Physical Society of Japan 72.2 (2003), pp. 411–418.
[21] Ludovic DC Jaubert et al. “Three-dimensional Kasteleyn transition: spin ice
in a [100] field”. Physical review letters 100.6 (2008), p. 067207.
[22] Zenji Hiroi, Kazuyuki Matsuhira, and Masao Ogata. “Ferromagnetic Ising spin
chains emerging from the spin ice under magnetic field”. Journal of the Physical
Society of Japan 72.12 (2003), pp. 3045–3048.
[23] Shun-ichi Yoshida, Koji Nemoto, and Koh Wada. “Ordered phase of dipolar
spin ice under [110] magnetic field”. Journal of the Physical Society of Japan
73.7 (2004), pp. 1619–1622.
[24] Claudio Castelnovo, Roderich Moessner, and Shivaji L Sondhi. “Magnetic
monopoles in spin ice”. Nature 451.7174 (2008), pp. 42–45.
[25] Hiroaki Kadowaki et al. “Neutron scattering study of dipolar spin ice Ho2Sn2O7:
frustrated pyrochlore magnet”. Physical Review B 65.14 (2002), p. 144421.
[26] S Rosenkranz et al. “Crystal-field interaction in the pyrochlore magnet Ho2Ti2O7”.
Journal of Applied Physics 87.9 (2000), pp. 5914–5916.
[27] MJ Harris et al. “Magnetic structures of highly frustrated pyrochlores”. Journal
of magnetism and magnetic materials 177 (1998), pp. 757–762.
[28] LJ Chang et al. “Low-temperature muon spin rotation studies of the monopole
charges and currents in Y doped Ho2Ti2O7”. Scientific reports 3 (2013).
[29] A Bertin et al. “Crystal electric field in the R2Ti2O7 pyrochlore compounds”.
Journal of Physics: Condensed Matter 24.25 (2012), p. 256003.
[30] Charles Kittel. Introduction to solid state physics. Wiley, 2005.
[31] Patrik Henelius et al. “Refrustration and competing orders in the prototypical
Dy2Ti2O7 spin ice material”. Physical Review B 93.2 (2016), p. 024402.
[32] Ludovic DC Jaubert and Peter CW Holdsworth. “Signature of magnetic monopole
and Dirac string dynamics in spin ice”. Nature Physics 5.4 (2009), pp. 258–
261.
[33] ST Bramwell et al. “Measurement of the charge and current of magnetic
monopoles in spin ice”. Nature 461.7266 (2009), pp. 956–959.
[34] Y Tajima, T Matsuo, and H Suga. “Phase transition in KOH-doped hexagonal
ice” (1982).
[35] LJ Chang et al. “Magnetic correlations in HoxTb2-xTi2O7”. Physical Review
B 83.14 (2011), p. 144413.
[36] LJ Chang et al. “Magnetic correlations in the spin ice Ho2-xYxTi2O7 as revealed
by neutron polarization analysis”. Physical Review B 82.17 (2010), p. 172403.
[37] S Scharffe et al. “Suppression of Pauling’s residual entropy in the dilute spin
ice (Dy1-xYx)2Ti2O7”. Physical Review B 92.18 (2015), p. 180405.
[38] G Balakrishnan et al. “Single crystal growth of rare earth titanate pyrochlores”.
Journal of Physics: Condensed Matter 10.44 (1998), p. L723.
[39] Varley F Sears. “Neutron scattering lengths and cross sections”. Neutron news
3.3 (1992), pp. 26–37.
[40] Heinz Maier-Leibnitz Zentrum.
[41] Juan Rodrı́guez-Carvajal. “Recent advances in magnetic structure determination
by neutron powder diffraction”. Physica B: Condensed Matter 192.1
(1993), pp. 55–69.
[42] Thomas Brückel et al. Neutron scattering lectures. Forschungszentrum Jülich
GmbH, 2015.
[43] Andrew J Studer, Mark E Hagen, and Terrence J Noakes. “Wombat: The highintensity
powder diffractometer at the OPAL reactor”. Physica B: Condensed
Matter 385 (2006), pp. 1013–1015.
[44] J Rodriguez-Carvajal. “FULLPROF: a program for Rietveld refinement and
pattern matching analysis”. satellite meeting on powder diffraction of the XV
congress of the IUCr. Vol. 127. Toulouse, France:[sn]. 1990.
[45] Gen Shirane, Stephen M Shapiro, and John M Tranquada. Neutron scattering
with a triple-axis spectrometer: basic techniques. Cambridge University Press,
2002.
[46] Dehong Yu et al. “Pelican—a Time of Flight Cold Neutron Polarization Analysis
Spectrometer at OPAL”. Journal of the Physical Society of Japan 82.Suppl.
A (2013), SA027.
[47] D Richard, M Ferrand, and GJ Kearley. “Lamp, the large array manipulation
program”. J. Neutron Research 4 (1996), pp. 33–39.
[48] Saulius Gražulis et al. “Crystallography Open Database–an open-access collection
of crystal structures”. Journal of Applied Crystallography 42.4 (2009),
pp. 726–729.
[49] KE Sickafus et al. “Radiation tolerance of complex oxides”. Science 289.5480
(2000), pp. 748–751.
[50] ST Bramwell et al. “Bulk magnetization of the heavy rare earth titanate
pyrochlores-a series of model frustrated magnets”. Journal of Physics: Condensed
Matter 12.4 (2000), p. 483.
[51] J Snyder et al. “Low-temperature spin freezing in the Dy2Ti2O7 spin ice”.
Physical Review B 69.6 (2004), p. 064414.
[52] Hui Xing et al. “Emergent order in the spin-frustrated system DyxTb2-xTi2O7
studied by ac susceptibility measurements”. Physical Review B 81.13 (2010),
p. 134426.
[53] J Snyder et al. “How ‘spin ice’freezes”. Nature 413.6851 (2001), pp. 48–51.
[54] J Snyder et al. “Quantum-Classical Reentrant Relaxation Crossover in Dy2Ti2O7
Spin Ice”. Physical review letters 91.10 (2003), p. 107201.
[55] K Matsuhira, Y Hinatsu, and T Sakakibara. “Novel dynamical magnetic properties
in the spin ice compound Dy2Ti2O7”. Journal of Physics: Condensed
Matter 13.31 (2001), p. L737.
[56] G Ehlers et al. “Evidence for two distinct spin relaxation mechanisms in‘hot’
spin ice Ho2Ti2O7”. Journal of Physics: Condensed Matter 16.11 (2004), S635.
[57] Kazuyuki Matsuhira et al. “A new macroscopically degenerate ground state in
the spin ice compound Dy2Ti2O7 under a magnetic field”. Journal of Physics:
Condensed Matter 14.29 (2002), p. L559.
[58] Georg Ehlers et al. “Direct observation of a nuclear spin excitation in Ho2Ti2O7”.
Physical review letters 102.1 (2009), p. 016405.
[59] OV Lounasmaa. “Specific heat of holmium metal between 0.38 and 4.2 K”.
Physical Review 128.3 (1962), p. 1136.
[60] HD Zhou et al. “Dynamic spin ice: Pr2Sn2O7”. Physical review letters 101.22
(2008), p. 227204.
[61] LJ Chang. “Unpublished” ().
[62] HaM Rietveld. “A profile refinement method for nuclear and magnetic structures”.
Journal of applied Crystallography 2.2 (1969), pp. 65–71.