Quantum versus classical nature of the low-temperature magnetic phase transition in ${\mathrm{TbAl}}_{3}{({\mathrm{BO}}_{3})}_{4}$

Quantum versus classical nature of the low-temperature magnetic phase transition in ${TbAl}_{3} {({BO}_{3})}_{4}$

T. Zajarniuk, A. Szewczyk, P. Wiśniewski, M. U. Gutowska, R. Puzniak, H. Szymczak, I. Gudim, V. A. Bedarev, M. I. Pashchenko, P. Tomczak, and W. Szuszkiewicz

Phys. Rev. B 105, 094418 – Published 14 March 2022

Abstract

Specific heat $C_{B}$ of a ${TbAl}_{3} {({BO}_{3})}_{4}$ crystal was studied for 50 mK $< T <$ 300 K with emphasis on $T < 1$ K where a phase transition was found at $T_{c} = 0.68$ K. Nuclear, nonphonon ( $C_{m}$ ), and lattice contributions to $C_{B}$ were separated. Lowering of $T_{c}$ with an increase in magnetic field parallel to the easy magnetization axis $(B_{∥})$ was found. It was established that $C_{m}$ and a Grüneisen ratio depend on $B_{∥}$ and $T$ in a way characteristic of systems in which a classical transition is driven by quantum fluctuations (QFs) to a quantum critical point at $T = 0$ by tuning a control parameter ( $B_{∥}$ ). The $B_{∥} - T$ phase diagram was constructed, and the dynamical critical exponent $0.82 \leq z \leq 0.96$ was assessed. Nature of the transition was not established explicitly. Magnetization studies point at the ferromagnetic ordering of ${Tb}^{3 +}$ magnetic moments, however, lowering of $T_{c}$ with increase in $B_{∥}$ is opposite to the classical behavior. Hence, a dominant role of QFs was supposed.

Received 23 November 2021
Revised 18 February 2022
Accepted 23 February 2022

DOI:https://doi.org/10.1103/PhysRevB.105.094418

Physics Subject Headings (PhySH)

Ferromagnetism Magnetic order Magnetic phase transitions Magnetism Phase transitions Quantum criticality Quantum phase transitions Specific heat

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

T. Zajarniuk ¹, A. Szewczyk ^1,*, P. Wiśniewski ², M. U. Gutowska ¹, R. Puzniak ¹, H. Szymczak ¹, I. Gudim ³, V. A. Bedarev ⁴, M. I. Pashchenko ^4,5, P. Tomczak ⁶, and W. Szuszkiewicz ¹

¹Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, PL-02668 Warsaw, Poland
²Institute of Low Temperature and Structure Research, Polish Academy of Sciences, ulica Okólna 2, PL-50422 Wrocław, Poland
³Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Krasnoyarsk 660036, Russia
⁴B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Nauky Avenue, UA-61103 Kharkiv, Ukraine
⁵Institute of Physics, Czech Academy of Sciences, Cukrovarnická 10, 162 00 Praha 6, Czech Republic
⁶Faculty of Physics, Adam Mickiewicz University, Uniwersytetu Poznańskiego 2, PL-61614 Poznań, Poland

^*szewc@ifpan.edu.pl

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Vol. 105, Iss. 9 — 1 March 2022

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Figure 1
Specific heat of the ${TbAl}_{3} {({BO}_{3})}_{4}$ single crystal. (a) Trigonal structure of ${TbAl}_{3} {({BO}_{3})}_{4}$ with lattice parameters $a$ = 9.2926(9) and $c$ = 7.2516(4) Å. Magnetic ${Tb}^{3 +}$ ions are located inside the deformed (blue) trigonal prisms, located along the threefold $c$ axis, and formed by six $O^{2 -}$ ions. (b) Total specific heat $C_{B}$ measured in $B = 0$ with a standard calorimeter (STD), vertical puck (VP), and dilution refrigerator (DR). The red solid line presents the estimated $C_{l}$ . In the inset, $C_{1}$ and $y_{0}$ coefficients of the curves defined by Eq. (5) and fitted to the experimental data shown in the panel (e) are presented as a function of $B_{∥}$ . (c) Nonphonon contribution $C_{m}$ to $C_{B}$ for several $B_{∥}$ values. (d) Experimental $C_{m} (T, B = 0)$ function (circles), estimated nuclear specific heat $C_{N}$ (blue dashed line), and fit of the $C_{N} (T) + C_{1} T^{y_{0}}$ function to $C_{m} (T, B = 0)$ (red solid line). Near 330 mK a sinusoidal apparatus effect is visible. (e) Determined $C_{mc} (T)$ (i.e., $C_{m} (T)$ corrected for the apparatus effect for $B_{∥} \leq 0.475 T$ and $C_{m} (T)$ for $B_{∥} > 0.475 T$ ), symbols, and $C_{m} (T) = C_{N} (T) + C_{1} T^{y_{0}}$ functions fitted to these curves (red solid lines). To maintain readability, the curves for different $B_{∥}$ values are shifted along the vertical axis by the values given in parentheses. (f) $B_{∥} - T$ phase diagram found based on $C_{m} (T, B_{∥})$ functions (symbols). The inset presents the schematic phase diagram in which a line of classical phase transition ends at QCP.
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Figure 2
Magnetization of the ${TbAl}_{3} {({BO}_{3})}_{4}$ single crystal (not corrected for demagnetizing effects). Main panel, $M$ versus $B_{∥}$ and $B_{⊥}$ at $T = 0.5$ K. The inset, $M / B$ versus $T$ for $B_{∥} =$ 0.01, 0.1, 0.2, 0.25, and 0.35 T. The arrow indicates evolution of the transition temperature with an increase in $B_{∥}$ .
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Figure 3
The Grüneisen ratio $Γ$ . (a) $Γ$ values as a function of $B_{∥}$ , symbols, and the theoretical $g | B - B_{c} |^{- s}$ functions fitted to them, solid red lines, for several $T$ values. Curves for different $T s$ are shifted along the $Γ$ axis by the values given in the legend in parentheses. The insets, $g$ and $s$ parameters as a function of $T$ . (b) $Γ (T)$ for several $B_{∥} s$ , symbols. The blue solid line is the $Γ (T, B_{∥} = 0.2 T)$ function calculated based on the $C_{mc} (T)$ curves. (c) Scaling behavior. $Γ {({\tilde{B}}_{c} - B_{∥})}^{- σ}$ as a function of $T / {({\tilde{B}}_{c} - B_{∥})}^{ε}$ for $ε = 0.48 (4), σ = 0$ , and ${\tilde{B}}_{c} = 0.57 (3)$ T for several $B_{∥} s$ . The black solid line is the approximation of the scaling function: $Φ (x) = 3.2 + 2.4 tanh [19 (x - 1.161)]$ . The curves for $B_{∥} \geq 0.3$ T overlap for the arguments 1.132–1.221.
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Figure 4
$C_{m} (T)$ functions for several fixed $B_{∥}$ (dashed curves) and applied intentionally perpendicularly to the $c$ axis $B_{⊥}$ (solid lines) values. In the legend, the expected sample tilting angles (calculated under assumption that only the $B_{∥}$ component influences the transition point) as well as corresponding to them $B_{∥}$ components are given.
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Physical Review B

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Quantum versus classical nature of the low-temperature magnetic phase transition in ${TbAl}_{3} {({BO}_{3})}_{4}$

T. Zajarniuk, A. Szewczyk, P. Wiśniewski, M. U. Gutowska, R. Puzniak, H. Szymczak, I. Gudim, V. A. Bedarev, M. I. Pashchenko, P. Tomczak, and W. Szuszkiewicz

Phys. Rev. B 105, 094418 – Published 14 March 2022

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Quantum versus classical nature of the low-temperature magnetic phase transition in TbAl3(BO3)4

T. Zajarniuk, A. Szewczyk, P. Wiśniewski, M. U. Gutowska, R. Puzniak, H. Szymczak, I. Gudim, V. A. Bedarev, M. I. Pashchenko, P. Tomczak, and W. Szuszkiewicz

Phys. Rev. B 105, 094418 – Published 14 March 2022

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Quantum versus classical nature of the low-temperature magnetic phase transition in ${TbAl}_{3} {({BO}_{3})}_{4}$