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[T Naka](https://orcid.org/0000-0002-0645-6952), J Valenta, T Nakane, S Ishii, M Nakayama, [H Mamiya](https://orcid.org/0000-0002-7840-3008), K Takehana, [N Tsujii](https://orcid.org/0000-0002-6181-5911), Y Imanaka, [Y Matsushita](https://orcid.org/0000-0002-4968-8905), H Abe, T Uchikoshi, H Yusa

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This is an author-created, un-copyedited version of an article accepted for publication/published in JOURNAL OF PHYSICS-CONDENSED MATTER. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-648X/ad12fc.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Phase transitions and slow spin dynamics of slightly inverted A-site spinel CoAl2−xGaxO4](https://mdr.nims.go.jp/datasets/0259e89c-46d9-484a-88a2-31bc7220c9d1)

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1    Phase transitions and slow spin dynamics of slightly inverted A-site spinel CoAl2−xGaxO4   T. Naka1, *, J. Valenta,1 T. Nakane,2 S. Ishii,3 M. Nakayama,1 H. Mamiya,4 K. Takehana,5  N. Tsujii,1 Y. Imanaka,5 Y. Matsushita,6 H. Abe,7 T. Uchikoshi,2 and H. Yusa1  1Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-2-1, Sengen, Tsukuba, Ibaraki 305-0047, Japan 2Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-2-1, Sengen, Tsukuba, Ibaraki 305-0047, Japan 3Department of Physics, Tokyo Denki University, Hatoyama, Saitama 350-0394, Japan 4Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science, 1-2-1, Sengen, Tsukuba, Ibaraki 305-0047, Japan 5Center for Basic Research on Materials, National Institute for Materials Science, 1-2-1, Sengen, Tsukuba, Ibaraki 305-0047, Japan 6Research Network and Facility Services Divisions, National Institute for Materials Science, 1-2-1, Sengen, Tsukuba, Ibaraki 305-0047, Japan  7Joint and Welding Research Institution, Osaka University, 11-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan   We report the properties of an A-site spinel magnet, CoAl2−xGaxO4, and analyze its anomalous, low-temperature magnetic behavior, which is derived from inherent, magnetically frustrated interactions. Rietveld analysis of the x-ray diffraction profile for CoAl2−xGaxO4 revealed that the metallic ions were randomly distributed in the tetrahedral (A-) and octahedral (B-) sites in the cubic spinel structure. The inversion parameter  could be controlled by varying the gallium (Ga) composition in the range 0.055 ≤  ≤ 0.664. The composition-induced Néel-to-spin-glass (NSG) transition occurred between 0.05 ≤  ≤ 0.08 and was verified by measurements of DC-AC susceptibilities  and thermoremanent magnetization (TRM) below the Néel transition temperature TN. The relaxation rate and derivative with respect to temperature of TRM increased at both TN and the spin glass (SG) transition temperature TSG.  The TRM decayed rapidly above and below these transitions. TRM was highly sensitive to macroscopic magnetic transitions that occurred in both the Néel and SG phases of CoAl2−xGaxO4. In the vicinity of the NSG boundary, there was a maximum of the TRM relaxation rate at Tmax < TN. With increasing inversion (x), the anomaly at Tmax merged with that of the Néel transition at a tricritical point (tc, Ttc) = (0.08, 4.0 K), where the paramagnetic, Néel, and SG states met. We successfully extracted the relaxation time  and other characteristic 2  parameters from the TRM isothermal temporal evolution based on the Weron function derived for a purely stochastic process. To distinguish the magnetic states, we compared our results with previously studied inversion-free A-site spinel, CoRh2O4, and CoGa2O4 cluster glass. We generated an inversion-temperature phase diagram based on the comprehensive measurements of DC and AC susceptibilities, TRM, and specific heat in the range 0.055    0.664 for CoAl2−xGaxO4. Based on this phase diagram, we speculate that a NSG quantum critical phase transition occurred at  = 0.050(6). Our findings are consistent with suppression of the long-range order antiferromagnetic state in CoAl2O4 revealed through neutron diffraction studies, even at T << TN.  *naka.takashi@nims.go.jp  3  I. INTRODUCTION Nanoscale objects (nano-objects) embedded in matrix materials induce remarkable thermomagnetic phenomena, such as exchange bias and thermoremanent magnetization (TRM) [1]. Magnetization of nano-objects in such materials make them behave as complex systems that depend on the type of experimental pretreatment (i.e., magnetic, thermal, and temporal histories) [2]. These magnetized nano-objects (magnetic clusters) create disorders in the matrix material that often qualitatively alter its ground state. A recent study has revealed that such magnetic clusters due to disorders (anti-site defects) in the magnetically frustrated A-site spinel antiferromagnet can act as highly sensitive probes for magnetic phase transitions [3]. Such disorders and magnetic interactions in A-site spinel magnets induce specific relaxation phenomena in both antiferromagnetic (AF) and spin glass (SG) states. From the temperature and time dependencies of TRM, important information can be extracted about magnetic phase transitions and spin dynamics in A-site spinel magnets with formula unit compositions of CoM2O4, where M = Al, Rh, and Ga correspond to strongly frustrated antiferromagnets, weakly frustrated antiferromagnets, and cluster glasses (CGs), respectively [3]. The derivative of the TRM intensity, MTR, with respect to temperature (1/MTR)(dMTR/dT) and the relaxation rate MTR(t)/MTR(0) are highly sensitive to magnetic transitions, irrespective of the magnetic state and the degree of magnetic frustration. In this paper, we demonstrate the application of TRM measurements to study magnetic disorder effects in gallium-substituted cobalt aluminate (CoAl2−xGaxO4), and we compare them with those of the disorder-free, A-site spinel antiferromagnet cobalt rhodate (CoRh2O4) [4] and a cobalt gallate CG (CoGa2O4) [5]. In the spinel oxide CoAl2O4, divalent Co2+ and trivalent Al3+ ions occupy the tetrahedral (A-) and octahedral (B-) sites, respectively. The A-site forms a magnetically bipartite diamond lattice consisting of two interpenetrated face-centered cubic (FCC) lattices that are shifted along the [111] direction. A nearest-neighbor interaction J1 is hence realized between Co2+-spins located at the A-sites of different FCC lattices. Simultaneously, a next-nearest-neighbor interaction J2 is established between Co2+ spins located at the A-sites within the same FCC lattice. Note that, therefore, a diamond lattice is bipartite and the A-site antiferromagnet exhibits a Néel-type antiferromagnetic spin structure when J1 >> J2. Contrary, a magnetic frustration is anticipated because of the competition of J1 and J2 interactions acting between Co2+ spins at the A-sites. Bergman et al. [6] have used a Monte Carlo simulation projected on the T-J2/J1 plane to derive a magnetic phase diagram for 4  the A-site spinel antiferromagnet: As the J2/J1 ratio increases, the Néel point (TN) decreases drastically, and a spin spiral (SS) state emerges above J2/J1 = 1/8. The paramagnetic-Néel (PN) and paramagnetic-SS (PSS) phase boundaries meet at the Lifshitz point (LP), and the SS phase penetrates below the Néel phase as a Néel-SS (NSS) boundary, which terminates at the quantum critical point (QCP) of J2/J1 = 1/8. Plausibly, the A-site spinel antiferromagnet CoAl2O4 realizes a Néel state in the vicinity of J2/J1 = 1/8 [6, 7, 8]. Consequently, the Néel transition temperature TN is greatly reduced due to the degree of magnetic frustration. Remarkably, a magnetic neutron diffraction measurement [8] has revealed that a long-range AF ordering is suppressed even at T << TN for CoAl2O4. MacDougall et al. [8] have investigated this peculiar observation for CoAl2O4, and have revealed the formation of a fragmented AF domain structure due to the kinetically inhibited AF domain wall motion below TN. For the spinel oxides, distribution of the divalent (D) and trivalent (T) cations at the A- and B-sites is defined by the chemical formula (D1−T)[T2−D]O4, where the parentheses and square brackets indicate the A-site and B-site occupancies, respectively, and  is the inversion parameter [5]. In our previous investigation of the A-site antiferromagnet CoAl2O4, we discovered that inversion could be tuned widely within the  range 0.02    0.153 [9]. As shown in Fig. 1, the magnetic state is highly sensitive to the value of . Hanashima et al. [10] have analyzed the magnetic states of CoAl2O4 quenched samples for inversion tuned in the range 0.0467    0.153: they have demonstrated that a Néel-spin-glass (NSG) phase transition occurs at  ~ 0.08 (Fig. 1). Melot et al. [11] have studied the properties of CoAl2−xGaxO4 for 0.09    0.63, and they have located the NSG boundary in the range 0.09    0.36. We therefore conducted an extensive analysis of the effect of  on magnetism of CoAl2−xGaxO4. We constructed CoAl2−xGaxO4 specimens with precise -values and focused particularly on the vicinity of the NSG phase boundary. We discovered that the SG and Néel states separated from the paramagnetic (P) state by the phase boundaries TSG() and TN(), respectively, in the inversion-temperature (-T) plane; all the states met at the tricritical point (tc, Ttc) = (0.08, 4.0 K). Remarkably, for  < tc, the SG state penetrated below the Néel state and thereby allowed PN and NSG phase transitions. The NSG phase boundary could be traced out as an anomalous peak of the TRM relaxation rate. The slow spin dynamics detected by the TRM time course indicated the existence of an -induced quantum phase transition and the suppression of AF long-range ordering. The latter has been observed in several previous studies [6, 7, 8, 10, 12].  5  II. METHODS Polycrystalline specimens of CoAl2−xGaxO4 (0  x  2) were synthesized by a solid-solid reaction using appropriate amounts of CoO(4N), Al2O3(4N), and Ga2O3(4N). The mixed powder was pelletized and calcinated at 1300C for 24 h in ordinary air and cooled to room temperature at a rate of 36C/h. Synchrotron powder x-ray diffraction (XRD) at room temperature were carried out using two wavelengths (0.620089 Å and 0.65296 Å) on the BL15XU beam line at SPring-8 (Harima, Japan) [13] and refined the crystal structure with Rietveld analysis software, RIETAN-FP [14]. The AC and DC magnetic susceptibilities were measured with a magnetic-properties measurement system (MPMS-XL; Quantum Design) with an AC excitation field of Hac = 1 Oe. The frequencies were fixed in the range 0.3–30 Hz because of the presence of significant extrinsic background signals above 300 Hz. The temperature dependence and relaxation rate of TRM were measured after field cooling (FC) with a DC excitation field of HFC = 100 Oe applied at T = 70 K. After reaching the target temperature, the sample temperature was held constant for a variety of waiting periods, tw, between 60 and 54,000 s. The applied magnetic field was then immediately reduced to zero, and the MTR was recorded as a function of temperature and time. Finally, we corrected for an extrinsic paramagnetic contribution, i.e., (T)Hres produced by the residual field Hres less than 0.7 Oe in this work. After each isothermal relaxation measurement, Hres was estimated by measuring (T)Hres in the paramagnetic region (T = 70 K), where the short-range magnetic order was diminished [5,9]. For measurements of magnetization at T = 1.75 K and B = 0H = ±15 T, we used a house made high-field magnetometer and the extraction method at the Tsukuba Magnet Laboratory, National Institute for Materials Science, Japan. The specific heat of the sample was measured with a physical-properties measurement system (PPMS Dynacool; Quantum Design). III. RESULTS A. Inversion-temperature (-T) phase diagram 1. Crystal structure Figure 2 shows the XRD profile for a powdered CoAl2−xGaxO4 sample with x = 0.80. Each synthesized sample had a single-phase cubic spinel structure (space group: Fd-3m) that was confirmed by identifying the corresponding XRD reflection peaks (Figs. S1 and S2). We refined the crystallographic parameters by using the Rietveld method for the pseudo-ternary spinel 6  compound CoAl2−xGaxO4: the supplementary information (SI) describes the appropriate constraints for the refinement of cation occupancies at the A- and B-sites in the spinel lattice (Table S-I). Whereas the values for the lattice constant a and oxygen positional parameter u [Fig. 3(a)] were in accordance with those reported by Melot et al. [11], the A-site occupancy of Al cations, gA(Al) [Fig. 3(b)], and the inversion parameter,  [Fig. 3(c)], were relatively small for x < 2.0. For the A-site occupancy of Co cations, gA(Co), the inversion, which is defined as  = 1 − gA(Co), increased monotonically with increasing x and exhibited saturation for x ≥ 1.2 [Fig. 3(c)]. For CoGa2O4 CG (x = 2.0), the observed inversion value of  = 0.664(8) revealed the random distribution of Ga cations at both A- and B-sites [5]. Note that for x  0.8, the A-site occupancy of Ga cations, gA(Ga), was directly proportional to x and was described by the  equation gA(Ga) = 0.56x (Fig. S3). We extrapolated this relationship for x < 0.2 to obtain a reasonable estimate of cation occupancies and derived the crystallographic parameters listed in Table S-II. The value of u, the oxygen positional parameter, decreased linearly with increasing x for x < 1.0 but was relatively independent of x for x ≥ 1.0 [Fig. 3(a)]. Empirical evidence has shown that u is a function of the ratio rB/rA for ternary spinel oxides, where rA and rB are the ionic radii of the cations occupying the A- and B-sites, respectively [15]. Indeed, the u-values obtained for 0 < x  2.0 were in reasonable agreement with the values computed using the abovementioned empirical relationship [Fig. 3(a)]. To adapt this relationship for the pseudo-ternary system CoAl2−xGaxO4, we assumed that the average cation radius, <ri>, was given by the equation <ri> = gi(Co)ri(Co2+) + gi(Al)ri(Al3+) + gi(Ga)ri(Ga3+), where ri(M) is the ionic radius of the cation (M = Co, Al, or Ga) occupying the tetrahedral (i = A) and octahedral (i = B) sites. Remarkably, the linear dependence of x on u for x < 0.6 and insensitivity of x to u for x > 1.2 were fairly well reproduced [Inset of Fig. 3(c) and Fig. S4]. The monotonic relationship between  and x was therefore consistent with the relationship between u and x in CoAl2−xGaxO4. 2. DC susceptibility Figure 4(a) shows the temperature dependence of the DC susceptibility (T) as a function of x for CoAl2−xGaxO4 after zero-field cooling (ZFC) and FC. Similarly to the results of Melot et al. [11], we identified the SG phase transition based on the temperature associated with the maximum of the derivative of the magnetic susceptibility with respect to temperature d(T)/dT, i.e. Tmax(d/dT) [Fig. 4(b)]. For CoAl2O4, this temperature coincided well with the temperature associated with the maximum of C/T, i.e. Tmax(C/T), where C is specific heat [9]. At the phase 7  boundary, the shapes of both (T) and d(T)/dT curves changed abruptly [Figs. 4(a) and (b)]. In this case and typically in SG systems, the (T) measured after ZFC exhibited a cusp at T = TSG. The bifurcation after ZFC and FC occurred at T < TSG. In the Néel state, there was a broad maximum of (T) for T > TN. In accordance with the values reported by Melot et al. [11], the effective magnetic moment peff remained nearly constant, and the Weiss temperature  increased as the value of x increased [Fig. 4(c)]. Figure 4(d) shows Tmax(d/dT) and TSG as functions of x. The hysteresis loops shown in Fig. 5 for x = 0, 0.2, and 0.6 at T = 1.75 K matched exactly with those of Melot et al. [11]. For x = 0.6, a coercive field of Hc = 2.5 kOe was applied, and the remanent magnetization was observed to be mr = 0.016 B, which was comparable to m0 = MTR(t = 0) = 9.6×10–3 B at T = 2 K. 3. Specific heat Figure 6 shows the temperature dependence of C/T for multiple x-values. As mentioned in previous literatures [3, 5, 6], even at x = 0, an anomaly in specific heat at the transition is somewhat broader. In contrast, a -type transition is indicated for the weakly frustrated AF CoRh2O4 [3]. The trend followed the power law C(T)  T  with  ~ 7/3 [3, 5, 6] for T < TN, i.e. before the Néel phase transition. Note that Tmax(C/T) corresponded with Tmax(d/dT) and an anomalous point of TRM for x < 0.05, as can be seen below. For x > 0.06, Tmax(C/T) was slightly higher than TSG, and the peak of C(T)/T vs. T became broader with increasing values of x (Fig. 6). Such discrepancies between TSG and Tmax(C/T) have often been observed in SG compounds [5, 9]. 4. TRM TRM is a unique probe for characterizing the magnetic state and identifying phase transitions, particularly anomalies at the AF and SG phase boundaries [3, 5]. In this subsection, we describe the magnetic phase diagram obtained using measurements of dynamic and quasi-static features of TRM. Examination of the x()-T plane revealed that the phase boundaries determined from the TRM measurements corresponded partially with those indicated using the DC-AC susceptibilities and specific heat [Fig. 4(d)]. Such verifications are necessary to investigate the anomalous magnetic behavior of CoAl2O4 that is expected to result from magnetic frustration and inversion. Figures 7(a) and (b) show the temperature dependence of TRM intensity, MTR(T), and the derivative with respect to temperature of TRM, 8  (1/MTR)(dMTR/dT). The exponential decay of MTR ceased for temperatures even slightly above the transition [Fig. 8(a)]. Between ti = 300 and tf = 6,300 s, the relaxation rate MTR/MTR = [MTR(tf) – MTR(ti)]/MTR(ti) shows a sharp peak at the magnetic transition [Fig. 8(b)], especially, an intense peak at the SG transition and an Arrhenius type decay MTR/MTR  exp(−E/kBT) observed below TSG where E and kB are the activation energy and Boltzmann’s constant, respectively [5]. The temperature derivative and relaxation rate curves were used in a complementary manner; the former detects both AF and SG phase transitions, whereas the latter is highly sensitive to the SG phase transition. Figures 9(a) and (b) show contour images of (1/MTR)(dMTR/dT) and –(MTR /MTR)/2 in the x()-T plane. It should be emphasized that for   tc = 0.08 [corresponding to x = 0.06 in the inset of Fig. 3(c)], where TN and TSG meet [Fig. 9(b)], there is an obvious peak of MTR/MTR at x = 0.04 to 0.05 at temperatures below 4.0 K. The peak temperature Tmax(MTR/MTR) decreased, and the peak broadened as x decreased below 0.06 [Fig. 9(b)]. A low-temperature upturn of the relaxation rate was observed for x < 0.04; this upturn suggests that it is possible to extrapolate the trend for Tmax(MTR/MTR) to low temperatures below 2 K. If we assume a Debye-type TRM relaxation, MTR(t) = m0exp(−t/D) (i.e., MTR(t)/MTR(0)  1/D), the low-temperature upturn corresponds qualitatively to an enhancement in the relaxation rate 1/ at T << TN for x = 0 (CoAl2O4) with  = 0.055 [3]. 5. AC susceptibility To confirm the realization of the SG state in the vicinity of the NSG boundary (i.e., for  < tc) suggested by the (T) and TRM measurements [Fig. 9(b)], we investigated the AC magnetic susceptibility ′(T) for x  0.10 (Fig. 10). The frequency dependence of ′(T) increased with increasing  and was apparent even at low temperatures (T < TN) for 0.02  x  0.06. The temperature region where   was frequency dependent gradually expanded toward high temperatures for increasing x-values. This trend was consistent with the TRM observation that the SG state was realized at T < Tmax and  < tc. However, for x = 0.05, no discernable kinks were observed in the ′(T) curve at T ≈ Tmax(MTR/MTR) < TN that would have enabled effective detection of the NSG phase boundary (Fig. 10). In contrast, as discussed below, the magnetic dynamics detected through TRM were much more sensitive to the NSG transition, and they revealed important information about the -dependent magnetic properties of CoAl2O4. B. Magnetic dynamics 9  1. TRM at phase boundaries Before identifying the NSG boundary in the -T plane for CoAl2−xGaxO4, we first discuss formation of the CG state at T < TN in the vicinity of the NSG boundary and the tricritical point. We analyzed the temporal evolution of TRM relaxation based on Weron’s generalized probabilistic function. The Weron function W(t) for universal relaxation is based on a purely stochastic theory and is defined as  𝜙𝑊 = [1 + 𝑘 (𝑡𝜏)𝛽]−1/𝑘,          (1) where k > 0 is the interaction parameter, and   (0,1) is the scaling parameter defining the general relation ( = v1/) for the relaxation time  of particles in a volume v [16]. The k-parameter relates to the collective nature of the interactions and tw. For k → 0 and k → ∞, Eq. (1) asymptotically approaches the stretched exponential and logarithmic functions, respectively. The -parameter relates to the fractal geometry and nature of relaxation. As shown in Figs. 11(a) and (b), the Weron function reproduced the experimentally obtained MTR(t) and a derivative with respect to logarithmic time of MTR, S(t), with quantitatively reasonable values. The logarithmic time derivative S(t) is defined as 𝑆(𝑡) = −𝑑 (𝑀𝑇𝑅𝐻𝐹𝐶) /𝑑ln𝑡,          (2) which is a bell-shaped curve with a broad maximum at t =  as shown in Fig. 11(b). Interestingly, S(t) provides us approximately the density of the relaxation time t =  [2]. For an SG sample with x = 0.08 and TSG = 3.8 K, the maximum point of S(t) corresponding to t = , and the peak width broadened significantly with decreasing temperature [Fig. 11(b)]. Numerically, it was easily demonstrated that the parameters k and  in the Weron function (Eq. 1) strongly affected the -density. Both an increment in k and decrement in  contribute to broadening of an S(t) peak profile and gradually lead to a logarithmic relaxation [MTR(t)  lnt]. For x = 0.08, the S(t) profile broadened with increasing k and decreasing temperature [Fig. 11(b)]. We used Eq. (1) to extract the macroscopic parameters , k, and  at various x-values and temperatures from the isothermal temporal evolution of MTR(t, T). Figures 12(a) and (b) show relaxation time  for 0  x  0.10 (0.055    0.10), as a function of the reduced 10  reciprocal temperatures TN/T and TSG/T, respectively. In the Néel state, for x = 0 and 0.02, there was a clear decrement of  with decreasing temperature. The negative gradient of  with the inverse of temperature was consistent with the observation of an upturn in the relaxation rate at low temperatures [Fig. 8(b)]. For the CoGa2O4 CG and disorder-free CoRh2O4,  seemed to obey an Arrhenius law (T)  exp(Ea/kBT), where Ea is an activation energy comparable with kBTi (i = N or SG) as mentioned previously [3]. Figures 13(a) and (b) show the variation of k with the reduced temperatures T/TN and T/TSG, respectively, for CoM2O4 (M = Al and Ga). After the TRM measurements described in subsection A4 had been made for x = 0.04 and 0.05, the magnetic states at T > TN, TN > T > Tmax, and T < Tmax were identified as P, long-range ordered AF, and SG, respectively [Fig. 9(b)]. On the basis of the relaxation parameters, we evaluated the magnetic transitions and states separated by the NSG phase boundary as well as those in the vicinity of the tricritical point for 0  x  0.10. For T < TSG = 3.9 K and x = 0.10, the sample developed an SG state. There was a sharp minimum of the relaxation time ()  at T = TSG. The relaxation time increased as the temperature decreased below TSG in accord with an Arrhenius law [Fig. 12(b)]. The interaction parameter k(T) decreased with increasing temperature (similarly to the trend of the CoGa2O4 CG), reached a minimum at around T ~ TSG, and increased rapidly for T > TSG [Fig. 13(b)]. This pattern of k(T) was maintained qualitatively for x = 0.04 and 0.05 as well, although the value of k(Tmax) increased systematically with decreasing . In the intermediate state for Tmax < T < TN, the value of k(T) increased and was comparable to that of the CoRh2O4 antiferromagnet for T slightly below TN. Based on the work of Tsallis [17] on non-extensive statistical mechanics, we know that the interaction parameter is directly related to a non-extensive parameter q via a simple equation, k = (q – 1)/(2 – q) [18]. In previous studies, the q for SG samples, metallic systems Cu1-xMnx and Au1-xFex [19] and CoGa2O4 [5], were experimentally obtained as a function of temperature and exhibited the following characteristics: for T/TSG << 1, q(T) → 2 (asymptotically) and q(TSG) = 5/3; for T/TSG >> 1, q(T) = 0. As shown in Fig. 14, the q(T) for x = 2 (CoGa2O4 CG) exhibited similar features. However, for x = 0.06 and 0.10, there was a broad minimum of q(T) for T < TSG, and q(T) increased rapidly for T > TSG. This recovery of q and k at high temperatures, which was observed for 0.04  x  0.10 [Fig. 13(b)], implies that an AF correlation is maintained for T > Tmax or T > TSG. Note that there was an abrupt increase of (T) in the vicinity of TSG for the SG states of x = 0.06 and 0.10 ( > tc). In addition, (T) was constant for T < TSG. The specific temperature variations of k(T) and (T) provided indications 11  of whether and where an SG transition had occurred. The fact that similar behavior of  k(T) and (T) were also observed at x = 0.04 and 0.05 ( < tc) strongly suggests that the NSG phase transition occurred at Tmax. 2. Ageing We extrapolated the phase boundary temperature Tmax(MTR/MTR) for x ~ 0.03 and T = 2 K [Fig. 9(b)]. In this subsection, we analyze the ageing behavior (i.e., the tw-dependence of TRM relaxation), which is often investigated for T < TSG in SG systems [2, 20]. Figure 15(a) shows the tw-dependence of  computed by fitting the relaxation curve to Eq. (1). For CoAl2−xGaxO4, (tw) curves followed an empirical relation  = a + (b−1 + tw−1) −1, where a and b are constants; b is a characteristic time that represents the upper limit to the time to observe ageing behavior. For b >> tw, the above relation can be simplified to  = a + tw. In the SG state of the x = 0.04 sample,  followed the above simplified relationship, where a was related to the effective time (tcool) required to cool the sample from T > TN or TSG to the target temperature by application of a magnetic field with an HFC equal to 100 Oe. Based on the TRM protocol used in this work, we estimated tcool to be less than 300 s. This time was much smaller than a = 3,000 s at T = 2 K (with T/TSG ≈ 0.4) for x = 0.60. Consequently, because  >> tw = 300 s, the experimentally obtained value of  reflected the magnetic and thermodynamic characteristics of the sample. In contrast, for tw >> a, (tw) asymptotically approached the line  = tw and thereby confirmed the ageing behavior of TRM relaxation [Fig. 15(a)]. The tw-dependence of  for x = 0.03 at T = 2 K was comparable to that for x = 0.04, although the dependence for x  0.02 was rather weak. For x = 0.02, the (tw) curve crossed the  = tw line at tw = 2,000 s. This discrepancy between the x = 0.02 and 0.03 cases might suggest that the NSG phase boundary coordinate located at (x, T) ~ (0.03, 2.0 K), which matched with the extrapolated value for Tmax(MTR/MTR) [Fig. 9(b)]. Both k(tw) and (tw) decreased with increasing tw. The slopes of k(tw) and (tw) with respect to log(tw) were observed to strongly depend on  [Figs. 15(b) and (c)].  IV. DISCUSSION AND CONCLUSIONS By measuring the DC and AC magnetic susceptibilities, TRM, and specific heat, we demonstrated that an inversion-induced NSG transition occurs in CoAl2−xGaxO4. We constructed a magnetic phase diagram in the inversion-temperature (-T) plane for 0.055 ≤  ≤ 12  0.10 and down to T = 2 K. The P, Néel (AF), and SG states met at the tricritical point (tc, Ttc) = (0.08, 4.0 K). We used TRM measurements to identify the low-temperature branch of the magnetic phase boundary, which we expected to be the NSG boundary, and we also verified the identification through an analysis of AC susceptibility. The relaxation rate MTR/MTR exhibited anomalous peaks that could be associated with the PN, PSG, and NSG boundaries. Remarkably, the SG state penetrated into the Néel state for  < tc and established an NSG phase boundary that was extrapolated as  ~ 0.068 (x ~ 0.03) at T = 2 K [Fig. 9(b)]. To rigorously confirm the presence of the NSG phase boundary at T = 2 K, we analyzed the relaxation time  and the stochastic parameters k and  as functions of tw and  [Fig. 15 and Figs. 16(b)–(d)].  We observed that at T = 2 K, the value of  for x = 0.02 ( = 0.064) and x = 0.03 ( = 0.068) exhibited a weak and strong dependence, respectively, on tw [Fig. 15(a)]. For x ≥ 0.03, the value of  followed a simplified empirical law, (tw) = a + tw, and it asymptotically approached the  = tw line for tw >> a (n.b., a is a constant that depends on the TRM measurement protocol) [21]. This strong dependence of TRM relaxation on tw for x ≥ 0.03 implied the realization of an SG state [2]. For x = 0.02, we observed a peculiar two-step crossover tw-dependence of  due to crucial reductions in a and b in the vicinity of the NSG phase boundary. To verify the realization of this two-step crossover, further investigation of the TRM relaxation will be required for T > 2 K. Both the stochastic parameters k and  decreased with increasing tw. The decrement in  with increasing tw has been previously reported for the short-range Ising SG system [21], although, in that study,  was extracted from the relaxation curves using a stretched exponential. The relaxation parameters exhibited certain anomalous behaviors as functions of inversion. The k parameter increased with decreasing , with a particularly steep increment for  < 0.08. At the same time, the  and 1/ parameters showed significant enhancements for 0.02 ≤ x ≤ 0.03 (0.064 ≤  ≤ 0.068) [Fig. 16(c) and (d)]; this anomalous region corresponds to the boundary that defined the strong or weak tw-dependence of  [Fig. 15(a)]. We also observed enhancements in 1/ and  in the vicinity of the thermally induced PSG phase transition [Figs. 12(b) and 13(c)]. The absence of anomalies in specific heat measurements at Tmax was due to the low magnetic entropy released at the NSG phase boundary compared to the PN and PSG phase boundaries.  13   Together with the anomalous value of  (0.066) at T = 2 K denoted as age, the linear extrapolation of Tmax (MTR/MTR) with  [Fig. 16(a)] enabled us to estimate a zero-temperature boundary at  = 0.050(6). For x  0.02 ( = 0.064), both MTR/MTR and 1/ increased with decreasing temperature [Figs. 8(b) and 12(a)]. These increases indicate a proximity to the NSG transition at low temperatures. We therefore postulate that the NSG boundary could be extrapolated for T < 2 K and that an NSG QCP is located at qc = 0.050(6). Assuming the existence of this QCP, we examined the -(J2/J1) plane for  < tc by taking advantage of the following observations. We noticed that the k(T/TSG) curve for 0.04  x  0.10 gradually approached the curve for a CoGa2O4 CG [5] and canonical SGs [19]. Therefore, we speculate that for  < tc, the SG state coexists with the Néel state, even below the NSG boundary (for T < TNSG). In other words, the transition at the NSG phase boundary could be a first-order transition. This argument follows from the existence of an LP where the P, uniform, and modulated spin states meet [22]. The Landau free energy allows the construction of a phase diagram containing a QCP and LP. Depending on whether the order parameter is a scalar or vector, the transition from a uniform phase to a modulated spin phase at the phase boundary between the QCP and LP would be a first-order or second-order transition, respectively. This situation has occurred on the J2/J1-T plane of an A-site spinel magnet [6]. Our experimentally obtained -T phase diagram is quite similar to the theoretically predicted J2/J1-T phase diagram for an A-site antiferromagnet. However, the CG state that was realized in the -T plane was unlike the spatially modulated spin spiral (SS) state realized in the J2/J1-T plane. We therefore speculate that the AF fragments could be quenched from the Néel state and frozen in a metastable SG state. This work contributes to the understanding of the controversial magnetic ground state of a slightly inverted A-site spinel magnet CoAl2O4, and it paves the way for further investigations, such as the construction of an -J2/J1 phase diagram for T << Ttc. The speculations based on the findings revealed in this study were led from mainly the macroscopic observations down to T = 2 K, and therefore, to confirm the speculations, especially, the realization of the NSG-QCP in a slightly inverted CoAl2O4, both macroscopic and microscopic investigations are required to conduct at low temperatures below 2 K. ACKNOWLEDGEMENTS This work was supported by World Premier International Research Center Initiative (WPI). This work was partially supported by a Grant-in-Aid for Scientific Research (KAKENHI Grants 14  No. 25246013, No. 16K13999, No. 21H01637, No. 19H05790, No. 19H05819, and No. 21F21322) and the Research Institute for Science and Technology of Tokyo Denki University (Grant No. Q23K-01). The synchrotron radiation experiments were performed using the BL15XU beam line at SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (Grants No. 2015B4505, 2016B4505, and 2018A4507).  References  [1] R. F. 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Transition temperatures denoted by TN(T*), TSG, and Tmax[C(T)] were determined by measurements of magnetic susceptibility [9–12,23,24,26,27], NMR [28,29], and specific heat [9,24–26,28]. Dashed lines and hashed areas aid visualization. P, AF and SG denote paramagnetic, antiferromagnetic (Néel) and spin-glass states, respectively. The figure was originally published in Ref. 9, with the addition of several data points recently reported [11,12,29].   17      FIG. 2. XRD profile for CoAl1.2Ga0.8O4. The vertical green lines show the positions of the Bragg reflections. The blue trace indicates the difference between the observed and computed intensities.   18     FIG. 3. Variations of (a) lattice constant a and oxygen positional parameter u, (b) A-site occupancies of Co, Al, and Ga ions, and (c) inversion parameter  as functions of x for CoAl2−xGaxO4. Dashed line in (a) represents the u-parameter computed based on the empirical relation formulated in a previous work [15]. Inset in (c) shows (x) in the range of 0  x  0.2, where the solid line was obtained by least squares (also see SI).  19   FIG. 4. (a) DC–magnetic susceptibility as a function of temperature for CoAl2−xGaxO4, measured at H = 0.1 kOe after zero-field cooling (ZFC) and field cooling (FC). Data points for x = 0 and 2.0 are taken from previous works [5,9]. (b) Temperature derivative of susceptibility dg/dT as a function of temperature; the arrows indicate the magnetic transition temperature TN. (c) Effective magnetic moment peff and Weiss temperature  as functions of x. (d) Temperature-composition (x – T) phase diagram. The hashed area around x = 0.06 represents the phase boundary between the Néel (antiferromagnetic, AF) and spin glass (SG) states. Previously reported values [11] are plotted in (c) and (d) for comparison.    20   FIG. 5.  Hysteresis loops for x = 0, 0.2, and 0.6 measured at T = 1.7 K for CoAl2−xGaxO4. Inset shows the minimized view for x = 0.6.   21    FIG. 6. Temperature-scaled specific heat as function of temperature for x = 0.03, 0.1, 0.2, 0.4, 0.6, and 1.2. The solid black line represents C(T)/T for CoAl2O4 taken from previous work [9]. Arrows indicate maximum points of C(T)/T at various gallium compositions.    22    FIG. 7. Temperature dependence of (a) thermoremanent magnetization intensity MTR(T) and (b) the derivative of its natural logarithm with respect to temperature (dMTR /dT)/MTR for 0  x  0.10. An activation field HFC = 100 Oe was applied during field cooling prior to the isothermal  measurement of thermoremanent magnetization.   23   FIG. 8. (a) Isothermal time evolution of thermoremanent magnetization intensity MTR at various temperatures for x = 0.03. (b)Temperature dependence of relaxation rate MTR/MTR = [MTR(tf) – MTR(ti)]/MTR(ti) for 0 ≤ x ≤ 0.10 with ti = 300 and tf = 6300 s.   24   FIG. 9. Contour images of (a) the derivative with respect to temperature of thermoremanent magnetization intensity (1/MTR)(dMTR/dT) and (b), –(MTR/MTR)/2 (relaxation rate scaled by the factor of –1/2 for plotting convenience) on the x()-T plane.  25   FIG.10. Temperature dependence of the real component of the alternating current susceptibility ′(T) for 0  x  0.10 at various frequencies.  26   FIG. 11. (a) Isothermal relaxation of thermoremanent magnetization intensity, MTR(t), measured at T = 2, 3.8, and 4.2 K for x = 0.08. Solid line represents a numerically fitted curve of Eq. (1). The relaxation curve is offset along the vertical axis with the value shown against each curve in the unit of emu/g to aid visualization. (b) Logarithmic time derivative of MTR(t), S(t), scaled by m0 = MTR(t = 0) as a function of logarithmic time. Solid lines represent the calculated S(t)/m0 using the fitting parameters m0, , k, and . The arrows indicate the positions for t = . Values of relaxation parameters k and  are represented in parentheses against each curve. 27   FIG. 12. Relaxation time  as a function of the inverse of the reduced temperatures (a) TN/T and (b) TSG/T or Tmax/T for CoAl2−xGaxO4. Here Tmax = Tmax(MTR/MTR). Note that, for x = 0.04 and 0.05, the magnetic anomalies are indicated both at T = TN and Tmax(MTR/MTR). Data points for CoM2O4 (M = Al, Ga, and Rh) are taken from previous works [3,5]. Blue and purple dashed vertical lines in (a) indicate Tmax/TN for x = 0.04 and 0.05, respectively. For x = 0.02, we chose  = 1.3, a value that we obtained experimentally at T = 2 K and tw = 300 s [Fig. 16(d)].  28   FIG. 13. Interaction parameter k as a function of reduced temperatures (a) T/TN and (b) T/TSG or T/Tmax. (c) Here Tmax = Tmax(MTR/MTR). Scaling parameter  as a function of T/TSG or T/Tmax. Blue and purple dashed vertical lines indicate TN/Tmax for x = 0.04 and 0.05, respectively. Data points for x = 0 and 2.0 are taken from previous studies [3,5].   29    FIG. 14. Non-extensive parameter q as a function of reduced temperature T/TN, T/Tmax, and T/TSG for 0 ≤ x ≤ 0.02, 0.04 ≤ x ≤ 0.05, and 0.06 ≤ x, respectively. Here Tmax = Tmax(MTR/MTR). Data points for x = 0 and 2.0 are taken from a previous study [5]. Dashed curve represents experimental results for spin glasses [5,19] (see text).   30   FIG. 15. Relaxation parameters (a)  (b) k, and (c)  extracted from the curve of thermoremanent magnetization intensity, MTR(tw, t), as a function of tw in the vicinity of x = 0.03 ( = 0.064) and at x = 0.60  ( = 0.342).    31   FIG. 16. Variations of (a) m0 = MTR(T = 2K, t = 0) and Tmax(MTR /MTR), (b) k,  (c)  and (d)  with inversion  for 0  x  0.10. The anomalous point (age, T) = (0.066, 2 K) observed in the tw-dependence of  in Fig. 15(a) is also plotted in (a). Black dashed line shows the linear variations of thermoremanent magnetization, MTR, and Tmax(MTR/MTR) with respect to  to give an estimation of the QCP cq at T = 0. In (b) – (d) ×’s represent the values for x = 0 with  = 0.6 [3]. Vertical arrow in (a) indicates the tricritical point tc.