1000 1 10 100 50 100 150 200 Δε ( B )/ ε (3 T) (%) P  ( µ C/m 2  ) T $^{ }$= 30 K 0.02 0.04 0.06 x $^{ }$= 0.08 (a) (b) Ba$_{0.5}$Sr$_{1.5}$Zn$_{2}$(Fe$_{1-}$$_{x}$Al$_{x}$)$_{12}$O$_{22}$ B$_{c,L}$ B$_{c,L}$ B$_{c,U}$ B //ab 0 0 0 0 0.02 0.04 0.06 x $^{ }$= 0.08 $^{x }$4 0 6 % B  (mT) 1000 1500 B  (mT) 1 10 100 2000 Paraelectric Ferroelectric Paraelectric 0.03 0.06 x $^{ }$= 0.08 0.04 0.00 0.01 0.02 0 50 100 150 200 250 300 T  (K) B$_{c}$$_{,L}$ B$_{c,U}$ (c) FIG. 2. (a) Electric polarization and (b) Δ ε ( B )/ ε (3 T) ≡ [ ε ( B ) - ε (3 T)]/ ε (3 T) curves for x = 0.02, 0.04, 0.06 and 0.08 at T = 30 K. (c) Ferroelectric phase boundaries determined from the Δ ε -maxima while ramping down B . example, at B = 0 T for x < 0.05 and B = 2.5 T for x > 0.05 at 70 K, and cooled to the target temperature in the FE state (at B = 1.2 T). An LCR meter was used for dielectric constant ε measurements at 4 MHz. While determined FE phase boundaries were confirmed to be independent of frequency from 100 - 10 MHz below 100 K, the specific frequency was chosen because the loss was more or less minimal ( < ∼ 0.3), and thus ε ( B ) peaks could be clearly identified even up to higher temperatures. BSZFO is one of the Y-type hexaferrites with alternat- ing layers of tetrahedral Fe/Zn and octahedral Fe sites (Fig. 1(a)) [12, 17]. It undergoes a thermal transition from a paramagnetic to a helical spin state at ∼ 337 K, in which alternating stacks of magnetic L and S blocks develop a proper-screw-type rotation of their magnetic moments ( ⃗ µ$_{L}$ and ⃗ µ$_{S}$ ) along [001]. In each magnetic block, the moment prefers to lie within the ab -plane rather strongly so that the moment remains so even un- der an in-plane B = 10 mT (modified helix phase, Fig. 1(b))[17]. When B is increased further, BSZFO exhibits a few successive intermediate phases (I, II, and III) and finally reaches a collinear ferrimagnetic phase around 2 T. Finite P develops only in the intermediate-III phase stabilized around 1 T [12], of which origin remains un- resolved. When a small amount of Al is substituted for Fe in BSZFO, we find remarkable features in curves of P vs. B and ε vs. B (Fig. 2). All the samples yield induced P in finite B -windows between a lower critical field ( B$_{c}$$_{,}$$_{L}$ ) and an upper critical field ( B$_{c}$$_{,}$$_{U}$ ) (Fig. 2(a)). As x in- creases, B$_{c}$$_{,}$$_{L}$ is greatly reduced, and thus a finite P is observed even at B ≦ 1 mT for x = 0.06 and 0.08. Mean- while, the maximum of the electric polarization ( P$_{max}$ ) gradually increases with x up to x = 0.08. When x is increased further, to 0.20, the turn-on behavior of P be- comes broader and P$_{max}$ is substantially reduced (not shown). Figure 2(b) shows the magnetodielectric (MD) effects Δ ε ( B )/ ε (3 T) ≡ [ ε ( B ) - ε (3 T)]/ ε (3 T) of the selected samples at 30 K. The Δ ε ( B )/ ε (3 T) curves ex- hibit maxima at positions concomitant with both B$_{c}$$_{,}$$_{L}$ and B$_{c}$$_{,}$$_{U}$ . Thus, the Δ ε -maxima can be traced to identify the FE phase boundaries in a wide temperature-magnetic field window (Fig. 2(c)). As is evident in Fig. 2(c), B$_{c}$$_{,}$$_{L}$ systematically decreases from ∼ 1 T ( x = 0.00) to ∼ 1 mT ( x = 0.08) at T = 90 K, with a step of B$_{c}$ ∼ 100 - 200 mT per Δ x = 0.01, and it becomes lower than 1 mT for x = 0.08 below 90 K. These results demonstrate that Al-substitution into BSZFO is one of the most effective ways to control B$_{c}$$_{,}$$_{L}$ , down to 1 mT or even lower. Another conspicuous feature of Al-substituted BSZFO is the realization of giant α$_{ME}$ at extremely low mag- netic fields. Figure 3(a) shows the B -dependent α$_{ME}$ curves at 30 K for different x and the inset summa- rizes their maximum values ( α$_{m}$ ). α$_{m}$ for the undoped sample ( x = 0.00) is about 4.3 × 10 1 ps/m. As x in- creases, α$_{m}$ increases to 2.0 × 10 4 ps/m at x = 0.08, 470 times larger than at x = 0.00, and then decreases for x > 0.08. Such a large enhancement in α$_{m}$ could be re- alized because the Al-substitution not only increases P but also induces P in a lower B -region. The present op- timal α$_{m}$ = 2.0 × 10 4 ps/m is ∼ 12 times larger than α$_{m}$ ∼ 1.6 × 10 3 ps/m of Ba 2 Mg 2 Fe 12 O 22 [15], and records the highest value among single-phase multifer- roics [18]; for comparison, ∼ 10 2 ps/m in TbMnO 3 [1] and TbMn 2 O 5 [2] and 4.2 ps/m in Cr 2 O 3 [18] have been reported. Ba 0 . 5 Sr 1 . 5 Zn 2 (Fe 1$_{−}$ $_{x}$Al$_{x}$ ) 12 O 22 also ex- hibits substantial MD effects at low temperatures. Fig- ure 3(b) shows Δ ε ( B )/ ε (3 T) curves for x = 0.08 below 130 K. The ε , showing a maximum around 0 mT, rapidly decreases within low B -ranges ( | B | < 50 mT), and B = 20 mT is sufficient to induce about 6 % MD effect. Vary- ing x , the MD effects reached as high as 12 % for x = 0.00 - 0.03 and 15 % for x = 0.04 - 0.12. The observed ferroelectricity induced at extremely low B is likely associated with a heliconical spin ground state. The emergence of spontaneous magnetization along [001] with the increase of x (Fig. 4(a)) supports the hypothesis that the Al-substitution progressively stabilizes the lon- gitudinal heliconical state (Fig. 1(c), left) as reported in Ba 2 Mg 2 Fe 12 O 22 [15, 16]. The estimated conical ordering temperature T$_{con}$ is ∼ 20 K for x = 0.02, and it increases 2