3 ( B )/ (3 T) (%) T  (K) 50 0 -50 100 50 0 2 4 6 x  = 0.08 (b) 7 % 0 B  (mT) B  (mT)  0 200 400 600 $_{α}$ME  (ps/m)  10 4 10 2 10 3 10 1  0.04 0.08  0.15 x  = 0.00  0.02 Ba$_{0.5}$Sr$_{1.5}$Zn$_{2}$(Fe$_{1-}$$_{x}$Al$_{x}$)$_{12}$O$_{22}$ T  = 30 K (a) 0 1 2 m   ( x  10 4  ) 0.00 0.06 0.12 x  800 0.02 (a) M c ( µ B /f.u.) 0 1 2 (b) (c) (d) 0 100 200 300 T  (K)$^{400}$ T  (K) 50 100 150 0.00 0.04 0.08 x Heliconical Helical Paramagnetic T$_{N}$ T$_{con}$ 700 0.0 0.2 Δ I  (arb. unit.) 0.4 720 740 x  = 0.08 0.06 0.04 0.02 x  = 0.10 0.04 0.08 Photon Energy (eV) M ab  ( µ B /f.u.) 10 0 10 0 10 Ba$_{0.5}$Sr$_{1.5}$Zn$_{2}$(Fe$_{1-}$$_{x}$Al$_{x}$)$_{12}$O$_{22}$ B  (mT) 1000 1 10 100 x  = 0.08 0.04 0.02 II III C II III C modified helix modified helix collinear  ( C ) III T  = 30 K intermediate-II 0 FIG. 3. (a) B -dependence of magnetoelectric susceptibility ( α ME ) at T = 30 K. The inset shows the maximum values of α ME ( α m ) for each x . (b) Intensity and 3 D-plots of mag- netodielectric effects (Δ ε ( B )/ ε (3 T)) of the x = 0.08 sample below 130 K in low B region ( | B | < 50 mT). FIG. 4. (a) The c -axis magnetization M c measured at zero B after field cooling at B = 3 T. (b) The magnetic phase evolution with x , estimated from M c ( T ) and M ab ( T ) curves. (c) X-ray absorption intensity for several x at the Fe L 2 , 3 - edge after subtraction of the intensity of x = 0.00 (Δ I = -[ I ( ω , x )- I ( ω , x = 0.00)]). The data for x ≥ 0.04 are shifted by every 3.5 eV for clarity. (d) M ab ( B ) curves at T = 30 K. The dashed lines represent the phase boundaries estimated for the down-sweep following the criteria in Ref. [17]. The black triangles represent B c , L for x = 0.02 and 0.04. up to ∼ 110 K for x = 0.06 and 0.08. In the phase di- agram of Fig. 2(c), B$_{c}$$_{,}$$_{L}$ values for x = 0.06 and 0.08 indeed decrease rather abruptly near their T$_{con}$ ∼ 110 K to become less than 10 mT, showing a close link between the heliconical spin state and the extremely low B$_{c}$$_{,}$$_{L}$ . Ac- cording to the spin current model [6] that has explained induced P in various spiral- or heli-magnets, ⃗ P ∝ ⃗ k$_{0}$ × [ ⃗ µ$_{S}$ × ⃗ µ$_{L}$ ]. In the present hexaferrite, ⃗ k$_{0}$ is along [001] and thus the model consistently explains why P becomes trivial in the helical or modified helix phases (Fig. 1(b)). Even for the longitudinal conical state (Fig. 1(c), left), P should go to zero as the [ ⃗ µ$_{S}$ × ⃗ µ$_{L}$ ] vector, parallel to the spin-rotation axis of each cone, is along [001]. Under a small in-plane B , only the spin-rotation axis of the lon- gitudinal conical state is expected to align easily along the B -field to form a transverse conical state (Fig. 1(c), right), which then generates a finite P in the ab -plane. Therefore, it is most likely that below T$_{con}$ the spin state changes from a longitudinal to a transverse conical type across B$_{c}$$_{,}$$_{L}$ in the x = 0.06 and 0.08 samples. that the in-plane magnetic anisotropy decreases with in- creasing x , allowing the nontrivial moment along [001] to develop at smaller B . Meanwhile, the location of B$_{c}$$_{,}$$_{L}$ for the x = 0.02 and 0.04 (black triangles) is not any- more coincident with the intermediate-III phase bound- ary but is well inside the intermediate-II or modified helix phases. Therefore, it is likely that the low x -samples also form a kind of transverse conical spin state across their B$_{c}$$_{,}$$_{L}$ , which can coexist with the helical phases, and gen- erate nontrivial ab -components of [ ⃗ µ$_{S}$ × ⃗ µ$_{L}$ ] and thus P . A previous study of Ba 0 . 4 Sr 1 . 6 Zn 2 Fe 12 O 22 supports this possibility through neutron scattering and mag- netic torque measurements [19]; near the intermediate-III phase boundary, a cone of easy magnetization, compati- ble with the transverse conical state in Fig. 1(c), develops and coexists with a basal plane of easy magnetization, compatible with the helical phases in Fig. 1(b). What can be then the spin state of the FE phases in the low x ( ≤ 0.04) or possibly above T$_{con}$ in the high x ( ≥ 0.06) samples, in which larger B$_{c}$$_{,}$$_{L}$ is observed? The in- plane magnetization M$_{ab}$ curves (Fig. 4(d)) reveal some clues to this puzzle. According to Ref. [17], the modified helix, intermediate-I, -II, -III and collinear ferrimagnetic phases coincide with the first, second and third plateau, the steep ascent, and the last plateau regions, respec- tively. With these criteria, the magnetic phase bound- aries can be roughly estimated from Fig. 4(d); for exam- ple, the x = 0.02 sample exhibits magnetic phase bound- aries similar to BSZFO [12, 17]. However, we notice that the B -range for the modified helix phase decreases progressively as x increases to 0.04. For x = 0.08 (also x = 0.06), the first two plateaus of M$_{ab}$ ( B ) no longer exist during the up-sweep, suggesting that the modified he- lix and intermediate-I phases change into or coexist with the transverse conical state. These observations suggest To understand the role of Al-substitution microscopi- cally, we have performed Fe L$_{2}$$_{,}$$_{3}$ -edge X-ray absorption spectroscopy. The spectra taken for each x were sub- tracted from that of x = 0.00 to plot the difference spec- tra as shown in Fig. 4(c). The systematic increase in the intensity of the difference spectra near the octahedral Fe L$_{2}$$_{,}$$_{3}$ -edge supports that Al replaces the octahedral Fe- sites in proportion to x . Furthermore, comparative neu- tron diffraction studies for x = 0.00 and 0.06 produced a