The anti-symmetric and anisotropic symmetric exchange interactions between electric dipoles in hafnia

The anti-symmetric and anisotropic symmetric exchange interactions between two magnetic dipole moments – responsible for intriguing magnetic textures (e.g., magnetic skyrmions) – have been discovered since last century, while their electric analogues were either hidden for a long time or still not known. It is only recently that the anti-symmetric exchange interactions between electric dipoles was proved to exist (with materials hosting such an interaction being still rare) and the existence of anisotropic symmetric exchange interaction between electric dipoles remains ambiguous. Here, by symmetry analysis and first-principles calculations, we identify hafnia as a candidate material hosting the non-collinear dipole alignments, the analysis of which reveals the anti-symmetric and anisotropic symmetric exchange interactions between electric dipoles in this material. Our findings can hopefully deepen the current knowledge of electromagnetism in condensed matter, and imply the possibility of discovering novel states of matter (e.g., electric skyrmions) in hafnia-related materials.

Here, via symmetry analysis and first-principles calculations, we identify hafnia (HfO 2 ) material as an ideal candidate that accommodates the exchange interaction (eDMI and eASEI) between electric dipoles.We show that HfO 2 has various polymorphisms (i.e., P 2 1 /c, P mn2 1 , P ca2 1 and P bca phases) demonstrating non-collinear alignment of electric dipoles.The noncollinear dipole patterns (NCDP) herein are interpreted by our phenomenological theories, further revealing that (i) eDMI and eASEI are simultaneously hosted by HfO 2 , and that (ii) both eDMI and eASEI stem from the structural distortions associated with O sublattice.
Before extracting the NCDP in P 2 1 /c, P mn2 1 , P ca2 1 and P bca phases, we analyze the possible structural distortions accommodated by the high-symmetric F m 3m phase of HfO 2 .For simplicity, we select the conventional cell of F m 3m HfO 2 -containing four formula units -as our reference structure [50].In such a con- ventional cell, there are four Hf ions and eight O ions, whose motions [51] constitute thirty-six structural order parameters.In the present work, the order parameters contributed by Hf sublattice are labelled by Hf U α (U = F, X, Y, Z; α = x, y, z).Here, the subscript α and superscript U denote that the atomic displacement for each Hf ion is either parallel or antiparallel to α direction, depending on the configuration labelled by U as defined in Figs.1(a)-1(d).For example, Fig. 1(a) depicts the Hf F α configuration whose atomic motions are homogeneous with respect to the four Hf ions in the cell.When endowing the Hf F α configuration with Hf's displacement along x direction, we arrive at the Hf F x order parameter sketched in Fig. 1 Starting from these thirty-six order parameters, we construct phenomenological theories that describe NCDP in HfO 2 .We notice that the combination of Hf U α and Hf V β order parameters naturally yields NCDP, when U = V and α = β.By symmetry arguments, Hf U α and Hf V β are possibly coexisting via the trilinear coupling, that mediated by the O W γ structural order parameter.As shown in Section I of the SM, we have derived four effective Hamiltonians (H 1 , H 2 , H 3 , and H 4 ) involving trilinear couplings of our aforementioned kind, summarized in Table I.We can verify the existence of the couplings in H l (l = 1 − 4) by first-principles numerical calculations, using the following strategy (see Section II of the SM for details): to verify the Hf U α Hf V β O W γ coupling, we (i) start from the F m 3m phase and impose a structural distortion according to O W γ with fixed amplitude, (ii) displace Hf ions following Hf V β mode with varying magnitude, and (iii) measure the first-principles-calculated forces acting on the Hf sublattice and associated with the Hf U α mode.The lin- ) are indicated in Fig. 1.The trilinear coupling associated with a specific phase of HfO2 is shown in the parentheses after the space group of that phase.
Hamiltonian Phases ear relationship between these forces (related to Hf U α ) and the distortion amplitudes (of Hf V β ) will corroborate the existence of the We now extract the NCDP in HfO 2 and link the NCDP with our phenomenological theories.In Section III of the SM, we analyze the structural distortions in P 2 1 /c, P mn2 1 , P ca2 1 and P bca phases of HfO 2 .In P 2 1 /c phase, the Hf X x Hf Z y O

Ay
x and Hf X z Hf Z y O Ay z trilinear couplings -see Table I -imply the (Hf X x , Hf Z y ) and (Hf X z , Hf Z y ) combinations, respectively.As sketched in Figs.3(a) and 3(g), the (Hf X x , Hf Z y ) and (Hf X z , Hf Z y ) combinations yield NCDP.The NCDP in P ca2 1 phase seems more complicated.To be specific, the (Hf Y y , Hf Z x ), (Hf Y y , Hf F z ) and (Hf F z , Hf Z x ) combinations [see Figs.  of the Hf U α Hf V β O W γ -type of trilinear couplings (U = V , α = β) towards the NCDP in HfO 2 's structural phases.Here, the central structural distortion is O W γ and is thus contributed by the O sublattice, mediating then the interaction between Hf U α and Hf V β distortions.In other words, the O W γ -type distortion is the structural origin of the NCDP in HfO 2 .
The anti-symmetric exchange interaction.-Thecorrelation between Hf U α Hf V β O W γ couplings (U = V , α = β) and NCDP opens a door to reveal the anti-symmetric exchange interactions (eDMI) of dipoles in HfO 2 oxide.To begin with, we recall that the magnetic exchange interaction is given by [3,8]  where (i) m i,α and m j,β (α, β = x, y, z) are α-and βcomponent of magnetic dipole moments centered on the i th and j th ions, respectively, and (ii) J ij,αβ characterizes the strength of coupling between m i,α and m j,β .Eq. ( 1) implies that the electric exchange interaction between u i,α and u j,β dipoles (if it exists) can be written as Here, u i,α and u j,β are atomic displacements, depicting the electric dipoles centered on i th and j th ions.To evaluate J ij,αβ in HfO 2 , we start from the F m 3m phase and work with a big supercell made of N conventional cells (see Section IV of the SM for details).Such a supercell contains 4N Hf ions with their atomic coordinates given by R m + r τ , where R m locates the m th conventional cell (m = 1, 2, ..., N ) and r τ is the coordinate of Hf inside the m th cell (τ = 1, 2, 3, 4, see Fig. S6 of the SM).Every Hf ion in the supercell can displace along the α direction (α = x, y, z) with respect to R m +r τ , creating a dipole u m,τ,α [53].We expand the Hf F α , Hf X α , Hf Y α , and Hf Z α order parameters by the u m,τ,α basis, and insert these expansions into our derived H l (l = 1 − 4), as demonstrated in Eqs.(S1)-(S6) of the SM.This yields the effective Hamiltonian as where J mτ m κ,αβ -a function of O W γ , m, m , κ, τ , α, and β -characterizes the coupling between u m,τ,α and u m ,κ,β dipoles.As a result, the J mτ m κ,αβ interaction associated with the H l Hamiltonian can be extracted via and the strength of the eDMI is evaluated by [54] A mτ m κ,αβ = 1 2 (J mτ m κ,αβ − J mτ m κ,βα ). (5) In the following, we focus on the eDMI associated with two neighbored Hf ions which belong to the same conventional cell (e.g., m = m , τ = κ).Our detailed derivation process is shown in Section IV of the SM; here, we omit the cell labels m amd m .With respect to each H l effective Hamiltonian, the A τ κ,αβ components -for the eDMI between Hf τ and Hf κ pair (τ, κ = 1, 2, 3, 4) -form a 3 × 3 anti-symmetric matrix.As shown in Tables S2 The anisotropic symmetric exchange interaction.-Wemove on to explore the eASEI that may be hosted by HfO 2 .Compared with the mAESI (see e.g., Refs.[3,8]), the eASEI between u i ≡ (u i,x , u i,y , u i,z ) and u j ≡ (u j,x , u j,y , u j,z ) dipoles (if it exists) can be defined by αβ S ij,αβ u i,α u j,β , where α, β = x, y, z, S ij,αβ = S ij,βα , and S ij,xx + S ij,yy + S ij,zz = 0. Working with Eqs. ( 2) and ( 4), the strength of the eASEI between u m,τ,α and u m ,κ,β is extracted by where δ α,β = 1 for α = β and δ α,β = 0 otherwise.The αβ-components of S mτ m κ,αβ form a 3 × 3 matrix that is symmetric and traceless.In Section IV and Tables S3 To complete our discussion on eASEI, we shall clarify the difference between eASEI and the interaction of local modes (in essence, electric dipoles).In ferroelectric theory, the interaction of local modes is written as H = i =j,αβ J ij,αβ µ i,α µ j,β [56], where J ij,αβ characterizes the strength of the interaction, and µ i,α is the amplitude of the local mode centered on the i th cell.Mathematically, the H interaction seems to resemble our proposed exchange interaction between electric dipoles [see Eq . 2].From physical point of view, however, the two interactions are different.To be specific, the interaction between local models is a bilinear coupling between atomic motions (J ij,αβ being independent of atomic motions), while the eASEI involves trilinear coupling among atomic motions-that is, J mτ m κ,αβ depends on O W γ structural distortion.
Summary and outlook.-Insummary, we show that various structural phases of HfO 2 exhibit NCDP.By constructing phenomenological theories, we interpret the NCDP in HfO 2 and further reveal that NCDP are rooted in the exchange interactions (eDMI and eASEI) of electric dipoles.This implies a possible marriage between HfO 2 -based oxides -high-profile materials in semiconductor technology because of their compatibility with silicon [44,[57][58][59][60][61][62][63][64][65][66] -and the topological textures of electric dipoles (e.g., electric skyrmions), which are desired states of matter towards the creation of novel information devices [21,[23][24][25][26][27][28]38].In other words, the HfO 2 and related materials [e.g., (Hf, Zr)O 2 and Y-doped HfO 2 ] may be ideal candidates to explore novel electric topological textures.Besides, we hope that our work will motivate the discoveries of a sequence of eASEI-based phenomena such as the dipole patterns as counterparts of the mASEI-driven skyrmionic states in two-dimensional magnets [8].This will deepen the current knowledge of electromagnetism in condensed matter systems such as ferroelectrics, magnets and multiferroics.
based on the conventional cell of F m 3m HfO3 can well describe the NCDP in P 21/c, P mn21, P ca21 and P bca phases.Using a larger cell, although captures more abundant structural distortions and NCDP, will significantly increase the difficulties for our symmetry analysis.
[51] The motion is referred to as the displacement of a ion with respect to its original position in the reference F m 3m phase.

FIG. 1 .
FIG. 1. Sketches of Hf and O motions in HfO2 with respect to the conventional cell of the F m 3m phase.In panels (a)-(d), the cyan and pink spheres indicate that the Hf's displacements centered on the corresponding spheres are along +α and −α directions, respectively.For displaying clarity, we do not show the periodic image of the Hf atoms in each cell.Panels (e)-(l) sketch various configurations associated with the O sublattice.In these sketches, the yellow and red spheres denote that the motions centered on the corresponding O sites are along +α or −α directions, respectively.Panels (m), (n) and (o) display three examples of atomic motions with the configurations defined by panels (h), (c) and (a) [the corresponding α directions being x, y and x], where pink and blue arrows indicate two motions of opposite directions.
(o).Another example is the Hf Y α configuration and the Hf Y y order parameter, shown in Figs.1(c) and 1(n).In the Hf Y α configuration, the displacements for Hf ions numbered by 1 and 3 (respectively, 2 and 4) are along +α (respectively, −α) orientations; the atomic motions along y direction associated with Hf Y α configuration is thus marked as Hf Y y .Similarly, we can define the other order parameters associated with Hf sublattice [see Figs.1(b) and 1(d)] and those contributed by the O sublattice [see Figs.1(e)-1(m)] in a self-explanatory manner.Following this convention, we have identified thirty-six order parameters for HfO 2 [see Section I of the Supplementary Material (SM) for details].

Fig. 2
indeed numerically confirms the existence of several selective trilinear couplings, namely, Hf X z Hf Y y O Az y , Az x .Interestingly, our derived Hf F z Hf Z x O Az x coupling coincides with the trilinear coupling that was claimed to drive the ferroelectricity of P ca2 1 HfO 2 (see Ref. [52]).
3(b), 3(e), and 3(f)] can yield the NCDP.These combinations come from the Hf Y y Hf Z x O Ax y , the NCDP in P mn2 1 phase [via (Hf F x , Hf Y y ) and (Hf F z , Hf Y y ) combinations, see Figs. 3(d) and 3(e)], while the Hf X z Hf Y y O Az y coupling gives rise to the NCDP in the P bca phase [via (Hf X z , Hf Y y ) combination, see Fig. 3(c)].Our aforementioned analysis thus emphasizes the importance

FIG. 2 .
FIG. 2. Forces on Hf sublattice in HfO2 as a function of Hf X z (a) and Hf F z (b) distortions.Purple square in panel (a): forces associated with Hf Y y mode (O Az y being fixed).Cyan diamond in panel (a): forces associated with Hf Z y mode (O Ay z being fixed).Purple square in panel (b): forces associated with Hf Y y mode (O Ay z being fixed).Cyan diamond in panel (b): forces associated with Hf Z x mode (O Az x being fixed).The dash lines in panels (a) and (b) display the linear fitting results, corresponding to Hf X z Hf Y y O Az y , Hf X z Hf Z y O Ay z , Hf F z Hf Y y O Ay z and Hf F z Hf Z x O Az x couplings, respectively.

FIG. 3 .
FIG. 3. Sketches of atomic motions associated with various trilinear couplings.The Hf X x Hf Z y O Ay x , Hf Y y Hf Z x O Ax y , Hf X z Hf Y y O Az y , Hf F x Hf Y y O Ay x , Hf F z Hf Y y O Ay z , Hf F z Hf Z x O Az x , S4, S6 and S8 of the SM, the A τ κ,αβ component is determined by the O W γ -type distortion contributed by the O sublattice.For example, we examine the interaction involving Hf 1 and Hf 2 ions, where r τ ≡ r 1 = 0 and r κ ≡ r 2 = 0a + 1 2 b + 1 2 c (a, b and c being the lattice vectors of F m 3m's conventional cell).The H 1 Hamiltonian suggests that A 12,xy ∝ −O Ay x [see Eq. (S9) and Table S2 of the SM].
, S5, S7 and S9 of the SM, we calculate the eASEI between u mτ and u mκ dipoles in HfO 2 , followingH l (l = 1 − 4).The S τ κ,αβ (cell label m being omitted) component is proportional to the O W γ -type distortion contributed by the O sublattice [55].So far, our discussion is based onHf U α Hf V β O W γ -type couplings (U = V , α = β) -as indicated in H 1 , H 2 , H 3 ,and H 4 -that are linked with NCDP.As a by-product, we additionally obtain seven other effective Hamiltonians H l (l = 5 − 11).In contrast to H l (l = 1 − 4) associated with NCDP, H l (l = 5 − 7) and H l (l = 8 − 11) are effective Hamiltonians with the types of HfU α Hf U β O W γ (α = β) and Hf U α Hf V α O W γ (U = V ), respectively, describing the collinear alignment of dipoles.As shown in Section IV of the SM, these Hamiltonians H l with l = 5 − 11 yield the eASEI as well (see Tables S10-S16 in SM), with the structural origin being the ω W γ -type distortion.

TABLE I .
Trilinear couplings resulting in NCDP for various phases of HfO2.Here, the definitions of the notations (e.g., Hf Y y and O Ax y