Non-Degenerate Sliding Ferroelectricity

• Non-degenerate sliding ferroelectricity is a phenomenon where sliding motions in layered materials create inequivalent polar states beyond simple ±P symmetry.
• A robust symmetry framework governs the switching processes, linking relative interlayer translations to distinct polarization and electronic behavior.
• Experimental and computational studies in bilayer and multilayer systems reveal asymmetric energy profiles and multistable pathways with implications for photovoltaics, spintronics, and slidetronics.

Non-degenerate sliding ferroelectricity denotes a class of sliding-ferroelectric phenomena in which the states connected by relative layer, sublayer, or stacking translation are not exhausted by a single symmetry-equivalent $\pm P$ pair. In the recent literature, the term is used in at least three partially overlapping senses: strict thermodynamic non-degeneracy of opposite-polarization states; inequivalent multistate stacking families with different polarization magnitudes or layer-resolved electronic structure; and sliding-induced removal of electronic degeneracies, even when the polar minima themselves remain symmetry-related. A general symmetry theory of slidetronics formalizes switching as $P' = G P$, with $G$ a point-group operator of the constituent layers but not of the stacked structure, so sliding switching is broader than complete inversion $P'=-P$ from the outset (Lee et al., 28 Feb 2025, Li et al., 30 Jul 2025).

1. Strict non-degeneracy and what it excludes

The strictest usage of non-degenerate sliding ferroelectricity requires opposite-polarization states with unequal energies and no symmetry relation that enforces equivalence. Several prominent sliding-ferroelectric systems do not satisfy that criterion. In bilayer CuF$_2$, the sliding coordinate $\{m_z|(t_x,t_y,d)\}$ produces two ferroelectric minima at $t_x=\pm a/6$, $t_y=b/2$, with $P_z=\pm 1.23~{\rm pC/m}$, polar space group $Pc$, and a switching barrier $P' = G P$0, but the two minima are explicitly described as “two equivalent FE states,” with no reported static total-energy splitting; the nonpolar intermediate state at $P' = G P$1, $P' = G P$2 instead restores the symmetry that suppresses $P' = G P$3 (Peng et al., 11 Mar 2026).

A closely related caution applies to bilayer $P' = G P$4 ($P' = G P$5 Ge, Sn, Pb). There, sliding from the high-energy $P' = G P$6 reference creates two polar stackings $P' = G P$7 and $P' = G P$8 with opposite out-of-plane polarization and modest barriers of $P' = G P$9, $G$0, and $G$1 for Ge, Sn, and Pb, but the paper explicitly states that $G$2 is “energetically equivalent to $G$3.” The reported Berry-phase polarizations are exactly opposite in sign and equal in magnitude, $G$4, $G$5, and $G$6, so the system exemplifies switchable sliding ferroelectricity coexisting with topology rather than strict non-degenerate bistability (Dong et al., 11 Jun 2025).

Bilayer WTe$G$7 under excitonic renormalization makes the same point from another angle. The glide-mirror-symmetric structure at $G$8 is nonpolar, and opposite displacements $G$9 create two “energetically equivalent ground states” with opposite out-of-plane polarization. Exciton condensation strongly deepens the polar minima relative to the nonpolar midpoint, increasing the effective energy scale from about $P'=-P$0 meV to $P'=-P$1–$P'=-P$2 meV, but the redressed double well remains symmetric: the work stabilizes sliding ferroelectricity without making the $P'=-P$3 pair intrinsically non-degenerate (D'Alessio et al., 1 Oct 2025).

2. Symmetry framework of sliding switching

The generator formalism of slidetronics gives the most compact classification of sliding-ferroelectric switching. For a layered structure $P'=-P$4, sliding and a point-group generator $P'=-P$5 are related by

$P'=-P$6

so that the polarization transforms as

$P'=-P$7

The allowed generators are

$P'=-P$8

subject to two requirements: $P'=-P$9 must be a symmetry of the constituent layers, and $_2$0 must not be a symmetry of the stacked structure as a whole. Within this scheme, type I slidetronics is generated by $_2$1, type II by $_2$2 or $_2$3, type III by $_2$4 or $_2$5, and type IV by $_2$6. The formal result with the strongest bearing on non-degenerate behavior is that complete polarization inversion $_2$7 is impossible in bilayers but becomes possible in multilayers such as PdSe$_2$8 trilayers (Lee et al., 28 Feb 2025).

This classification immediately separates exact inversion from more general switching. In h-BN bilayers, $_2$9 generates out-of-plane-only reversal; in cellulose bilayers, $\{m_z|(t_x,t_y,d)\}$0 yields type III switching $\{m_z|(t_x,t_y,d)\}$1; in As$\{m_z|(t_x,t_y,d)\}$2S$\{m_z|(t_x,t_y,d)\}$3/As$\{m_z|(t_x,t_y,d)\}$4Se$\{m_z|(t_x,t_y,d)\}$5 heterobilayers, $\{m_z|(t_x,t_y,d)\}$6 gives type II switching $\{m_z|(t_x,t_y,d)\}$7; and only in PdSe$\{m_z|(t_x,t_y,d)\}$8 trilayers does $\{m_z|(t_x,t_y,d)\}$9 produce $t_x=\pm a/6$0 (Lee et al., 28 Feb 2025). This suggests that non-degenerate sliding ferroelectricity should not be identified solely with asymmetric $t_x=\pm a/6$1 wells: bilayers are naturally predisposed to switching between non-$t_x=\pm a/6$2 partners, whereas multilayers can host either partial switching or full inversion.

3. Biased and quasi-nondegenerate sliding landscapes

An explicit realization of non-degenerate sliding ferroelectricity is proposed for monolayer Janus In$t_x=\pm a/6$3S$t_x=\pm a/6$4Se. The two switchable sliding-ferroelectric phases, $t_x=\pm a/6$5 and $t_x=\pm a/6$6, have unequal OOP polarizations

$t_x=\pm a/6$7

are connected by a NEB barrier of $t_x=\pm a/6$8 meV, and differ qualitatively in electronic structure: $t_x=\pm a/6$9 is indirect-gap with $t_y=b/2$0 eV, whereas $t_y=b/2$1 is direct-gap with $t_y=b/2$2 eV. The photocurrent maxima are $t_y=b/2$3 and $t_y=b/2$4, and the paper assigns different photovoltaic roles to the two phases: $t_y=b/2$5 benefits from stronger polarization in the visible region, whereas $t_y=b/2$6 shows a red-shifted and enhanced infrared photocurrent because of its smaller direct gap and higher mobility (Li et al., 30 Jul 2025).

A different route to effective non-degeneracy appears in the amphidynamic crystal $t_y=b/2$7. There the total polarization decomposes as

$t_y=b/2$8

with

$t_y=b/2$9

Because geometric ferroelectricity from frozen rotators already breaks inversion, the sliding degree of freedom is biased: at fixed geometric polarity, the sliding coordinate has a single-well rather than a double-well profile, and the conventional double well is recovered only when geometric and sliding components reverse together. In that precise sense, the sliding mode is effectively non-degenerate even though the full crystal still has switchable macroscopic $P_z=\pm 1.23~{\rm pC/m}$0 states (Miao et al., 2023).

Pressure-driven CuInP$P_z=\pm 1.23~{\rm pC/m}$1S$P_z=\pm 1.23~{\rm pC/m}$2 supplies a third form of asymmetry. In the low-pressure monoclinic $P_z=\pm 1.23~{\rm pC/m}$3 phase, Cu$P_z=\pm 1.23~{\rm pC/m}$4-dominated and Cu$P_z=\pm 1.23~{\rm pC/m}$5-dominated configurations are inequivalent polar states with estimated polarizations of about $P_z=\pm 1.23~{\rm pC/m}$6 and $P_z=\pm 1.23~{\rm pC/m}$7. Hydrostatic pressure first drives a continuous increase in Cu$P_z=\pm 1.23~{\rm pC/m}$8 occupancy and polarization, then near $P_z=\pm 1.23~{\rm pC/m}$9 GPa triggers a first-order interlayer-sliding transition to a trigonal phase with a lateral shift of $Pc$0, an abrupt $Pc$1 contraction of about $Pc$2, and loss of the Cu–S4 interlayer bond that had stabilized Cu$Pc$3. The calculated sliding potential is described as an asymmetrical double well, so the transition proceeds between inequivalent stackings and inequivalent Cu configurations rather than between symmetry-opposite FE partners (Zhou et al., 2024).

Graphene-based heterolayers illustrate a weaker, quasi-degenerate regime. In across-layer sliding ferroelectricity, the relevant states are not exactly symmetry-equivalent but differ by energy splittings much smaller than the sliding barriers. For bilayer graphene with a $Pc$4-twisted top BN interface, the $Pc$5 and $Pc$6 states differ by $Pc$7 while the barrier exceeds $Pc$8, with $Pc$9 and $P' = G P$00 and a switchable polarization of $P' = G P$01. For bilayer graphene on BN, $P' = G P$02, the barrier again exceeds $P' = G P$03, and $P' = G P$04 and $P' = G P$05. For benzene/graphene/BN, $P' = G P$06 and the switchable polarization is $P' = G P$07 (Yang et al., 2022).

Photoexcitation offers a nonequilibrium route to inequivalent polar states even in systems whose parent sliding pair is degenerate. In 3R bilayer MoS$P' = G P$08, the low-fluence “original” and “inverted” $P' = G P$09 structures remain mirror-related with equal-magnitude opposite-sign polarization and sliding barriers of $P' = G P$10–$P' = G P$11, but at $P' = G P$12 the photoinduced $P' = G P$13 phase supports opposite polarizations of nonequivalent magnitude, $P' = G P$14 and $P' = G P$15, and at $P' = G P$16 the corresponding values are $P' = G P$17 and $P' = G P$18 (Gao et al., 2024).

4. Multistate sliding ferroelectricity and inequivalent stacking families

A major development beyond bistability is the emergence of inequivalent multistate sliding-ferroelectric manifolds. Quadrilayer CuF$P' = G P$19 is a paradigmatic example. The paper identifies four coupled polarization–spin–layer states,

$P' = G P$20

which separate into two inequivalent polarization sectors with

$P' = G P$21

for the large-polarization class and

$P' = G P$22

for the smaller-polarization class. The large-$P' = G P$23 states place the valence-band maximum on the two outer layers, whereas the small-$P' = G P$24 states place it on the two inner layers. Opposite-polarization partners within a class remain glide-related, but the two classes are not symmetry-equivalent and therefore realize multistability beyond ordinary FE bistability (Peng et al., 11 Mar 2026).

Multilayer $P' = G P$25-InSe extends this idea to a combinatorial sliding problem. For trilayer $P' = G P$26-InSe, $P' = G P$27, $P' = G P$28, and $P' = G P$29 correspond to $P' = G P$30, $P' = G P$31, and $P' = G P$32, and are nearly degenerate, with $P' = G P$33 and $P' = G P$34 only about $P' = G P$35 and $P' = G P$36 meV above $P' = G P$37. The general $P' = G P$38-layer counting gives

$P' = G P$39

with $P' = G P$40 switching steps. The low-energy trilayer path has intermediate barriers of $P' = G P$41, and experimentally the multistate character is reflected in multiple current jumps in C-AFM. The same system was integrated into graphene/$P' = G P$42-InSe/graphene tunneling devices with a photovoltaic current density of $P' = G P$43 and a photoresponsivity of $P' = G P$44, both attributed to multiple sliding steps and multistate polarization control (Liang et al., 2024).

Cluster-assembled trilayer qHP C$P' = G P$45 shows that elemental systems can also host multiple polar minima. The six distinct stackings include two OP+IP ferroelectric families, $P' = G P$46 and $P' = G P$47, both with point group $P' = G P$48, but with different out-of-plane polarizations,

$P' = G P$49

different SHG spectra, and different switching pathways. Their total-energy differences are on the order of $P' = G P$50, so the states are nearly degenerate energetically, yet they are structurally and optically distinct. The direct OP reversal barrier is about $P' = G P$51 for $P' = G P$52 and about $P' = G P$53 for $P' = G P$54, while multistep paths lower these to about $P' = G P$55 and $P' = G P$56, respectively (Wang et al., 2024).

5. Electronic non-degeneracy induced by sliding ferroelectricity

In several magnetic and metallic systems, the essential non-degenerate aspect is not unequal FE-state energy but the lifting of an electronic degeneracy by the sliding-ferroelectric distortion. Bilayer Fe$P' = G P$57GeTe$P' = G P$58 is the clearest example. The nonpolar $P' = G P$59 phase is an A-type AFM metal with spin-degenerate bands protected by the combined symmetry $P' = G P$60 and nearly zero net magnetization. Sliding to the ferroelectric $P' = G P$61 or $P' = G P$62 stacking breaks $P' = G P$63, lifts the spin degeneracy throughout the Brillouin zone, and produces ferrimagnetism with net moments $P' = G P$64 and $P' = G P$65 per unit cell. The same FE switching simultaneously reverses polarization, spin splitting, and net magnetization,

$P' = G P$66

with a climbing-image NEB barrier of $P' = G P$67 and $P' = G P$68. The resulting anomalous responses are substantial: $P' = G P$69, AHC about $P' = G P$70 at the Fermi level, ANC peaks around $P' = G P$71 at 100 K and $P' = G P$72 at 300 K, Kerr rotation $P' = G P$73, and Faraday rotation $P' = G P$74 deg·cm$P' = G P$75 (Guo et al., 11 Aug 2025).

Bilayer H$P' = G P$76-Co$P' = G P$77CF$P' = G P$78 realizes a related but symmetry-richer situation. The paraelectric $P' = G P$79 bilayer with magnetic space group $P' = G P$80 has nonrelativistically spin-degenerate bands protected by $P' = G P$81. Interlayer sliding to the $P' = G P$82 ferroelectric state breaks $P' = G P$83, lowers the symmetry to $P' = G P$84, and yields $P' = G P$85. The FE phase is a compensated ferrimagnet with Zeeman-like nonrelativistic spin splitting already without SOC, while SOC adds valley inequivalence, Rashba coupling, and “alternating” spin-polarized bands. The ferro-valley polarization is

$P' = G P$86

the visible-range Kerr angle reaches about $P' = G P$87, and with a near-zero-index substrate $P' = G P$88 it can be enhanced to about $P' = G P$89 (Xinfeng et al., 9 Jul 2025).

Sliding ferroelectricity can also lift degeneracies in topological channel space. In bilayer ScI$P' = G P$90, the sliding-induced polarization potential $P' = G P$91 makes layer-locked spin channels evolve asynchronously through topological phase transitions. At $P' = G P$92, strain drives concurrent SOTI–QSHI–NI evolution, whereas finite sliding polarization produces the sequence

$P' = G P$93

The material realization is supported by a valence-band splitting $P' = G P$94 and a corner charge $P' = G P$95, and the paper proposes anomalous Nernst conductivity as an experimental discriminator of the resulting layer/spin-selective non-degeneracy (Yang et al., 2 Jun 2025).

Bilayer CuF$P' = G P$96 exemplifies the altermagnetic variant of this theme. Although its FE-I and FE-II minima are symmetry-equivalent, the sliding-induced polarization directly couples to the layer-locked $P' = G P$97-wave altermagnetic spin splitting. The FE states carry a layer polarization energy splitting of $P' = G P$98, and the coupled states are encoded as $P' = G P$99 and $G$00. Reversing $G$01 therefore reverses the energetic ordering of spin- and layer-resolved states and changes the sign of the transverse spin conductivity (Peng et al., 11 Mar 2026).

6. Methods, observables, and current boundaries of the concept

The recent literature combines a relatively stable methodological core with highly heterogeneous observables. Polarization in insulating systems is commonly computed by the Berry-phase method, as in bilayer CuF$G$02, $G$03, qHP C$G$04, and $G$05; metallic or photoexcited systems instead use real-space dipole-density approaches, as in bilayer Fe$G$06GeTe$G$07 and photoexcited bilayer MoS$G$08. Sliding barriers are typically obtained by NEB or climbing-image NEB, while electronic and transport responses are extracted from Wannier-based tight-binding, Berry-curvature calculations, Boltzmann transport, or NEGF-DFT photocurrent formalisms (Peng et al., 11 Mar 2026, Dong et al., 11 Jun 2025, Wang et al., 2024, Miao et al., 2023, Guo et al., 11 Aug 2025, Gao et al., 2024, Yang et al., 2 Jun 2025).

Experimentally, the field is no longer limited to indirect nanoscale signatures. Direct $G$09–$G$10 hysteresis, pyroelectric current, SHG, and NMR/XRD were used to resolve the coupled geometric-plus-sliding ferroelectricity in $G$11, while high-pressure XRD, Raman, dielectric, and polarization measurements mapped the sliding-mediated phase transition of CuInP$G$12S$G$13. In rhombohedral $G$14-InSe, DART-PFM, KPFM, and C-AFM provided evidence for room-temperature multistate switching and electrically tunable photovoltaic response (Miao et al., 2023, Zhou et al., 2024, Liang et al., 2024).

A persistent interpretive issue is that sliding ferroelectricity, multistability, and electronic non-degeneracy do not coincide automatically. Bilayer $G$15 and bilayer WTe$G$16 under excitonic stabilization retain symmetry-equivalent $G$17 partners despite topological or many-body richness; bilayer 1T$G$18-WTe$G$19 with tunable spin Hall effects also treats opposite FE partners as symmetry-related even though different stackings carry different polarization magnitudes and spin Hall coefficients (Dong et al., 11 Jun 2025, D'Alessio et al., 1 Oct 2025, Wu et al., 20 Jun 2026). The most precise contemporary usage therefore distinguishes at least four regimes: symmetry-degenerate bistable sliding ferroelectricity, biased sliding coordinates, inequivalent multistate stacking families, and electronically non-degenerate polar phases. Non-degenerate sliding ferroelectricity in the strict sense belongs only to a subset of these, but the broader family of inequivalent sliding-polarized states is now large enough to connect ferroelectric switching to photovoltaics, spin transport, anomalous Hall and Nernst effects, magneto-optics, and layer-resolved topology (Li et al., 30 Jul 2025, Guo et al., 11 Aug 2025).