Introduction

Modern magnetic memories primarily rely on a fundamental physical effect that the magnetoresistance of a magnetic tunnel junction (MTJ) can be significantly changed by switching its magnetic state. Controlling the magnetic state by an external electric field (voltage), rather than a current or magnetic field, has been a long-sought goal for developing next-generation, low-dissipation and high-efficiency spintronic devices, such as voltage-controlled magnetic random access memories1,2. Interestingly, a giant tunneling magnetoresistance has also been observed in two-dimensional (2D) van der Waals (vdW) magnetic layers3,4,5, since the discovery of 2D magnets6,7,8,9,10,11,12. Such a magnetoresistance can be significantly modulated by altering the interlayer magnetic state through various external factors, such as magnetic fields13, mixed magnetic and electric fields14,15,16,17, ferroelectric fields18,19, strain20, pressure21,22, twisting23,24,25, and spin injection26. These advancements have opened up a new avenue toward vdW spin-filter MJT27. However, pure electric-field switching of interlayer magnetic order remains a significant challenge owing to the lack of an effective magnetoelectric coupling mechanism.

Generally, magnetoelectric coupling could be induced through spin-orbit, spin-lattice and/or spin-charge interactions28. Although plenty of 2D vdW magnets have been discovered so far, electrical switching between interlayer spin-parallel (SP) and spin-antiparallel (SAP) orders has only been reported in the CrI3 bilayer29,30,31,32, driven by the charge transfer mechanism. As shown in Fig. 1a, extra electron or hole carriers transfer from the substrate to the vdW magnetic layers upon electrostatic gating, which increases the density of states at the Fermi level and typically leads to an SAP-to-SP order transition33,34. However, the spin-charge interaction induced by interlayer charge transfer is commonly rather weak, so that the switching of interlayer magnetic order still requires the assistance of a strong external magnetic field. Therefore, a general mechanism to enhance the coupling between spin order and external electric field is urgently needed for achieving pure electric-field-driven interlayer magnetic order switching in 2D vdW magnets.

Fig. 1: Interfacial magnetoelectric coupling mechanism.
Fig. 1: Interfacial magnetoelectric coupling mechanism.
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Schematic diagrams for electric-field switching of interlayer magnetic order driven by (a) charge transfer and (b) spin-electric potential mechanism. Gray and yellow slabs represent magnetic vdW monolayers and non-magnetic substrates, respectively. Red and Blue arrows represent spin moments (M). Green arrows represent electric dipole moments (P).

Here, we propose a general spin-electric-potential mechanism that in an asymmetric vdW magnetic bilayer, the interlayer polar spin interaction could lead to distinct different out-of-plane electric dipole moments for different interlayer spin orders. Consequently, under an external out-of-plane electric field, the SP and SAP orders gain very different electric dipole potential energies, which could reverse their relative stability (Fig. 1b). Based on a tight-binding dimer model, we reveal that the remarkable interplay between spin order and electric dipole moment originates from the disparity in occupation and spatial distribution of spin charges between SP and SAP states. Then, utilizing first-principles calculations, we predict a series of asymmetric vdW magnetic heterobilayers that exhibit strong magnetoelectric coupling driven by the spin-electric-potential mechanism. Especially, for the CrI3/MnSe2 heterobilayer, an SP-to-SAP phase transition could be realized by applying a feasible out-of-plane electric field around 0.1 V/Å.

Results and discussion

The concept of spin-electric potential

It is known that an electric dipole can be reversed by external electric fields because of the electric dipole potential energy given by W = − P ∙ Ɛ, where P is the electric dipole moment and Ɛ is the electric field intensity (Fig. 2a). On the other hand, for a magnetic system, the relative stability of the SP and SAP orders is determined by their energy difference, defined as Eex = ESAP – ESP (here ESP and ESAP are total energies of SP and SAP states, respectively). A positive (negative) value of Eex indicates that the SP (SAP) states is the magnetic ground state. The Eex can be a function of external magnetic fields owing to the magnetic potential energy, so that the SAP order could be switched to SP order by applying external magnetic fields (Fig. 2b). However, electric-field-switching of magnetic order is generally rather difficult because there is no direct interaction between a spin and an electric field.

Fig. 2: Spin-electric potential and driven magnetoelectric coupling mechanism.
Fig. 2: Spin-electric potential and driven magnetoelectric coupling mechanism.
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Schematic diagrams of (a) electric dipole potential, (b) magnetic potential, and (c) spin-electric potential. P, M, Ɛ and B represent electric dipole moment, magnetic moment, electric field intensity and magnetic field intensity, respectively. PSAP and PSP represent electric dipole moments of spin-antiparallel (SAP) and spin-parallel (SP) states, respectively. Red and blue arrows represent electron spins. Green arrows represent electric dipole moments. d Schematic diagram of electric-field switching of interlayer magnetic order driven by spin-electric potential. Ɛz represents the out-of-plane electric field, Ɛc represents the coercive electric field required to switch the interlayer magnetic order. e Tight-binding dimer model for the origin of spin-order-dependent electric dipole moment. Δdd represents the on-site energy difference between magnetic atoms. Light green arrows indicate electric dipole moments (p) contributed by bonding and antibonding orbitals. f Energy difference between SP and SAP orders (Eex = ESAP – ESP) as a function of external electric fields with Δdd = 0, − 0.1, − 0.3 and − 0.5 eV derived from the tight-binding model. Inset: PSAP and PSP, and their difference (Pex = PSAP – PSP) for the magnetic dimer as a function of Δdd.

Let’s assume that there is an asymmetric magnetic dimer, which is inherently polar (Fig. 2c). The electric dipole potential energies for the SP and SAP states of the dimer are given by WSP = − PSP ∙ Ɛ and WSAP = − PSAP ∙ Ɛ, respectively (here, PSP and PSAP are electric dipole moments for SP and SAP states, respectively). If PSAP ≠ PSP, then there will be a difference between WSP and WSAP, which can be written as ΔW = WSAP − WSP = − (PSAP – PSP) ∙ Ɛ = − Pex ∙ Ɛ (here Pex = PSAP – PSP). Interestingly, owing to the presence of ΔW, the Eex becomes a function of electric fields, namely, Eex(Ɛ) = Eex(Ɛ = 0) + ΔW = Eex(Ɛ = 0) − Pex ∙ Ɛ, where Eex(Ɛ = 0) represents the intrinsic Eex under zero electric field. This equation can also be rewritten as ΔEex/ΔƐ = −Pex [here, ΔEex = Eex(Ɛ’) − Eex(Ɛ) and ΔƐ = Ɛ’ − Ɛ]. According to this equation, as long as the magnitude of Pex is considerable, the relative stability of SP and SAP orders can be significantly manipulated by applying external electric fields. Here, we name such an additional spin-order-dependent electric dipole potential energy that modifies the relative stability of spin orders for an asymmetric magnetic system as spin-electric potential.

Now we consider an asymmetric vdW magnetic bilayer, which possesses out-of-plane electric polarizations and intralayer ferromagnetic orders in each layer. If the interlayer SP and SAP states have distinctly different out-of-plane electric dipole moments, namely PSP ≠ PSAP. Then, owing to the spin-electric potential, the relative stability of the interlayer SP and SAP orders could be reversed by applying an out-of-plane electric field when the Pex is large enough (Fig. 2d). However, the Pex and corresponding magnetoelectric effects have never been reported in a vdW magnetic system. Whether and how an electric dipole moment interplays with the spin order is still unclear.

Tight-binding dimer model

To explore this, we firstly introduce a tight-binding dimer model35 to explore the interplay between electric dipole moment and spin order. For simplicity, we adopted the one-dimensional Gaussian functions as the atomic basis functions to calculate the electron density and electric dipole moment (see supplementary section 1 for details). The general Hamiltonian of the dimer is written as

$$\widehat{H}={\sum}_{i,l,\sigma }{\epsilon }_{{il}}{\hat{d}}_{{il}\sigma }^{{{\dagger}} }{\hat{d}}_{{il}\sigma }+{\sum}_{i\ne j} {\sum}_{l,{l}^{{\prime} },\sigma }\left({t}_{{il}\sigma,{{jl}}^{{\prime} }\sigma }{\hat{d}}_{{il}\sigma }^{{{\dagger}} }{\hat{d}}_{j{l}^{{\prime} }\sigma }+h.c.\right)+{\sum}_{i,l}\frac{1}{2}{U}_{i}{m}_{i}\cdot {S}_{{il}}$$
(1)

where i and j are site (atom) index, l and l’ are orbital index, and σ is spin index. The first, second and third terms represent the on-site energy of each orbital, the interatomic hopping and the spin-splitting under the mean field approximation, respectively.

In the magnetic dimer, each atom has two orbitals and one localized spin (Fig. 2e). If we introduce an on-site orbital energy difference (Δdd) between the two magnetic atoms, the polar spin interaction will induce an electric dipole moment, which is totally contributed by the spatial disproportion of spin charges and, thus, could be sensitive to the spin order. In the SAP order, the occupied eigenstates are both bonding states (Supplementary Fig. 1d). Whereas in the SP order, the lowest eigenstate is a bonding state while the second-lowest eigenstate is an antibonding state (Supplementary Fig. 1e). This disparity in occupation states leads to a significant redistribution of spin charges and, thus, induces a distinct Pex. The magnitude of Pex is positively correlated with the magnitude of Δdd (inset in Fig. 2f). Consequently, the Eex is more sensitive to the electric field when the Δdd is larger (Fig. 2f) according to the equation ΔEex/ΔƐ = − Pex. Furthermore, if we ignore the electric field effects on the Δdd and Pex, then we obtain Ɛc = Eex(Ɛ=0)/Pex, where Ɛc is defined as the critical electric field that makes the Eex become zero, namely Eex(Ɛ = Ɛc) = 0. Supplementary Fig. 5b shows that the Eex(Ɛ = 0) is only slightly affected by Δdd. Therefore, the magnitude of Ɛc could be reduced by increasing the magnitude of Δdd (Fig. 2f and Supplementary Fig. 5c). On the other hand, the Pex, Eex(Ɛ = 0), and Ɛc are also affected by the interatomic hopping integral (t) (Supplementary Fig. 2c and Fig. 5). Overall, the model analysis results indicate that the Pex and spin-electric potential originate from the polar spin interactions in an asymmetric magnetic system with nonzero Δdd.

Magnetic heterodimer

One alternative way to introduce an intrinsic large Δdd in a realistic magnetic system is choosing different transition metal atoms to construct a magnetic heterostructure. The simplest magnetic heterostructure is the transition metal heterodimer, e.g., MnCr dimer. Our first-principles calculation results (Supplementary Fig. 6) show that, for the MnCr dimer, the PSAP is indeed distinctly different from the PSP, resulting in a large Pex (> 0.06 eÅ) when the interatomic distance (r) is ranging from 3.2 to 4.2 Å. Such a large Pex is mainly contributed by the direct polar spin interactions between Mn- and Cr-dz2 orbitals with a Δdd of ~5.1 eV (see supplementary section 2 for more details). These calculation results are consistent with the tight-binding model analysis results.

vdW magnetic heterobilayer

A vdW all-magnetic heterostructure, in which each layer has a different magnetic transition metal atom, may also possess a sizable Δdd. Therefore, here we focus on a vdW all-magnetic heterobilayer, i.e., CrI3/MnSe2, to explore the interplay between out-of-plane electric dipole moment and interlayer spin order. The CrI3 (Fig. 3a) and MnSe2 (Fig. 3b) monolayers have both been predicted and experimentally confirmed to be 2D ferromagnetic materials7,10. Individually, they are both centrosymmetric and non-polar. Since the lateral lattice constants of CrI3 (6.98 Å) are much larger than that of MnSe2 (3.61 Å), we used a 2 × 2 supercell of MnSe2 to construct the CrI3/MnSe2 heterobilayer (with a lattice mismatch of ~ 3.4%). Figure 3c shows the optimal stacking structure for both interlayer SP and SAP orders (see supplementary section 3 for details), which has in-plane c3 symmetry and, thus, an out-of-plane electric polarization. Notably, the PSAP and PSP are calculated to be 0.0202 and −0.0073 eÅ/u.c., respectively, with a positive value indicates the direction pointing from MnSe2 to CrI3. The Pex is 0.0275 eÅ/u.c. Note that, although the system is metallic in the periodic (in-plane) direction, electric polarization along the non-periodic (out-of-plane) direction is well-defined and is tunable by applying out-of-plane electric fields, as demonstrated in the metallic WTe2 multilayers36.

Fig. 3: CrI3/MnSe2 vdW heterobilayer.
Fig. 3: CrI3/MnSe2 vdW heterobilayer.
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Top (upper panels) and side (lower panels) views of the optimized atomic structures for (a) CrI3, (b) MnSe2 monolayers and (c) CrI3/MnSe2 heterobilayer. Blue, orange, purple and yellow balls represent Cr, I, Mn and Se ions, respectively. d Projected density of states for the SP and SAP states of the CrI3/MnSe2 heterobilayer. The gray dashed line represents the Fermi level. e Out-of-plane electric dipole moment (Pz) and z-component of magnetic moment (Mz) per unit cell as a function of the angle (θ) between the spins in the two magnetic layers. f The PSP and PSAP as a function of the change of interlayer distance (d) with respect to the equilibrium state.

To understand the origin of such a large Pex for CrI3/MnSe2, we firstly compare the atomic structures of SAP and SP states and find that the structural difference is quite small (see supplementary section 3 for details). Even though we use the same atomic structure for SAP and SP states without structural optimization, the obtained PSAP, PSP and Pex are 0.0216, − 0.0056 and 0.0272 eÅ/u.c., respectively, very close to those of the optimized CrI3/MnSe2. Therefore, the small structural distortion (ionic polarization) is not the main origin of the large Pex. Next, we investigate the electronic polarization driven by the interlayer polar spin interactions (see supplementary section 3 for details). Different from the MnCr dimer, the ligands on the inner surfaces of the CrI3/MnSe2 also play an important role. The charge accumulation in the interlayer space (Supplementary Fig. 10) implies an interlayer covalent interaction, which is dominated by Se- and I-pz orbitals (Supplementary Figs. 11 and 14). This provides a sizable indirect d-p-p-d super-exchange interaction between Cr and Mn ions. On the other hand, Fig. 3d shows a distinct energy difference between Cr-3d and Mn-3d orbitals, implying a large Δdd, which is estimated to be ~ 1.41 eV (Supplementary Fig. 14c). Therefore, the Pex of the CrI3/MnSe2 can also be qualitatively explained by the aforementioned dimer model. Nevertheless, the Pex of the CrI3/MnSe2 is distinctly smaller than that of the MnCr dimer. This is because the indirect interlayer spin interactions in the CrI3/MnSe2 are generally weaker than the direct interactions in the MnCr dimer.

The spin-order-dependent electric dipole moments have also been observed in other vdW magnetic heterobilayers (see supplementary section 4 for details). These results indicate that the magnitude of Pex can be manipulated by changing the types of transition metal and/or ligand ions. For instance, the CrI3/MnSe2 exhibits a much larger |Pex | (0.0275 eÅ/u.c.) than the CrI3/CrSe2 (0.0015 eÅ/u.c.) does. This is because the Δdd between the same TM atoms (e.g., Cr) in different layers is generally much smaller than that between different TM atoms (e.g., Cr and Mn). Besides, the CrI3/MnX2 (X = S, Se, Te) systems show an increasing magnitude of Pex (0.0040, 0.0275 and 0.0713 eÅ/u.c.) from S to Se, and to Te. This can be understood as the fact that (i) the 3p, 4p and 5p orbitals for the S, Se and Te atoms are more and more delocalized as the atomic size increasing. A more delocalized orbital usually manifests a larger spatial charge disproportion, thereby, leads to a larger electric polarization. (ii) As the electronegativity decreases from S to Se, and to Te, the metal-chalcogen covalent interactions become stronger (Supplementary Fig. 17), which enhances the interlayer super-exchange interactions and enlarges the Pex. Similar behavior also occurs in the CrGeTe3/MnX2 (X = S, Se, Te) systems.

To directly show the magnetoelectric effect in CrI3/MnSe2, we considered a possible spin-order-transition process characterized by the collective rotation of spins in each layer. By gradually changing the angle (θ) between the spins in the two magnetic layers from 0 to 180° (θ = 0 and 180° corresponding to SP and SAP states, respectively), we obtain a continuous change of out-of-plane electric dipole moment (from − 0.0073 to 0.0202 eÅ/u.c.) and total magnetic moment along z-axis (from 18 to − 6 μB/u.c.) with respect to the θ (Fig. 3e), which indicates a strong interfacial magnetoelectric coupling. Besides, from the relative energy profiles (Supplementary Fig. 18), we find a rather small activation barrier ( < 0.5 meV/u.c.) for the spin-order-transition, suggesting that the transition from a high-energy to the ground-state spin order could occur under ambient conditions.

The relation between interlayer distance and magnetoelectric property of the CrI3/MnSe2 has also been investigated. Supplementary Fig. 15 shows a Bethe–Slater-curve-like relation between Eex and interlayer distance, which has also been observed in other magnetic bilayers (e.g., CrSe2)37. Figure 3f shows that the PSAP monotonously increases, while the PSP first decreases and then increases as the interlayer distance decreasing. These profiles can also be reproduced by our dimer model by combining the contribution of non-magnetic and magnetic atoms to the electric dipole moments (Supplementary Fig. 2e).

Electric field effect on magnetic order

Now we explore the response of Eex to external out-of-plane electric fields and its relation with Pex for vdW magnetic bilayers. As shown in Fig. 4a and Supplementary Fig. 19, the ΔEex/ΔƐz (Ɛz is the out-of-plane electric field intensity), namely the slope of the profiles, is distinctly different for different magnetic bilayers and is closely related with the Pex. For instance, when the Ɛz increases from 0 to 0.2 V/Å (positive values of Ɛz indicate the direction pointing from MnSe2 to CrI3), the |ΔEex| of CrI3 bilayer is only 0.14 meV, while that of CrI3/MnSe2 is as large as 6.91 meV. Correspondingly, the Pex of CrI3 bilayer and CrI3/MnSe2 are zero and 0.0275 eÅ/u.c., respectively. Therefore, a sizable Pex can significantly increase the magnitude of ΔEex/ΔƐz. Besides, for a symmetric magnetic bilayer (e.g., CrI3 bilayer) with zero Pex, the Eex(Ɛz) and Eex(− Ɛz) are equal (see the inset in Fig. 4a). While for an asymmetric magnetic heterobilayer with a sizable Pex, the Eex is basically a monotonic function of Ɛz. More interestingly, by fitting the calculation results of a series of vdW magnetic heterobilayers, we obtained a nearly linear relationship between the ΔEex/ΔƐz and the Pex (Fig. 4b), namely ΔEex/ΔƐz ≈ − Pex, which demonstrates our aforementioned hypothesis. These indicate that the spin-electric potential mechanism is generally applicable and has a dominant effect on the magnetoelectric response for an asymmetric vdW magnetic system with a sizable Pex (see supplementary section 5 for details).

Fig. 4: Electric field effects on interlayer magnetic coupling.
Fig. 4: Electric field effects on interlayer magnetic coupling.
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a The change of Eex as a function of Ɛz for different vdW magnetic bilayers. A positive value of Ɛz indicates the out-of-plane direction pointing from MnX2 (X = S, Se, Te) to CrI3 layer. b The nearly linear relation between ΔEex/ΔƐz and Pex. Without loss of generality, here we take ΔEex/ΔƐz = [Eex(Ɛz = 0.1) − Eex(Ɛz = − 0.1)]/0.2. The gray line indicates the linear fitting of the calculation results. c Eex as a function of Ɛz for CrI3/MnSe2 bilayer without and with an in-plane biaxial tensile strain of 1%. d Comparison between charge transfer and spin-electric-potential mechanism in the CrI3/MnSe2.

The Eex(Ɛz = 0) is also an important parameter for realizing electric-field-driven magnetic order switching. Supplementary Table 3 shows that the magnitude of Eex(Ɛz = 0) for a vdW magnet is usually ranging from several to dozens of meV/u.c. Such a large Eex(Ɛz = 0) could make the electric-field, even the magnetic-field control of interlayer magnetic order very difficult. However, according to the equation Ɛc = Eex(Ɛz = 0)/Pex derived from the spin-electric-potential mechanism, as long as the Pex reaches several 0.01 eÅ/u.c., the Ɛc could be reduced to the order of magnitude of 0.1 V/Å, which is feasible in experiments. Besides, we find that the Eex(Ɛz = 0) is usually sensitive to interlayer stacking22,38,39,40 (Supplementary Fig. 9b) and in-plane strain20,41 (Supplementary Fig. 20), while the Pex is not. For instance, for the CrI3/MnSe2, the Eex(Ɛz = 0) greatly changes from 13.10 to − 35.85 meV/u.c., while the Pex slightly changes from 0.0201 to 0.0298 eÅ/u.c., when the in-plane strain increases from − 4% to 4%. Therefore, the Ɛc can also be manipulated by controlling the interlayer stacking or applying an external in-plane strain.

Owing to the large Pex of 0.0275 eÅ/u.c., the Eex of the CrI3/MnSe2 is very sensitive to the Ɛz (Fig. 4c). A Ɛz greater than ~ 0.1 V/Å will change its Eex from 2.66 meV/u.c. to a negative value and, thus, switch the interlayer spin order from SP to SAP (Supplementary Fig. 18b). The relation between these values is consistent with the equation Ɛc = Eex(Ɛz = 0)/Pex = 0.00266 eV/0.0275eÅ ≈ 0.1 V/Å. This mechanism can cause not only an SP-to-SAP, but also an SAP-to-SP phase transition. For instance, a tensile in-plane strain of 1% changes the interlayer spin order of CrI3/MnSe2 to SAP with Eex(Ɛz = 0) = − 4.46 meV/u.c while the Pex barely changes. Then, a negative Ɛz greater than 0.16 V/Å in magnitude could switch the interlayer spin order from SAP to SP (Fig. 4c).

It is worth noting that the charge transfer mechanism cannot explain the electric-field switching of interlayer magnetic order in the CrI3/MnSe2. Generally, charge transfer (electrostatic doping) benefits the SAP-to-SP phase transition. In the CrI3/MnSe2, the inherent interlayer charge transfer for the SAP state is − 0.018 e/u.c., with negative values indicate a net electron transfer from CrI3 to MnSe2 layer (Fig. 4d). In this case, applying a negative out-of-plane electric field (pointing from CrI3 to MnSe2 layer) suppresses the interlayer charge transfer, however, benefits the SP phase. This violates the charge transfer mechanism. Therefore, our proposed spin-electric-potential mechanism provides a new angle of view to understand the electric-field switching of magnetic order.

In summary, utilizing a tight-binding dimer model and first-principles calculations, we have proposed a general mechanism of realizing electric-field switching of interlayer magnetic order in a vdW magnetic system by introducing the concept of spin-electric potential. The spin-electric potential is an additional spin-order-dependent electric dipole potential energy that modifies the relative stability of spin orders. It originates from the polar spin interaction (with Δdd ≠ 0) in an asymmetric magnetic structure. The polar spin interaction causes a distinct difference on occupation and spatial distribution of spin charges between SAP and SP states, leading to a sizable Pex. The nearly linear relation between the ΔEex/ΔƐz and the Pex, namely, ΔEex/ΔƐz = − Pex, demonstrates that the magnetoelectric effect is dominated by the spin-electric-potential mechanism. Consequently, by inducing a sizable Pex via constructing a vdW all-magnetic heterostructure, a pure electric-field switching of interlayer magnetic order could be achieved, as predicted in the CrI3/MnSe2 and other vdW magnetic heterobilayers. These findings will be of great interest for developing pure voltage-controlled vdW spin-filter MTJs. We look forward to experimental observation in the future.

Methods

Computational methods

The first-principles calculations based on density functional theory were carried out using the Vienna Ab initio Simulation Package42. The PBE + U calculations43 with Ueff = 3 eV for 3 d electrons of transition metal atoms (e.g., Cr and Mn) according to the Dudarev’s method44 was employed to treat the exchange-correlation and strong electronic correlation. We have also repeated our calculations on the CrI3/MnSe2 using different calculation methods, i.e., the PBE + U method with Ueff values ranging from 2 to 4 eV for Cr- and Mn-3d orbitals, the meta-GGA SCAN and the hybrid HSE06 functionals. The results show that the choice of different calculation methods barely affects the main conclusion (Supplementary Table 2). The energy cutoff for the plane wave45 was 500 eV. A Γ-centered 12 × 12 × 1 Monkhorst-Pack46 grid was used to sample the Brillouin zone. The vdW correction47 was included. To avoid the incorrect periodic interactions along the c axis, a vacuum slab of ~ 30 Å and the dipole-dipole interaction corrections48 were adopted. The energy and force convergence criteria were 1 × 10–6 eV and 0.005 eV/Å, respectively. The out-of-plane electric dipole moment was calculated by integrating the electron density times the coordination over the whole unit cell.