Hyperferroelectrics: proper ferroelectrics with persistent polarization
HHyperferroelectrics: proper ferroelectrics with persistent polarization
Kevin F. Garrity, Karin M. Rabe and David Vanderbilt
Department of Physics and AstronomyRutgers University, Piscataway, NJ 08854 (Dated: November 1, 2018)All known proper ferroelectrics are unable to polarize normal to a surface or interface if the re-sulting depolarization field is unscreened, but there is no fundamental principle that enforces thisbehavior. In this work, we introduce hyperferroelectrics, a new class of proper ferroelectrics whichpolarize even when the depolarization field is unscreened, this condition being equivalent to insta-bility of a longitudinal optic mode in addition to the transverse-optic-mode instability characteristicof proper ferroelectrics. We use first principles calculations to show that several recently discoveredhexagonal ferroelectric semiconductors have this property, and we examine its consequences both inthe bulk and in a superlattice geometry.
Ferroelectrics, which are materials with a non-zerospontaneous polarization that can be switched by an ex-ternal electric field, have been extensively studied bothexperimentally and theoretically. Much of the work onferroelectrics has focused on proper ferroelectrics, such asBaTiO . These have a non-polar reference structure thatis related to the ferroelectric ground state by a polar dis-tortion that lowers the energy in zero macroscopic electricfield, corresponding to an unstable transverse optic (TO)mode. However, a slab of a typical proper displacive fer-roelectric with insulating surfaces will not spontaneouslypolarize with polarization normal to the surface, becauseat quadratic order in the polarization the energetic costof the resulting depolarization field is larger than the en-ergy gain from freezing in the distortion [1]. In order topolarize, the depolarization field must be screened, as forexample by a metallic electrode placed on the surfaces ofthe ferroelectric slab [2].In contrast to proper ferroelectrics, improper ferro-electrics do not have an unstable polar distortion in theirhigh-symmetry structure. Instead, these materials haveone or more unstable non-polar distortions. However,when these distortions assume non-zero values they breakinversion symmetry in the material, resulting in a non-zero polarization [3–6]. Because the primary energy-lowering distortion in an improper ferroelectric is non-polar, the depolarization field is too weak to prevent theinstability. Thus, a slab cut from such a material can de-velop a non-zero polarization normal to the surface [7].In this work, we demonstrate a new class of “hyper-ferroelectrics.” These are proper ferroelectrics in whichthe polarization persists in the presence of a depolariza-tion field. Using first-principles calculations, we iden-tify hyperferroelectrics in the recently discovered class ofhexagonal ABC semiconducting ferroelectrics [8]. Us-ing first-principles-based modeling, we show that hyper-ferroelectrics have an electric equation of state that isqualitatively different from those of both proper and im-proper ferroelectrics, resulting in persistent polarizationregardless of screening and unique dielectric behavior. Finally, we discuss the potential applications of hyper-ferroelectrics, whose ability to polarize in ultra-thin lay-ers may allow the creation of highly tunable thin-filmor superlattice structures displaying ultra-fast switchingbehavior.We perform first-principles density functional theory(DFT) calculations [15, 16] within the local-density ap-proximation [17] using the Quantum Espresso code [18].We use ultrasoft [19] pseudopotentials from the GBRVhigh-throughput pseudopotential set [20, 21]. Phononfrequencies, Born effective charges, and electronic dielec-tric constants are calculated using DFT perturbation the-ory [22–24], and polarization is calculated using the Berryphase method [25].We begin by reviewing the properties of normal properferroelectric materials, which in their high-symmetryphase have at least one unstable TO mode, specifically,a Γ mode that is unstable under zero macroscopic elec-tric field ( E = 0) boundary conditions. The frequency ofthis mode can be obtained from first-principles compu-tation of the force-constant matrix with the usual peri-odic boundary conditions. The longitudinal optic (LO)modes can then be obtained by adding to the force-constant matrix a non-analytic long-range Coulomb termthat schematically takes the form ( Z ∗ ) /(cid:15) ∞ , where Z ∗ are the Born effective charges and (cid:15) ∞ is the electroniccontribution to the dielectric constant, generating thewell-known LO-TO splitting [9]. For normal proper ferro-electrics, this non-analytic term is sufficiently large thatall the LO polar modes are stable; in other words, the de-polarization field resulting from the long-range Coulombinteraction will prevent the ferroelectric from polarizingunder fixed D = 0 boundary conditions. For typical per-ovskite oxides, the strength of the depolarization fieldmust be weakened by at least 90% to allow for a non-zero polarization with D = 0 [1].While large-band-gap oxide ferroelectrics, which typi-cally have large Z ∗ ’s and small (cid:15) ∞ ’s, have all LO modesstable, there is no fundamental principle that enforcesthis stability. In fact, as we demonstrate in detail be- a r X i v : . [ c ond - m a t . m t r l - s c i ] D ec F r equen cy ( c m - ) a) b) c) K Γ ALO TO
FIG. 1: Structures of a) high-symmetry ( P /mmc ) and b)polar ( P /mc ) ABC ferroelectrics. The large green atom isthe ‘stuffing’ atom. c) Phonon spectrum of high-symmetryLiBeSb, from K ( π/ a, π/ a,
0) to Γ(0 , ,
0) to A (0 , , π/ c )(imaginary frequencies are plotted as negative numbers). low, unstable LO modes can be found in semiconductinghexagonal ABC ferroelectrics. The crystal structure isshown in Fig. 1(a-b) (space group P mc , LiGaGe struc-ture type). The high-symmetry phase of these materialsconsists of layers of two atoms in an sp -bonded honey-comb lattice separated by layers of a third ‘stuffing’ atom,as shown in Fig. 1(a-b). The polar phase is reached bya single Γ − phonon mode, which consists primarily ofa buckling in the honeycomb layers as the atoms movefrom an sp environment towards sp bonding, resultingin polarization in the z direction [8].In Table I, we report the lowest TO and LO phononfrequencies, dielectric constants, as well as band gaps,∆ Z ∗ zz = (cid:112)(cid:80) m ( Z ∗ zz ) m /N , and polarizations for a varietyof ABC ferroelectrics; those with imaginary LO frequen-cies are, by definition, hyperferroelectrics. The relativelysmall ∆ Z ∗ zz ≈ (cid:15) ∞ ) zz ≈ −
20 both contributeto the weak depolarization fields in these materials (forreference, cubic perovskites typically have ∆ Z ∗ zz ≈ (cid:15) ∞ ) zz ≈ ABC ferroelectrics are consequencesof the covalent bonding and resulting small band gaps ofthese semiconductors. In Fig. 1(c), we show the phonondispersion for the hyperferroelectric LiBeSb, which is apreviously synthesized material [10, 11]. We see that thelowest frequency phonon mode for q → u , as the buckling of the honeycomblayer, which varies from zero in the high symmetry struc-ture to one in the polar structure at E = 0. Then, weexpand the free energy up to second order in E , and upto sixth order in u , with the E = 0 polarization includedup to first order in u , F ( u, E ) = − au + bu + cu − P s u E − χ e ( u ) E , (1)where F is the free energy, χ e ( u ) = (cid:15) ∞ ( u ) − u , and P s , ABC ω TO ω LO ( (cid:15) ∞ ) zz ∆ Z ∗ zz Gap P ( E =0) P ( D =0) (cm − ) (cm − ) (eV) (C/m ) (C/m )LiZnP 134 i
49 13.3 3.0 1.27 0.80 0NaMgP 131 i
150 10.6 2.9 0.89 0.52 0LiZnAs 118 i i i i i i i i ABC hexagonal ferroelectrics. Com-pounds are listed with the stuffing atom first. First-principlesresults for high-symmetry phase: ω TO and ω LO are frequen-cies of unstable polar modes approaching Γ along ˆ q = (100)and (001) respectively; L c is defined in the text; ( (cid:15) ∞ ) zz is the zz electronic dielectric constant; ∆ Z ∗ zz is the RMS zz Borneffective charge; ‘Gap’ is the band gap. P ( E =0) is the first-principles polarization at E = 0. P ( D =0) is the polarizationcomputed from the model of Eqs. (1-2) at D = 0. E ne r g y ( e V / u . c . ) Polar mode ( u ) Electric field (C/m ) P o l a r i z a t i on ( C / m ) Displacement field (C/m ) P o l a r i z a t i on ( C / m ) D i s p l a c e m en t f i e l d ( C / m ) Electric field (C/m ) a) b)c) d) FIG. 2: Computed energy landscape and electric equationsof state for normal ferroelectric NaMgP. a) Energy vs. polarmode u . Dots are first principles; line is a fit to the model.b) P vs. (cid:15) E . c) P vs. D . d) D vs. (cid:15) E . Dashed red lines arelocally unstable at fixed E ; solid red lines are locally stable;solid black lines are globally stable. a , b , and c are constants. The polarization, P , is then P ( u ) = − ∂F∂ E = P s u + χ e ( u ) E , (2)which allows us to identify P s as the spontaneous polar-ization of the ground-state structure at zero electric field( u = 1, E = 0), justifying the notation for this constant.We fit this model to our materials by running a series ofcalculations with E = 0 and u fixed between 0 and 1.1, al-lowing all of the other internal degrees of freedom as wellas the lattice vectors to relax. In addition, we calculate (cid:15) ∞ ( u ) for each structure, which we fit to a cubic spline. E ne r g y ( e V / u . c . ) Polar mode ( u ) Electric field (C/m ) P o l a r i z a t i on ( C / m ) Displacement field (C/m ) P o l a r i z a t i on ( C / m ) D i s p l a c e m en t f i e l d ( C / m ) Electric field (C/m ) a) b)c) d) FIG. 3: Computed energy landscape and electric equations ofstate for hyperferroelectric LiBeSb. Details as in Fig. 2.
Using the model of Eqs. (1-2), we can parametricallyplot E ( u ), P ( u ) and D ( u ) = (cid:15) E + P versus each other,which we do for both the normal ferroelectric NaMgPand the hyperferroelectric LiBeSb in Figs. 2-3. In bothcases, we indicate regions that are locally unstable, lo-cally stable, and globally stable under fixed- E boundaryconditions. In locally unstable regions ( ∂P/∂ E < P as a function of E is multi-valued at E = 0, indicating thatNaMgP is ferroelectric, with spontaneous polarization asgiven in Table I. However, P vs. D is single-valued, in-dicating that NaMgP will not polarize under fixed D = 0boundary conditions, and thus is a normal proper ferro-electric. In contrast, for the hyperferroelectric LiBeSb,both P vs. E and P vs. D are multi-valued, so thatthe material will spontaneously polarize under both fixed E = 0 and fixed D = 0 boundary conditions. In addition,the slope of D vs. E indicates (cid:15) = ∂D/∂ E| E =0 >
0, de-spite the unstable polar mode. As shown in Table I andFig. 3(c), the D = 0 polarization of hyperferroelectrics, P ( D =0) , which we compute with the model of Eqs. 1-2, is small compared to P ( E =0) ; however, the amplitudeof the polar mode remains surprisingly large. The po-lar distortions of the materials at D = 0 are 25 −
75% oftheir E = 0 values, but the resultant ionic polarizationis largely canceled by the electronic polarization χ e ( u ) E induced by the depolarization field, resulting in a smallnet polarization.To emphasize the difference between hyperferro-electrics and improper ferroelectrics, we briefly reviewa model of an improper ferroelectric. In the simplest im-proper ferroelectrics, the primary order parameter, v , isnon-polar, but it couples to a stable polar mode u withthe form uv . Then u , which appears only to quadratic E ne r g y Non-polar mode ( v ) Electric field P o l a r i z a t i on Displacement field P o l a r i z a t i on D i s p l a c e m en t f i e l d Electric field a) b)c) d)
FIG. 4: Energy landscape and electric equations of state forimproper ferroelectric model of Eq. (3). a) Energy vs. non-polar mode v . b) P vs. E . c) P vs. D . d) D vs. E . All regionsare locally stable at fixed E ; globally stable regions in black;other regions in red. order, can be minimized over analytically, resulting in aneffective coupling between v and E : F ( v, E ) = − av + bv − cv E − χ e E . (3)In Fig. 4, we plot P vs D and D vs E for this modelwith typical parameters. Similar to hyperferroelectrics,improper ferroelectrics allow for a non-zero polarizationat D = 0; however, the overall shape of the curves isvery different. In particular, improper ferroelectrics lacka structure with D = P = 0. This reflects the fact ourmodel of improper ferroelectrics always has a barrier tohomogeneous switching via external field ( ∂P/∂ E > D = 0 boundary conditions. At this tempera-ture, T D , the LO mode becomes unstable and the mate-rial becomes a hyperferroelectric. As a hyperferroelectricgoes through T D under D = 0 boundary conditions, theinverse dielectric constant will diverge, rather than thedielectric constant, which can be understood by compar-ing the D vs. E plots of normal and hyperferroelectrics inFigs. 2(d) and 3(d), respectively. In order to transitionfrom the normal to the hyperferroelectric state, the slopeat the origin of the D vs. E curve, which is equal to thedielectric constant, must pass through zero.In order to demonstrate the consequences of the mostnotable quality of hyperferroelectrics, their ability topolarize under fixed D = 0 boundary conditions, weplace our ABC ferroelectrics in superlattice configura-tions with thick slabs of non-polar
ABC materials. Weexpect that normal ferroelectrics will not polarize in thisgeometry if there are no free charges, as a sufficientlythick non-polar layer will have P = 0, which enforces D = 0 boundary conditions on the ferroelectric, but hy-perferroelectrics will still polarize under these conditions.We consider superlattices consisting of ferroelectric ABC materials combined with non-polar hexagonal
ABC semiconductors, specifically normal ferroelectricNaMgP with non-polar KZnSb and hyperferroelectricLiBeSb with non-polar NaBeSb, as shown in Fig. 5. Weepitaxially strain each superlattice to the in-plane latticeconstant of the non-polar material, allowing the z latticeconstant to relax.As shown in Table II, the normal ferroelectric NaMgPhas essentially no polarization when in a superlattice withan insulating material. We attribute the tiny 10 meVenergy lowering of the 1/7 NaMgP/KZnSb superlatticeto interface effects, as the interfaces between NaMgP andKZnSb consist of single layers of NaZnSb, which as shownin Table I is itself a hyperferroelectric. On the otherhand, a single polarized layer of the hyperferroelectricLiBeSb interfaced with NaBeSb has a significantly lowerenergy and reduced band gap relative to an unpolarizedlayer. A second LiBeSb layer already provides sufficientpolarization to cause the system to become metallic, dueto dielectric breakdown, a field-induced overlap of con-duction and valence bands leading to charge transfer.As already demonstrated, hyperferroelectrics can re-main polarized down to single atomic layers even wheninterfaced with normal insulators. Such quasi-2d ferro-electric systems could have a variety of unusual proper-ties. First, by adjusting the spacing of layers in a su-perlattice, the polarization, well depth, band gap, andinternal electric field could all be tuned. More specu-latively, these superlattice systems could display noveldomain-wall motion or super-fast switching behavior, asthey consist of weakly-coupled ferroelectric layers whichmay allow for easier domain nucleation, and they supporthead-to-head and tail-to-tail domain walls. Also, unlikea normal ferroelectric, which requires asymmetric screen-ing charges to remain polarized, a hyperferroelectric canswitch between two states without the motion of screen-ing charges between its surfaces or interfaces, allowinghyperferroelectric slabs which are terminated by vacuumor by non-polar insulators to be switched via an externalfield. In addition, in contrast to improper ferroelectrics,the primary order parameter of hyperferroelectrics cou-ples directly to an applied electric field, which may allowfor easier switching. Finally, ABC materials could beused to build an all-semiconducting ferroelectric field ef-fect transistor, side-stepping many of the materials diffi-
Ferro. Non-polar Period ∆ E Gap(HS) Gap(FE) P ( E =0) (eV) (eV) (eV) C/m NaMgP KZnSb 1/7 –0.01 0.32 0.35 0.007NaMgP KZnSb 2/6 0 0.69 − − − m m LiBeSb NaBeSb 3/7 –0.25 0.28 m m
LiBeSb NaBeSb 4/6 –0.50 0.32 m m
LiBeSb NaBeSb 1/3 –0.08 0.77 0.46 0.07LiBeSb NaBeSb 2/2 –0.41 0.61 1.02 0.56TABLE II: Properties of superlattices. An n/m
ABC/A (cid:48) B (cid:48) C (cid:48) superlattice consists of n BC atomic layers separated by Aatomic layers, and m B (cid:48) C (cid:48) atomic layers separated by A (cid:48) atomic layers, with A layers at both interfaces. ∆ E is theenergy gained by allowing a polar distortion. ‘Gap(HS)’ and‘Gap(FE)’ are the band gaps for the non-polar and polarphases respectively; m indicates a metal. For insulators, P ( E =0) is the polarization for E = 0 boundary conditions. Li a) b) SbBeNa
FIG. 5: Interfacial region of a) non-polar and b) polar phasesof 1/7 LiBeSb/NaBeSb superlattice. The full supercell hasthree additional unpolarized NaBeSb layers. culties and interface effects that have hampered attemptsto interface ferroelectric oxides with semiconductors [12–14].In conclusion, we have introduced a new class of ferro-electrics, hyperferroelectrics, and we have identified ex-amples among the
ABC hexagonal semiconducting fer-roelectric family. These new ferroelectrics have a varietyof interesting and potentially useful properties, both inthe bulk and as thin films. Furthermore, this work high-lights the benefits of looking beyond well-studied mate-rials systems in the search for functional materials withnovel properties.
Acknowledgments
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