Effect of a built-in electric field in asymmetric ferroelectric tunnel junctions
EEffect of a built-in electric field in asymmetric ferroelectric tunnel junctions
Yang Liu,
1, 2, ∗ Xiaojie Lou, Manuel Bibes, and Brahim Dkhil Multi-disciplinary Materials Research Center, Frontier Institute of Science and Technology,Xi’an Jiaotong University, Xi’an 710054, People’s Republic of China Laboratoire Structures, Propri´et´es et Mod´elisation des Solides, UMR 8580 CNRS-Ecole Centrale Paris,Grande Voie des Vignes, 92295 Chˆatenay-Malabry Cedex, France Unit´e Mixte de Physique CNRS/Thales, 1 Av. A. Fresnel, Campus de l’Ecole Polytechnique,91767 Palaiseau and Universit´e Paris-Sud, 91405 Orsay, France
The contribution of a built-in electric field to ferroelectric phase transition in asymmetric fer-roelectric tunnel junctions is studied using a multiscale thermodynamic model. It is demon-strated in details that there exists a critical thickness at which an unusual ferroelectric-“polarnon-ferroelectric”phase transition occurs in asymmetric ferroelectric tunnel junctions. In the “polarnon-ferroelectric”phase, there is only one non-switchable polarization which is caused by the compe-tition between the depolarizing field and the built-in field, and closure-like domains are proposed toform to minimize the system energy. The transition temperature is found to decrease monotonicallyas the ferroelectric barrier thickness is decreased and the reduction becomes more significant for thethinner ferroelectric layers. As a matter of fact, the built-in electric field does not only result insmearing of phase transition but also forces the transition to take place at a reduced temperature.Such findings may impose a fundamental limit on the work temperature and thus should be furthertaken into account in the future ferroelectric tunnel junction-type or ferroelectric capacitor-typedevices.
INTRODUCTION
Ferroelectric (FE) tunnel junctions (FTJs) that arecomposed of FE thin films of a few unit cells sandwichedbetween two electrodes (in most cases the top and bottomelectrodes are different) have attracted much more atten-tion during the last decade. [1–3] It is generally believedthat the interplay between ferroelectricity and quantum-mechanical tunneling plays a key role in determiningtunnel electroresistance (TER) or tunneling current andTER effect usually takes place upon polarization rever-sal. Due to the strong coupling of FE polarization andthe applied field, the electric-field control of TER or tun-neling current, [1–11] spin polarization, [12–26] and elec-trocaloric effect [27] can be achieved, which makes FEspromising candidates for nondestructive FE storage, [1–10] FE memristor, [11] spintronics (magnetization), [12–26] or electrocaloric [27] devices. Meanwhile, another me-chanically (including strain or strain gradient) inducedTER is found recently, which also shows their potentialapplications in mechanical sensors, transducers and low-energy archive data storage decices. [28] Note that havingdifferent electrodes for the FTJs (some experiments useconductive atomic force microscope tips instead of thetop electrodes) is usually required for a large effect atlow bias voltage though the FTJs with same electrodesmay also display interesting performances. [2, 4, 9, 18]Also note that all the functionalities in these devicesare strongly related to the thermodynamic stability andswitching ability of FTJs. [1–28] Therefore, a fundamen-tal understanding of ferroelectricity of FTJs, especiallytheir size effects, is crucial at the current stage of re-search.Unfortunately, no consensus has been achieved on whether there exists a critical thickness h c below whichthe ferroelectricity disappears in FTJs, especially forthose with different top/bottom electrodes. It is believedthat an electrostatic depolarizing field caused by dipolesat the FE-metal interfaces is responsible for the size ef-fect. [29–34] However, recent theoretical studies suggestthat the choice of electrode material may lead to smearingof size effect or even vanishing of h c . [35–39] For exam-ple, it was reported that choosing Pt as electrodes wouldinduce a strong interfacial enhancement of the ferroelec-tricity in Pt/BaTiO (BTO)/Pt FTJs, where h c is only0.08 BTO unit cell. [35] In addition, the results of a mod-ified thermodynamic model [36, 37] and first-principlescalculations [38, 39] both indicate that BTO barrier withdissimilar electrodes, i.e. Pt and SrRuO (SRO) elec-trodes, might be free of deleterious size effects. In con-trast, it has been reported that asymmetric combinationof the electrodes (including the same electrodes with dif-ferent terminations) will result in the destabilization ofone polarization state making the asymmetric FTJs non-FE. [33, 40] And the up-to-date studies reported thatthe fixed interface dipoles near the FE/electrode inter-face is considered the main reason for that detrimentaleffect. [41, 42] Considering the importance of the physicsin FTJs with dissimilar top and bottom electrodes, weare strongly motivated to investigate the size effect insuch asymmetric FTJs.It was pointed out as early as 1963 that the con-tribution of different electronic and chemical environ-ments of the asymmetric electrode/FE interfaces wouldinduce a large long-range electrostatic built-in electricfield (cid:126)E bi in FE thin films. [43] (cid:126)E bi becomes more sig-nificant in asymmetric FTJs and should be taken in toaccount. [33, 34] In this study, we use a multiscale ther- a r X i v : . [ c ond - m a t . m t r l - s c i ] J un modynamic model [27, 33, 34] to investigate the effectof such built-in electric field on the phase transition ofasymmetric FTJs by neglecting the short-range interfacedipoles. As a result, we discover an unusual FE-“polarnon-FE”phase transition in asymmetric FTJs. Then, wemake detailed analysis of the contribution of the built-inelectric field to FE phase transition, i.e. h c , what hap-pens below h c , transition temperature T c , and tempera-ture dependence of dielectric response of the asymmetricFTJs. MULTISCALE THERMODYNAMIC MODEL FORTHE FTJS
FIG. 1. Schematic configurations of the system consideredin the present calculations for the the asymmetric FTJs: P + state (a); P − state (b); (cid:126)E bi (c) and the corresponding poten-tial profile at zero polarization (red line) (d). We concentrate on a short-circuited (001) single-domain FE plate of thickness h sandwiched between dif-ferent electrodes. The FE films are fully strained andgrown on thick (001) substrate with the polar axis lyingnormal to the FE-electrode interfaces. [32–34] We denotethe two interfaces as 1 and 2, with surface normals (cid:126)n and (cid:126)n = − (cid:126)n pointing into the electrodes. The config-urations are schematically shown in Fig. 1. The exactvalue and direction of (cid:126)E bi can be determined as [33, 34] (cid:126)E bi = − ∆ ϕ − ∆ ϕ h (cid:126)n = − δϕh (cid:126)n and (cid:126)n = (cid:126)n = − (cid:126)n , (1)where ∆ ϕ i which is the work function steps for FE-electrode i interface at zero polarization is simply definedas the potential difference between the FE and the elec-trode i . [33, 34] With the help of first-principles calcula-tions, one could easily obtain ∆ ϕ i through the analysisof the electrostatic potential of FTJs where FE films arein the paraelectric (PE) state. [33, 34] Then, the free energy per unit surface of the FE layeris presented as [33, 34] F = h Φ + Φ S = ( 12 α ∗ P + 14 α ∗ P + 16 α P + 18 α P + u m S + S − (cid:126)E dep · (cid:126)P − (cid:126)E bi · (cid:126)P − (cid:126)E · (cid:126)P ) h + ( ζ − ζ ) (cid:126)n · (cid:126)P + 12 ( η + η ) P , (2)where α ∗ i are Landau coefficients. [27] u m is the epitaxialstrain and S mn are the elastic compliances coefficients. ζ i and η i are the first order and second order coeffi-cients of the surface energy Φ S expansion for the twoFE-electrode interfaces. [33, 34] (cid:126)E is the applied electricfield along the polar axis. (cid:126)E dep is the depolarizing fieldwhich can be determined from the short-circuit conditionsuch that: [33, 34] (cid:126)E dep = − λ + λ hε + ( λ + λ ) ε b (cid:126)P , (3)where ε is the permittivity of vacuum space, and ε b in-dicates the background (i.e. without contribution of thespontaneous polarization) dielectric constant. λ i are theeffective screening lengths of the two interfaces and aredependent on the polarization direction if the electronicand chemical environments of FE/electrode interfaces aredifferent. [33, 34] For the two opposite polarization ori-entations, the direction dependence of λ i will induce theasymmetry in potential energy and hence will producethe TER effect, besides the depolarizing field effect dueto the polarization difference between two opposite ori-entations. [1–3] However, we ignore such an effect due tothe lack of information about the direction dependenceof λ i and we mainly focus on the role of the built-in fieldin this study. Note that δϕ and δζ = ( ζ − ζ ) are thick-ness and polarization independent and (cid:126)E bi is indeed along-range internal-bias field which has the effect of pol-ing the FE film. [33, 34, 43] In asymmetric FTJs, suchasymmetry parameters δϕ and δζ can introduce a po-tential energy profile difference and therefore induce theTER effect. [1–3]The equilibrium polarization can be derived from thecondition of thermodynamic equilibrium: ∂F∂P = 0 . (4)The dielectric constant ε under an applied field E whose direction is along the polar axis can be determinedas: [37] ε = 1 hε ( ∂ F∂ P ) − . (5)The multiscale thermodynamic model used in thisstudy combines first-principles calculations and phe-nomenological theory and its detailed description canbe found elsewhere. [33, 34] In the previous study, itis reported that (cid:126)E bi could result in a smearing of thephase transition and an internal-bias-induced piezoelec-tric response above T c in asymmetric FTJs. [33] However,adding to the forgoing controversy on the size effects, fur-ther analysis of the effect of built-in field on the FE tran-sition in asymmetric FTJs is still absent. Inserting Eq.(1) into Eq. (2) results in a term that encompasses an oddpower of the polarization: (cid:126)E bi · (cid:126)P , which leads to asym-metric thermodynamic potentials. We shall show thatthis term which behaves mathematically as identicallyas the phenomenological term suggested by Bratkovskyand Levanyuk [44] will result in an unusual FE-“polarnon-FE”phase transition in asymmetric FTJs. RESULTS AND DISCUSSIONSize effects
For a quantitative analysis, we consider a fully strainedBTO film sandwiched between Pt (electrode 2) and SRO(electrode 1) epitaxially grown on (001) SrTiO sub-strates. We neglect the energy difference of the asym-metric surfaces, i.e. by setting ζ = ζ , η = η , to in-sure that the effect of (cid:126)E bi is clearly observable from thecalculations since it is reported that surface effects aregenerally much smaller than that of (cid:126)E bi . [36, 37] All theparameters we used are listed in Ref. 65. We first exam-ine the effect of (cid:126)E bi on the ferroelectricity of asymmetricFTJs. Previous studies indicate that the direction of (cid:126)E bi in asymmetric Pt/BTO/SRO FTJs points to Pt electrodewith higher work function. [36–38] All recent results showindeed that a strong preference for one polarization statenamely P + while P − disappears at “ h c ”. [36–42] Accord-ing to the definition of ferroelectricity, the spontaneouspolarization of the FE materials is switchable under anac electric field. [45] However, knowing that the sponta-neous polarization of FE materials is switchable underan ac electric field, [45] recent reports [36–39] are ratherconfusing and remain incomplete on this point. Indeed,in addition to the forementioned divergence in the size ef-fects, two different transition temperatures at which thetwo polarization states reach zero are obtained (see Ref.37), which may be confusing since there should be onlyone finite phase transition temperature for disappearanceof ferroelectricity. In order to avoid such confusions, weused the classical definition of ferroelectricicity [45] in thefollowing parts.We make further analysis of the physical formulationof h c in asymmetric FTJs. Note that Eq. (4) is a non-linear equation and yields “at most”three solutions P ,two of them corresponding to minima and the other oneto a saddle point (unstable state). Whether the solutionis a minimum, a maximum or a saddle point can be re-vealed through inspecting the eigenvalues of the Hessian FIG. 2. Schematic representation of the variation in totalfree energy with respect to polarization with different BTObarrier thicknesses in asymmetric Pt/BTO/SRO tunnel junc-tions with/without consideration of the built-in field in zeroapplied field. matrix of the total free energy F . Because the asymmet-ric FTJ is internally biased, i.e., the energy degeneracybetween positive P + and negative P − is lifted, one of theminima corresponds to the equilibrium state (the globalminimum) of the system (the direction of which is along (cid:126)E bi ) and the other minima corresponds to a metastablestate (a local minimum) of the system. It means thatthe presence of two different electrodes in asymmetricFTJs results in a preferred polarization orientation ofthe FE plate. Having found all P solutions as a func-tion of h , one can clearly see that metastable state andunstable state solutions become closer to each other andcoincide at finite h , henceforth the number of solutions P drops from three to one. According to the bistableproperty of FE materials, this finite h is just h c . [45] Aslong as there are three P solutions: two of these threesolutions correspond to stable/metastable polarizationsso that two orientations of polarization are possible inthe BTO layer and thus it is FE. Switching the asym-metric FTJ into its unfavoured high energy polarizationmay be difficult. If there is the only P solution corre-sponding to the unstable state, although it attains a fi-nite value, it is not FE anymore and may be called “PE”.Indeed it would be more appropriate to consider it as“polar non-FE”since P has a unique finite value. [46–48]FTJs with no built-in field δϕ = 0 will exhibit two en-ergetically equivalent stable polarization states ( P + and P − ) along with an unstable polarization state at P = 0below h c . All the forgoing discussions can be clearlyand easily understood in the schematic representation of F − P curves with different BTO thicknesses as shown inFig. 2 which is quite similar with the results of FE thinfilms with/without consideration of the fixed interfacedipoles near the asymmetric FE/electrode interface [40–42] or FE superlattices with/without interfacial spacecharges. [48] Together with previous results, [41, 42] weconclude that no matter the (cid:126)E bi is considered as a long-range field or a short-range surface one, it cannot inducethe vanishing of h c in asymmetric FTJs, which is in con-trast with other works. [36–39]Note that “polar non-FE”phase is actually a pyroelec-tric phase because there is a non-switchable polarizationin this phase. This kind of phase transition has oncebeen reported in FE thin films with asymmetric elec-trodes [33, 40–42] or FE superlattices with interfacialspace charge. [46–48] As we discussed in the formationof h c , “polar non-FE”phase indeed always correspondsto the unstable state (see Fig. 2) and this kind of non-switchable polarization may not be stable at all. How-ever, breaking up the system into 180 domain stripes isunambiguously ruled out due to the long-range pinnedfield (cid:126)E bi . In plane vortex formation [50, 51] is also in-hibited because the large compressive strain favors more180 domain stripes. [51] The ferromagneticlike closuredomains are predicted to form in ultrathin FE films orFE capacitors even below h c [52–54] and are experimen-tally confirmed well above h c recently. [55, 56] However,typical FE closure domains [52–56] are also not expectedin “polar non-FE”phase where 180 domains in the clo-sure domain structure should be suppressed. But localrotations of non-switchable polarization ( < ) are stilllikely to occur and result in a closure-like domain struc-ture since the local change of the direction of the non-switchable polarization especially near the FE/electrodeinterface is helpful to minimize the system energy. [52–54] Although such closure-like domains can be favoredbelow h c ( < h c cannot be used forFE memory applications in which two thermodynamicstable polarization states are needed to encode “0”and“1”in Boolean algebra. [29–34, 45] However based on ourcalculation, one should expect a resistance change below h c between the non-switchable polarization state and theother one being ferroelectrically dead. This result agreeswell with recent works on Pt/BTO/Pt FTJs that evenbelow h c , the resistance of the FTJ would change by afactor of three due to the interface bonding and barrierdecay rate effects. [4] We argue that the TER effect below h c suggested in our work may be essentially attributed tothe asymmetric modification of the potential barrier bythe nonzero barrier height ( − δϕ ) (see Eqs. (1)-(4)) whicheven exists at zero polarization as shown in Fig. 1(d).Further theoretical and experimental efforts should be made to confirm these predictions. FIG. 3. Spontaneous polarization of the asymmetricPt/BTO/SRO tunnel junctions as a function of BTO layerthickness with δϕ =0 V, -0.092 V, -0.2 V, -0.3 V and 0.4 Vin zero applied field at 0 K, respectively. The results of sym-metric SRO/BTO/SRO and Pt/BTO/Pt tunnel junctions at0 K [27] are also added for comparison. The quantitative results of the forgoing analysis are di-rectly given in Fig. 3. It can be seen that h c exists regard-less of symmetric or asymmetric structures. As expected,the curves of P + and P − are symmetric with respect to P = 0 at δϕ = 0 where h c is about 1 nm which is smallerthan that of SRO/BTO/SRO, i.e. 1.6 nm. [27, 35] When δϕ (cid:54) = 0, the supposed degeneracy between P + and P − occurs, i.e. P + is enhanced while P − is reduced so thecoordinate of the center of the hysteresis loop along thepolarization axis [1/2( P + + P − )] is shifted along the di-rection of P + . It is shown that such a displacement of thehysteresis loop along the polarization axis becomes moresignificant as the strength of (cid:126)E bi increases. It may be at-tributed to the imprint caused by (cid:126)E bi such that the wholeshape of the hysteresis loop will shift along the directionof the field axis which is antiparallel to the direction of (cid:126)E bi . [45] Besides, it is found that as δϕ increases, h c in-creases, which indicates that (cid:126)E bi can enhance the size of h c . Thus, whether h c of Pt/BTO/SRO junction is largeror smaller than that in the SRO/BTO/SRO counterpartstrongly depends on exact value of δϕ as shown in Fig. 3.For the symmetric structures (SRO/BTO/SRO andPt/BTO/Pt FTJs), one can easily see in Fig. 3 that singledomain in the FE layer destabilizes as the film thicknessis decreased due to the depolarizing field effect. [27, 29–34] And it is shown in Fig. 3 that Pt/BTO/Pt FTJ whose h c is merely 0.08 BTO unit cell is nearly free of deleteri-ous size effects, [27] which agrees well with the result offirst-principles calculations. [35] h c of SRO/BTO/SROFTJ is about four BTO unit cells, which is consistentwell with our previous results. [27] The qualitative re-sult that h c of Pt/BTO/Pt FTJ is smaller than thatof SRO/BTO/SRO FTJ in this work is consistent wellwith those of first-principles calculations [35] and latticemodel. [57] However, our results are in contrast with pre-vious works [31, 36, 37] predicting h c of SRO/BTO/SROFTJ to be smaller than that of Pt/BTO/Pt FTJ. Inthese previous works, [36, 37] Mehta et al ’ electrostatictheory about the depolarizing field ( (cid:126)E dep = − (cid:126)Pε b (1 − h/ε b l s /ε e + l s /ε e + h/ε b ) where l s and l s are Thomas-Fermiscreening lengths and ε e and ε e are dielectric constantsof electrode 1 and 2) is used [29] while in our work weused the “effective screening length”model to describethe depolarizing field (see Eq. (3)). Note that we usedthe same parameters as Refs. 36 and 37 except for themodel of depolarizing field. [65] The distinct results areunderstandable since it is generally accepted that imper-fect screening should be characterized by effective screen-ing length (See Eq. (3)) rather than Thomas-Fermi onein Mehta et al ’ model. [49] In fact, the effective screen-ing length at Pt/BTO interface is only 0.03 ˚A [35] muchsmaller than that of Thomas-Fermi one ∼ h c in Pt/BTO/Pt FTJs. Pre-vious study attributes this freedom of size effects in thePt/BTO/Pt structure to the “negative dead layer”nearthe Pt/BTO interface, [35] while we argue that it mayresult directly from the fact that the effective screeninglength of Pt electrode is extremely small since Bratkovskyand Levanyuk suggested the “dead layer”model is totallyequivalent as to consider an electrode with a finite screen-ing length. [58] Here we ignore the effect of the extrinsic“dead layer”formed between metal electrode (i.e. Au orPt) and a perovskite FE (i.e. Pb(ZrTi)O or BTO). In-deed, Lou and Wang found that the “dead layer”betweenPt and Pb(ZrTi)O is extrinsic and could be removedalmost completely by doping 2% Mn. [59] Experimen-tally, many researchers found that SRO/BTO/SRO ca-pacitors (as well as other perovskite FE structures withconductive oxide electrodes) are free from passive lay-ers. [31, 60, 61] Recently, a very interesting experimentalresult demonstrates that the RuO /BaO terminations atBTO/SRO interface, which is assumed as many pinnedinterface dipoles and plays a detrimental role in stabiliz-ing a switchable FE polarization, can be overcome by de-positing a very thin layer of SrTiO between BTO layerand SRO electrode. [41, 42] Nonetheless, it is still un-clear whether such pinned interface dipoles are intrinsicand can be found in other FE/electrode interfaces (i.e.SRO/PbTiO and Pt/BTO).In the asymmetric structures in Fig. 3, it is shown thatin comparison with δϕ = 0 in asymmetric Pt/BTO/SROFTJs, h c is significantly enhanced, as δϕ increases, whichis in good agreement with the recent results regarding (cid:126)E bi as short-range interface field, [41] and is similar withthe previous results. [37] Note that δϕ is intrinsic anddetermined strictly by the electronic and chemical envi- FIG. 4. (a): Polarization state P ± of the asymmetricPt/BTO/SRO tunnel junctions as a function of BTO layerthickness with δϕ =-0.1 V and -0.2 V in zero applied field at0 K, respectively. The dash lines mark the boundary betweenpolar non-FE and FE phases for different values of δϕ ; (b):Dependence of the strengths of the built-in field E bi and de-polarizing field for different directions, E dep + and E dep − onthe BTO layer thickness with δϕ =-0.1 V at 0 K. ronments of FE/electrode interfaces but not by any po-tential drop through the FTJ which “creates”an appliedfield. [33, 34, 43, 44] Changing δϕ is simply due to thelack of its exact value and for the purpose of studyingthe effect of (cid:126)E bi in asymmetric FTJs, which is similar tothe previous method. [36, 37] This method [36, 37] in-deed does not mean that any asymmetric electrodes areconsidered here, since the electrode is replaced, the elec-trode/FE interface parameters in Eqs. (1)-(3) such as λ i and other interface parameters will also change. Thevariation of P + in Pt/BTO/SRO FTJs as a function ofthe whole BTO layer thickness with δϕ =-0.1 V and -0.2V at 0 K is shown in Fig. 4(a). It is found that below thecritical thickness P + state shows an interesting recoveryof a polar non-FE polarization, in contrast to P − state(see Fig. 4(a)), becoming less significant when δϕ ∼ < -0.2V. Note that such recovery has been reported in FE su-perlattices with asymmetric electrodes and demonstratedto be independent of the interfacial space charge. [46]Although such a recovery of polar non-FE polarizationin BTO layer does not mean the recovery of ferroelec-tricity as it is not switchable, it is necessary to realizeits origin. We plot the build-in field E bi and depolar-izing field for different directions, E dep + and E dep − , asa function of the BTO film thickness considering δϕ =-0.1 V as an example in Fig. 4(b). For the condition of P − state as schematically illustrated in Fig. 1(b), E dep − shows the typical behavior as the FTJ with the same elec-trodes, [30–32, 57] which means that E dep − plays a keyrole forcing the single domain in the FE layer to destabi-lize as the film thickness is decreased. E bi with the samedirection of E dep − helps then to speed up such destabi-lization, therefore enhancing the critical thickness. Forthe P + state, E dep + and E bi are in the opposite direc-tions, as depicted in Fig. 1(a), and both the strengths of E dep + and E bi increase as the BTO layer thickness is de-creased (Fig. 4(b)), which means that E dep + is partiallycancelled by E bi . The strength of this partial compensa-tion becomes stronger with the film thickness decreasing(see the slopes of E bi − h and E dep + − h curves)(Fig. 4(b)).Therefore, E bi is fighting against E dep + allowing the po-larization to recover into a polar non-FE polarization.This recovery of polar non-FE polarization forces the sys-tem to a higher energy state which strongly supports ourforgoing predictions of local rotations of non-switchablepolarization ( < ) and the formation of closure-like do-main structure to minimize the system energy. FIG. 5. The variation in critical thickness h c in epitaxialasymmetric Pt/BTO/SRO tunnel junctions as a function of − δϕ at three different temperatures: 0 K, 300 K, and 600 K,respectively ( E =0 kV/cm). The critical thickness h c under different ambient tem-peratures T as a function of ( − δϕ ) in asymmetricPt/BTO/SRO FTJs is shown in Fig. 5. It can be seenthat h c decreases with T increasing. And it is found thatfor other T , the asymmetric Pt/BTO/SRO FTJs show asimilar behavior of enhancement of h c by increasing thestrength of (cid:126)E bi as shown in Fig. 3 at 0 K. Transition temperature and dielectric response
The transition temperature T c of the asymmetric FTJsis extremely important, especially for the device appli-cations. Fig. 6 summarizes T c as a function of BTOthickness in epitaxial asymmetric Pt/BTO/SRO FTJsat various values of δϕ . It is shown that T c in asymmet-ric Pt/BTO/SRO FTJs monotonically decreases with theBTO layer thickness decreasing, which is similar to thebehavior of symmetric SRO/BTO/SRO or Pt/BTO/PtFTJs. [27] Moreover, T c decreases more significantly for thinner BTO barrier layer thickness (see the slope of T c − h curves in Fig. 6). At a given BTO layer thickness,it is found in Fig. 7 that T c decreases as δϕ becomesmore negative, which means a larger built-in field canforce the phase transition to occur at lower temperatures.The transition temperature T c is strongly sensitive to the δϕ change especially for the thinner BTO barrier (see theslope of T c − ( − δϕ ) curves in Fig. 7). It can be clearly seenthat the FE transition temperature is suppressed as thebuilt-in field is increased for different BTO thicknesses.Usually, the TER effect is always significantly larger forthicker barrier with larger polarization. [3, 5] Here wefind that a fundamental limit (which is more drastic forthinner FE barrier thickness) on the work temperatureof FTJ-type or capacitor-type devices should also be si-multaneously taken into account together with the FEbarrier thickness or polarization value. In addition andinterestingly, since the electrocaloric effect is always thestrongest close to the FE-PE transition, [62] such tun-ing of T c by (cid:126)E bi should be also considered in potentialasymmetric FTJs for the room temperature solid-staterefrigeration. [27] Moreover, the fact that large tunnel-ing current in asymmetric FTJs [6] results in significantJoule heating should also be included in the design offuture devices. FIG. 6. The transition temperature T c as a function of BTOlayer thickness in epitaxial asymmetric Pt/BTO/SRO tunneljunctions at various values of δϕ with no applied field. Theresults of symmetric SRO/BTO/SRO and Pt/BTO/Pt tunneljunctions [27] are also provided for comparison. The dielectric response ε + ( (cid:126)E is parallel to (cid:126)E bi ) and ε − ( (cid:126)E is antiparallel to (cid:126)E bi ) of Pt/BTO/SRO FTJs (con-sider the 5-nm-thick BTO film as an example) as a func-tion of T at different δϕ is shown in Figs. 8(a) and (b).Several key parameters with different δϕ in Fig. 8(a) areextracted in Table. I: T max corresponds to the tempera-ture where ε + reaches its maximum, ε max + ; ε min + sim-ply means the minimal value of ε + ; δε d is in somehow thediffuseness of the transition. It can be seen that when FIG. 7. The transition temperature T c as a function of ( − δϕ )in epitaxial asymmetric Pt/BTO/SRO tunnel junctions withthree different BTO layer thicknesses: 2 nm, 4 nm and 6 nm,respectively ( E =0 kV/cm).FIG. 8. Dielectric constants ε + (a) and ε − (b) as a func-tion of temperature T at various values of δϕ in asymmetricPt/BTO/SRO tunnel junctions where BTO layer thickness h is 5 nm ( E =100 kV/cm). The dash lines mark the boundarybetween polar non-FE and FE phases for different values of δϕ . δϕ = 0, ε + shows a sharp peak near T c . However, agradual decrease in ε max and δε d is seen upon increasing ε + which is well consistent with the results of smearingof T c by increasing ε + in Fig. 6 (see the slope of T c − h curves in Fig. 6). The diffusive transition response in ε + clearly shows smearing of the phase transition as aresult of (cid:126)E bi , which verifies the predictions of Tagantsev et al [33, 34] and Bratkovsky et al . [44] In addition, it isshown T max is shifted to higher temperatures due to (cid:126)E bi .As the strength of (cid:126)E bi increases, the smearing of phasetransition and the shift of T max becomes more signifi-cant. On the other hand, the applied field cannot fullycompensate the built-in field, resulting in a discountin- TABLE I. The different parameters extracted from Fig. 8(a). δϕ (V) T max (K) ε max + ε min + δε d = ( ε max + − ε min + ) /
20 1086 1001.4 24.8 513.1-0.1 1223 428.5 23.0 225.7-0.2 1345 285.8 21.4 153.6-0.3 1465 218.3 20.1 119.2-0.4 1574 178.1 18.9 98.5 uous phase transition from FE phase to polar non-FEphase with temperature increasing as depicted in dielec-tric response ε − in Fig. 8(b), which is distinct from thecountinuous counterpart of ε + as shown in Fig. 8(a). P − abruptly changes its sign near the transition point re-sulting a dielectric peak and a similiar smearing of ε − byincreasing the strength of (cid:126)E bi is found. Furthermore, itis found that though the transition temperatures for twodirections are different, they both decrease as the built-infield increases which is consistent with the results with-out any external field (See Fig. 6), which indicates thatthe built-in field forces the transition to take place at areduced temperature. Comments on the built-in field effect
We make further comments on the built-in field ef-fect in asymmetric FTJs. The main assumption in thisstudy is that δϕ does not change during the polariza-tion reversal. [33, 34] The presence of δϕ which results inan asymmetric potential energy and barrier height dif-ferences by switching the polarization will induce theTER effect. [1–3] Note that the switching of the polar-ization in the asymmetric FTJs may change the value of δϕ . [4, 18, 40, 63, 64] However, according to our anal-ysis, the variation in δϕ (even changing its sign occursduring the polarization reversal) does not alter the mainresults of this study due to its induced broken spatialinversion symmetry of FTJs. In addition to the built-infield, if the surface term δζ = ( ζ − ζ ) is nonzero, themain conclusions of this paper will not change as well. CONCLUSIONS
In summary, on the basis of a multiscale thermody-namic model, a detailed analysis of the changes broughtby the built-in electric field in asymmetric FTJs is made.It is demonstrated that the critical thickness does existin asymmetric FTJs. Below the critical thickness, it isfound that there is a recovery of polar non-FE polariza-tion due to strong cancelling of the depolarizing field bythe built-in field, and closure-like domains are proposedto form to minimize the system energy. It is found thatthe built-in electric field could not only induce imprintand a behavior of smearing of the FE phase transitionbut also forces the phase transition to take place at areduced temperature. A fundamental limit of transitiontemperature dependence of the barrier layer thickness onthe work temperature of FTJ-type or FE capacitor-typedevices is proposed and should be simultaneously takeninto account in the further experiments. Hopefully, ourresults will be helpful to the fundamental understandingsof phase transitions in asymmetric FTJs.This work is supported by the Ministry of Science andTechnology of China through a 973-Project under GrantNo. 2012CB619401. The authors gratefully thank Dr.X. Y. Wang, Dr. M. B. Okatan, and Prof. S. P. Alpayfor their fruitful suggestions. Y. Liu is thankful to theMultidisciplinary Materials Research Center (MMRC) atXi’an Jiaotong University for hospitality during his visit.Y. Liu and B. Dkhil wish to thank the China Schol-arship Council (CSC) for funding YL’s stay in France.Y. Liu, M. Bibes and B. Dkhil also acknowledge theAgence Nationale pour la Recherche for financial supportthrough NOMILOPS (ANR-11-BS10-016-02) project. X.J. Lou would like to thank the “One Thousand YouthTalents”program for support. ∗ [email protected][1] E. Y. Tsymbal, A. Gruverman, V. Garcia, M. Bibes, andA. Barth´el´emy, MRS Bulletin , 138 (2012).[2] H. Kohlstedt, N. A. Pertsev, J. Rod´ıguez Contreras, andR. Waser, Phys. Rev. B , 125341 (2005).[3] M. Y. Zhuravlev, R. F. Sabirianov, S. S. Jaswal, and E.Y. Tsymbal, Phys. Rev. Lett. , 246802 (2005).[4] J. P. Velev, C.-G. Duan, K. D. Belashchenko, S. S.Jaswal, and E. Y. Tsymbal, Phys. Rev. Lett. , 137201(2007).[5] V. Garcia, S. Fusil, K. Bouzehouane, S. Enouz-Vedrenne,N. D. Mathur, A. Barth´el´emy, and M. Bibes, Nature(London) , 81 (2009).[6] A. Gruverman, D. Wu, H. Lu, Y. Wang, H. W. Jang,C. M. Folkman, M. Y. Zhuravlev, D. Felker, M. Rz-chowski, C. B. Eom, and E. Y. Tsymbal, Nano Lett. ,3539 (2009).[7] A. Crassous, V. Garcia, K. Bouzehouane, S. Fusil, A. H.G. Vlooswijk, G. Rispens, B. Noheda, M. Bibes, and A.Barth´el´emy, Appl. Phys. Lett. , 042901 (2010).[8] A. Chanthbouala, A. Crassous, V. Garcia, K. Bouze-houane, S. Fusil, X. Moya, J. Allibe, B. Dlubak, J. Grol-lier, S. Xavier, C. Deranlot, A. Moshar, R. Proksch, N. D.Mathur, M. Bibes, and A. Barth´el´emy, Nat. Nanotech-nol. , 101 (2011).[9] D. I. Bilc, F. D. Novaes, J. ´I˜niguez, P. Ordej´on, and P.Ghosez, ACS Nano , 1473 (2012).[10] D. Pantel, H. D. Lu, S. Goetze, P. Werner, D. J. Kim, A.Gruverman, D. Hesse, and M. Alexe, Appl. Phys. Lett. , 232902 (2012). [11] A. Chanthbouala, V. Garcia, R. O. Cherifi, K. Bouze-houane, S. Fusil, X. Moya, S. Xavier, H. Yamada, C.Deranlot, N. D. Mathur, M. Bibes, A. Barth´el´emy, andJ. Grollier, Nat. Mater. , 860 (2012).[12] M. Y. Zhuravlev, S. S. Jaswal, E. Y. Tsymbal, and R. F.Sabirianov, Appl. Phys. Lett. , 222114 (2005).[13] C.-G. Duan, S. S. Jaswal, and E. Y. Tsymbal, Phys. Rev.Lett. , 047201 (2006).[14] S. Sahoo, S. Polisetty, C.-G. Duan, S. S. Jaswal, E. Y.Tsymbal, and C. Binek, Phys. Rev. B , 092108 (2007).[15] C.-G. Duan, Julian P. Velev, R. F. Sabirianov, W. N.Mei, S. S. Jaswal, and E. Y. Tsymbal, Appl. Phys. Lett. , 122905 (2008).[16] M. K. Niranjan, J. D. Burton, J. P. Velev, S. S. Jaswal,and E. Y. Tsymbal, Appl. Phys. Lett. , 052501 (2009).[17] J. D. Burton and E. Y. Tsymbal, Phys. Rev. B ,174406 (2009).[18] J. P. Velev, C.-G. Duan, J. D. Burton, A. Smogunov, M.K. Niranjan, E. Tosatti, S. S. Jaswal, and E. Y. Tsymbal,Nano Lett. , 427 (2009).[19] M. Y. Zhuravlev, S. Maekawa, and E. Y. Tsymbal, Phys.Rev. B , 104419 (2010).[20] V. Garcia, M. Bibes, L. Bocher, S. Valencia, F. Kronast,A. Crassous, X. Moya, S. Enouz-Vedrenne, A. Gloter, D.Imhoff, C. Deranlot, N. D. Mathur, S. Fusil, K. Bouze-houane, and A. Barth´el´emy, Science , 1106 (2010).[21] M. Hambe, A. Petraru, N. A. Pertsev, P. Munroe, V.Nagarajan, and H. Kohlstedt, Adv. Funct. Mater. ,2436 (2010).[22] S. Valencia, A. Crassous, L. Bocher, V. Garcia, X. Moya,R. O. Cherifi, C. Deranlot, K. Bouzehouane, S. Fusil, A.Zobelli, A. Gloter, N. D. Mathur, A. Gaupp, R. Abrudan,F. Radu, A. Barth´el´emy, and M. Bibes, Nat. Mater. ,753 (2011).[23] H. L. Meyerheim, F. Klimenta, A. Ernst, K. Mohseni, S.Ostanin, M. Fechner, S. Parihar, I. V. Maznichenko, I.Mertig, and J. Kirschner, Phys. Rev. Lett. , 087203(2011).[24] L. Bocher, A. Gloter, A. Crassous, V. Garcia, K. March,A. Zobelli, S. Valencia, S. Enouz-Vedrenne, X. Moya, N.D. Marthur, C. Deranlot, S. Fusil, K. Bouzehouane, M.Bibes, A. Barth´el´emy, C. Colliex, and O. St´aphan, NanoLett. , 376 (2012).[25] J. M. L´opez-Encarnaci´on, J. D. Burton, Evgeny Y.Tsymbal, and J. P. Velev, Nano Lett. , 599 (2011).[26] D. Pantel, S. Goetze, D. Hesse, and M. Alexe, Nat.Mater. , 289 (2012).[27] Y. Liu, X.-P. Peng, X. J. Lou, and H. Zhou, Appl. Phys.Lett. , 192902 (2012).[28] X. Luo, B. Wang, and Y Zheng, ACS Nano , 1649(2011); H. Lu, D. J. Kim, C.-W. Bark, S. Ryu, C. B.Eom. E. Y. Tsymbal, and A. Gruverman, Nano Lett. ,6289 (2012).[29] R. R. Mehta, B. D. Silverman, and J. T. Jacobs, J. Appl.Phys. , 3379 (1973).[30] J. Junquera and P. Ghosez, Nature (London) , 506(2003).[31] D. J. Kim, J. Y. Jo, Y. S. Kim, Y. J. Chang, J. S. Lee,J.-G. Yoon, T. K. Song, and T. W. Noh, Phys. Rev. Lett. , 237602 (2005).[32] N. A. Pertsev and H. Kohlstedt, Phys. Rev. Lett. ,257603 (2007).[33] G. Gerra, A. K. Tagantsev, and N. Setter, Phys. Rev.Lett. , 207601 (2007). [34] A. K. Tagantsev, G. Gerra, and N. Setter, Phys. Rev. B , 174111 (2008).[35] M. Stengel, D. Vanderbilt, and N. A. Spaldin, NatureMater. , 392 (2009).[36] Y. Zheng, W. J. Chen, C. H. Woo, and B. Wang, J. Phys.D: Appl. Phys. , 139501 (2011).[37] Y. Zheng, W. J. Chen, X. Luo, B. Wang, and C. H. Woo,Acta Materialia , 139501 (2012).[38] M.-Q. Cai, Y. Zheng, P.-W. Ma, and C. H. Woo, J. Appl.Phys. , 024103 (2011).[39] M.-Q. Cai, Y. Du, and B.-Y. Huang, Appl. Phys. Lett. , 102907 (2011).[40] Y. Umeno, J. M. Albina, B. Meyer, and C. Els¨asser,Phys. Rev. B , 205122 (2009).[41] X. H. Liu, Y. Wang, P. V. Lukashev, J. D. Burton, andE. Y. Tsymbal, Phys. Rev. B , 125407 (2012).[42] H. Lu, X. Liu, J. D. Burton, C.-W. Bark, Y. Wang, Y.Zhang, D. J. Kim, A. Stamm, P. Lukashev, D. A. Felker,C. M. Folkman, P. Gao, M. S. Rzchowski, X. Q. Pan, C.-B. Eom, E. Y. Tsymbal, and A. Gruverman, Adv. Mater. , 1209 (2012).[43] J. G. Simmons, Phys. Rev. Lett. , 10 (1963).[44] A. M. Bratkovsky and A. P. Levanyuk, Phys. Rev. Lett. , 107601 (2005).[45] J. F. Scott, Ferroelectric Memories (Springer, Berlin,2000).[46] Y. Liu and X.-P. Peng, Applied Physics Express ,011501 (2012).[47] Y. Liu and X.-P. Peng, Chinese Physics Letters ,057701 (2012).[48] M. B. Okatan, I. B. Misirlioglu, and S. P. Alpay, Phys.Rev. B , 094115 (2010).[49] J. Junquera and P. Ghosez, J. Comput. Theor. Nanosci. , 2071 (2008).[50] I. I. Naumov, L. Bellaiche, and H. X. Fu, Nature (Lon-don) , 737 (2004).[51] I. Naumov and A. M. Bratkovsky, Phys. Rev. Lett. ,107601 (2008).[52] I. Kornev, H. X. Fu, and L. Bellaiche, Phys. Rev. Lett. , 196104 (2004).[53] P. Aguado-Puente and J. Junquera, Phys. Rev. Lett. , 177601 (2008).[54] T. Shimada, S. Tomoda, and T. Kitamura, Phys. Rev. B , 144116 (2010).[55] C. T. Nelson, B. Winchester, Y. Zhang, S.-J. Kim, A.Melville, C. Adamo, C. M. Folkman, S.-H. Baek, C.-B. Eom, D. G. Schlom, L.-Q. Chen, and X. Q. Pan, NanoLett. , 828 (2011).[56] C.-L. Jia, K. W. Urban, M. Alexe, D. Hesse, and I. Vre-joiu, Science , 1420 (2011).[57] X. Y. Wang, Y. L. Wang, and R. J. Yang, Appl. Phys.Lett. , 142910 (2009); Y. L. Wang, X. Y. Wang, Y.Liu, B. T. Liu, and G. S. Fu, Physics Letters A, ,4915 (2010).[58] A. M. Bratkovsky and A. P. Levanyuk, J. Comp. Theor.Nanosci. , 465 (2005).[59] X. J. Lou and J. Wang, Journal of Physics: CondensedMatter , 055901 (2010).[60] Y. S. Kim, J. Y. Jo, D. J. Kim, Y. J. Chang, J. H. Lee,T. W. Noh, T. K. Songa, J.-G. Yoon, J.-S. Chung, S. I.Baik, Y.-W. Kim, and C. U. Jung, Appl. Phys. Lett. ,072909 (2006).[61] H. Z. Jin and J. Zhu, J. Appl. Phys. , 4594 (2002).[62] J. F. Scott, Annu. Rev. Mater. Res. , 229 (2011).[63] M. Stengel, P. Aguado-Puente, N. A. Spaldin, and J.Junquera, Phys. Rev. B , 235112 (2011).[64] F. Chen and A. Klein, Phys. Rev. B , 094105 (2012).[65] We used the following set of parameters (in SI units): u m = − . Q = − . ε = 8 . × − , ε b ≈ ε , S = 8 . × − , S = − . × − , η SRO = 0 . η Pt = 0 . λ SRO = 1 . × − , λ Pt = 3 × − . Landau coefficients α ∗ i (the expressionscan be origianlly found in N. A. Pertsev, A. K. Tagant-sev, and N. Setter, Phys. Rev. B , R825 (2000)), elec-trostrictive coefficients, and elastic compliances of BTOat room temperature we used are the same as those inRefs. 27, 36 and 37. The effective screening length λ i and the coefficients of the surface energy expansion η i are taken from Refs. 33, 34 and 35, respecticely. Thebackground dielectric constant ε b is taken from Refs. 36and 37 which is distinct from 7 ε used in Refs. 33 and34. In fact, there is no consensus on its exact value, es-pecially by the theoretical researchers. In this study, wehave made some comparisons with the previous studiesin Refs. 36 and 37, so selecting 50 ε0