Domain wall contribution to negative capacitance

The 180^◦domains appear as a response to the presence of a depolarization field which destabilizes the polarization. A typical example of imperfect screening is present in superlattices where the ferroelectric layers are sur-rounded by dielectric layers. As discussed in Chapter 5, the thickness of each individual dielectric layer is crucial and governs whether the polar-ization of the SL is continuous throughout or whether domains appear in the ferroelectric slabs. The idea here is to model the dielectric response of a polydomain structure. Three models were used: 1) a Landau-Kittel model [49,50] for mobile abrupt (thin) domain walls, 2) Ginzburg-Landau theory for static and soft (with a gradual change in the polarization pro-file) domain walls and 3) first principles calculations. The two first models were developed by A. Sené and I. Luk’yanchuk, and the first principles cal-culations were carried out by by J.C. Wojdel and J. Íñiguez.

Landau-Kittel model

The Landau-Kittel model allows the contribution from domain-wall mo-tion to the dielectric permittivity to be determined. For an isolated ferro-electric slab of thicknessl_f in zero field, the up- and down-oriented 180^◦

5Note that the larger the domain wall density the lower theT_C^ih. This is due to the fact that when the sample transitions to a polydomain ferroelectric state, two energies are competing:

the energy gain due to the appearance of the spontaneous polarization and the energy cost of the domain walls. The higher the domain wall density, the lower the temperature at which the energy gain from the development of the spontaneous polarization outweights the domain wall energy cost.

6.2 Negative Capacitance

b

a U

P c

P P

TCh TCih

T0 ϵf-1

T 0

ϵd ϵf T<TCh

T<TCih

TCih

<T<T0

Figure 6.4: Phenomenological description of negative capacitance. (a) Sketch of the a ferroelectric-dielectric bilayer system. The dielectric, ferro-electric and metallic layers are shown in green, blue and gray respectively.

(b) Free energy as a function of polarization. The total energy is shown in purple, whereas the individual contributions from the dielectric and ferro-electric layers are shown in green and blue respectively. (c) Temperature dependence of the local dielectric stiffness of the ferroelectric layer cal-culated from phenomenological models with homogeneous polarization state (blue), and with inhomogeneous polarization with static and soft domain walls (red, dashed) and for mobile and abrupt domain walls (red, solid). The dotted part corresponds to the breakdown of the Landau-Kittel model. T_C^ih andT_C^h are the ferroelectric transition temperatures to inho-mogeneous and hoinho-mogeneous states, respectively.T is the bulk transition

domains are of equal widthω, given by the Landau-Kittel square root de-pendence [49, 50]. For high- dielectrics ω ≈ (_⊥/_k)^1/4p

ζ·λ·2ξ·l_f, where_kand_⊥are the bulk lattice dielectric constants parallel and per-pendicular to the polarization, respectively, ξ is the coherence length, λ= 1 +d/(_k⊥)^1/2 andζ≈3.53(see Refs. [42, 55, 81–83]). It requires the dielectric layers to be thick enough⁶, at least thicker than the domain width to allow interfacial stray fields to decay sufficiently. Under the ap-plication of a field, the polarization increases due to the lattice (_k) and domain wall response. The domain wall displacements [81, 82, 85] create a net polarization that leads to a depolarization field. The key result that emerges from the calculations is that this depolarizing field dominates the total local field and therefore leads to NC. The resulting effective dielectric constant of the ferroelectric can be expressed as [82]

f =_k− π

where the first term is the lattice response and the second term is the negative contribution from domain-wall motion. Within the limits of the Landau-Kittel theory (lf/ω large), the second term is dominant. On a microscopic scale, this corresponds to the redistribution of the interfacial stray fields resulting in a negative net contribution to the free energy and thus to the dielectric constant.

The results of the Landau-Kittel model are shown in Fig. 6.4(c) as the full red curve (the dashed region corresponds to the region close toT_C^ih where the model is no longer applicable). Interestingly, the negative di-electric region is enhanced with respect to the monodomain model.

Notice that, although the Landau-Kittel model captures the process of domain wall breathing, the theory breaks down close toT_C^ih since the do-main profile becomes soft. In other words, close toT_C^ih, the hypothesis of abrupt domain walls is not valid anymore, since the width of the domain is comparable to the lateral size of the wall. Although the theory for mobile domain walls in this regime is challenging, the lattice response of a poly-domain structure can be calculated analytically from a Ginzburg-Landau model (at least far fromT_C^ih).

6Note, that as a starting point, the ferroelectric layer should be thick enough for the Kittel approximation to be valid [84].

6.2 Negative Capacitance

Ginzburg-Landau model

The Ginzburg-Landau model [43, 86] allows us to obtain the contribution of a static and soft domain structure to the dielectric response and the results are shown in Fig. 6.4(c) by the red dashed line. The presence of soft domain structure results in a qualitatively different⁻¹_f (T)curve, with its minimum value pushed below the actualT_C^ih. It shows a reduced region of negative response with respect to the monodomain Landau model.

Atomistic model

In addition to the Landau-Kittel and Ginzburg-Landau models, atomistic calculations were performed. The details of the calculations are beyond the scope of this thesis; here we discuss only the main results. A first-principles-based effective model was used permitting the treatment of thermal effects. As the superlattice is cooled from high temperatures, thec/aratio of the PbTiO₃ layers evidences an elastic transition at about 490 K. This reflects the onset of local instantaneous ferroelectric order.

Then, for temperatures in between 490 K and 370 K, a state characterized by strong fluctuating ferroelectric domains is observed. This fluctuating phase could be indicative of temperature-induced domain melting. A sim-ilar phase has been observed in high-temperature superconductors where the vortex lattice was found to melt [87]. It is then only below 370 K that the static multidomain ferroelectric state freezes in. In addition, calcula-tions of the local dielectric response and susceptibility were performed. It is worth mentioning here that, while the typical definition of the macro-scopic dielectric susceptibility is χ = ¹

0

∂hPi

∂Eext, we might as well define the local susceptibilityχ_i as being the local response to an external field which induces polarization changes that are parallel to it: χi = ¹

0

∂hPii

∂E_ext. This quantity is always positive confirming the expectation that an applied external field will create dipoles parallel to it. By contrast, the local di-electric constanti=^∆D

0∆E_i, measures the response to a local field, taking into account depolarization fields resulting in a more challenging physi-cal interpretation as discussed in Ref. [81]. Calculations of the lophysi-cal di-electric constants show thati = tot/(tot−χi), withtot the dielectric constant of the whole system. Therefore for χi > tot, i is found to be negative. Physically, this means that regions of negative capacitance are those which are more responsive than the system as a whole. The results are displayed in Fig. 6.5(a) at a temperature of 320 K. The susceptibility is found to be much larger at the domain walls than at the domains. In

Figure 6.5: Results of Monte Carlo simulations of a first-principles-based model for the (8,2) superlattice. (a) Local dielectric susceptibilityχi map at 320 K. The arrows indicate the dipole component within the (-110) plane. (b)1/f of the PbTiO3layers.

other words, the field-induced polarization of the walls, resulting from the growth/shrinkage of the domains, dominates the response. Additionally, the response is also found to be enhanced in the vicinity of the interfaces with the SrTiO3layers. Hence, the domain-wall region near the interfaces are found to dominate the negative capacitance of the PbTiO3layers. The calculated temperature behavior of the local dielectric constant⁷ in the PbTiO3layers is shown in Fig. 6.5(b) confirming the presence of a region of negative capacitance.

As shown in Fig. 6.5(a), the dipoles form closure domains, indicat-ing that they almost do not penetrate into the SrTiO3. In other words, it corresponds to the vanishing of the polarization at the surface of the poly-domain ferroelectric.

In spite of the very different assumptions of these methods, all the results display a region of negative capacitance. Let’s now discuss the experimental results and then compare them to the theoretical work.

7Note that this local dielectric constant is an average over the entire PbTiO3layers. The locally resolved response has been calculated and is found to show a negative character in the vicinity of the the PbTiO3-SrTiO3 interfaces at about 550 K way before any ordering;

while the whole layer continues to be positive.

6.2 Negative Capacitance