Polarization of Perovskite Ferroelectrics

1 Introduction
Ferroelectric films have drawn great interest due to their ability to withstand a macroscopic polarization that can be switched by the action of an electric field. Many applications have been developed based on this property. Particularly, the demands for smaller industrial components have made ferroelectric thin film more and more important, the thickness can be small as 10s of nanometers. Perovskites, as a subclass of ferroelectrics, has aroused lots of concerns due partly to its structure simplicity, with ABO3 form, A and B sitting at the corners and centers of a cube and oxygen atoms at the face center forming a regular octahedron.
Polarization comes from two contributions, electronic contribution and ionic contribution . The latter can be calculated from the point charges directly, while the former part can be now calculated using theories related to polarization. Electric polarization is a key quantity for computing and understanding technologically relevant effective charges, dielectric, and piezoelectric responses that are the derivatives of polarization with respect to atomic displacement, electric field, and strain, respectively. It was also found that the polarization is strongly dependent on strain, response of the polarization dispersion structure to strain could also be an interesting problem.
This film perovskites has a wide application in nanocapacitors, however, the capacitance in reality is much smaller than the expected value from the bulk dielectric constant. Investigation about the polarization as well as dielectric constant was conducted to find out the reason the leads to the capacitance loss, finally a investigation about "deal layer effect" was presented according to the DFT calculations [4].

2 DFT calculations of electric polarization based on KSV formalism

2.1 KSV theory
In paper [1], King-Smith and Vanderbilt proposed that the electrical contribution of polarization corresponds to a geometrical phase of the valence electron states, by
Eq1.png (1)
Eq2.png (2)
is the Berry phase of occupied Bloch wave functions. unk . In Eq. (1), the integral is carried out by a weighted summation of the phases at sampled discrete k points. Φ(k) can be calculated as the phase of the determinant formed by valence states at two neighboring k' s on the kparallel string by:
Eq3.png (3)
Then the total polarization can be determined by both the electronic part and ionic part as the ionic part can be calculated directly from the point charges. Here the dependence of Φ(k) on k is defined as “polarization structure” or “polarization phase structure”, as it describes the Berry’s phase contribution of to the electronic polarization from individual points that are on the Brillouin-zone plane perpendicular to the direction of polarization, density-functional calculations, can be used to study the polarization structure in ferroelectrics. The meaning to investigate polarization structure is multifold.
i. The field-induced variation in Φ(k) enlarges the k dependent polarization current. As a result, the relevance of polarization structure to electronic polarization is similar to the band structure to electronic properties.
ii. Second, Fundamentally, Φ(k) is determined by the Bloch wave functions, not in the ordinary sense of spatial distribution but through the interesting aspects of the Berry’s phase of occupied manifold of electron states, thus it can provide better understanding of electron states, as well as the rather intriguing connection between these states and their contributions to polarization in insulator solids.
iii. From a computational point of view, The work can help find out which k may generate a phase not in the principal range, one in principle should compute the whole dispersion structure of polarization and then map out all the points based on the assumption that the phase is a continuous function of wave vector, which makes it important to study the properties of the phase as a function of wave vector .
2.2 .1 DFT calculations about PbTiO3
Take PbTiO3 for example. As a tetragonal with a1=a2=a and a3=c, polarization is long the c direction, perpendicular to the k plane. Firstly, density functional calculations were carried out on the polarization structure. The calculation were finished within the local density approximation and mixed basis employing the Troullier-Martins types pseudopotential set were used for the pseudopotential method, the cutoff of energy is 100 Ry. The optimized cell structure and atomic positions are determined by minimizing the total energy and the Hellmann-Feynman forces. LDA-calculated in-plane lattice constant for unstrained PbTiO3 (PT) is a=3.88 Å, with c/a=1.04, both agreeing well with other existing calculations.
Figure 1. (a) The 2D Brillouin zone for the kp plane; (b) Berry’s phase at different k points for PbTiO3 at equilibrium.
The above figure shows the reduced 2D Brillouin zone (BZ) that the k points sample over. The calculated phases at individual points along the Γ→X1→X2→Γ path are given in Fig. 1(b). Reciprocal-space coordinates of X1 and X2 are (∏/a,0) and (∏/a, ∏/a) respectively. The dispersion curve is rigidly shifted such that the phase at Γ is taken as the zero reference.
One needs to notice that the shape of this k dependent phase curve is translation invariant, this can be proved by analyzing the change in the phase when one displaces the solid arbitrarily, the phase differences between an original system and displaced system is a constant independent of k .
Three direct conclusions can be drawn directly from figure 1(b). The first is that the largest phase polarization comes from the X2 point which lies at the far end of the BZ instead of coming from the zero center. Second, along the Γ-X1 line, there is only a small dispersion of polarization, while when it comes to Γ-X2 line, the dispersion becomes very large. Thirdly, when k comes to very high symmetry points, the phase curve shows zero slope. Finally, polarization dispersion shows very little details, e.g. there is a local maximum along Γ-X1 which makes X1 a local minimum. It can also be very easily obtained from Figure 1 that the dispersion width which is always less than 0.6 is much smaller than 2∏. Which, from the previous texts, tells us that the phase contributions from different k are quite close and that it gets rid of the difficulty to determine which phase branch of a specific k vector should be assigned.
Despite all the advantages of the calculations, it may still introduce errors to the total polarization. Detailed explanations can be seen in Figure 2. Figure 2a addresses the PbTiO3 problems with all five atoms displaced along the c axis by the same distance. The results of total polarization(including both ionic and electronic part ) as a function of displacement show that discontinuities exist although it is easily assumed that total polarization should be uniquely determined due to the transformation invariance mentioned above. Figure 2(b) depicts the phase contributions from individual k . A clear periodicity can be seen along the c axis, which explains the periodic discontinuity in Figure 2(a). It can be seen that electrons is 22, thus the periodicity is C/22, resulting in a 0.0455c of the phase in real space, which also explains the transformational invariance by Eq(4).
Eq4.png (4)
Along the G’’ direction, whenever the vector is an integer times c over NbandOCC , phase A and B differ 2∏n from each other. Spurious discontinuity occurs whenever phases exceed 2∏, which induces the spurious results in the calculation. Based on the previous experiences, spurious polarization may come from two conditions: Materials have very large polarization, such as BiScO3 and atoms in the unit cell are translationally shifted. Fortunately, the spurious results can be avoided by using different vectors in the reciprocal space for small band width dispersions and for dispersions larger than 2∏, avoiding the spurious results may have to depend on the mapping out the periodic phases.
Figure 2 (a) Total polarization in strained PbTiO3 of in-plane lattice constant a=3.72 Å as a function of the uniform displacement of five atoms; (b) the phases at six Monkhorst-Pack sampling K points as a function of z0.

2.2.2 Strain dependence of polarization structure
Because the response of polarization dispersion structure can be greatly affected due to the fact that phase at different k may depend on the in-pane strain very differently, which also makes the interaction of atoms both in the in-plane and out of plane very complicated. This problems is investigated by fixing the in-plane lattice constant a at different values and relaxing the out of plane the lattice constant c, the resulting structure is shown in Figure 3. Fully relaxed atomic positions can be obtained by minimizing the DFT total energy.
Figure 3 Tetragonal perovskites at room temperature, in which the atoms are constrained in the x-y plane due to the presence of the substrate, cubes appear only at high temperature and high pressure.

The result is plotted in Figure 4. For different in-plane lattice constants, the phase for different K is obtained for the conditions with fully relaxed atoms. From the plot, it can be seen that (a)The phase difference between Φ(Γ) and Φ(X1) is very small, while the phase difference between Φ(Γ) and Φ(X2) is quite huge.(b)While the strain increases from zero (a=3.88 Å), there is little change of phase dispersion at first(comparison between a=3.88 Å and a=3.84 Å ). The phase dispersion change when a becomes 3.72 Å, the change is very huge. But later the change turns back to be small when a changes from 3.72 to 3.65. (c) The curvature also show subtle changes at different in-plane lattice constants among different K points.
Figure 4. The phases of different k__ points for PbTiO3 under different in-plane lattice constants. The inset shows the total polarization in PbTiO3 as a function of in-plane lattice constant.

A similar calculation to the one conducted with PbTiO3 was conducted to BaTiO3 for comparison. Stain increase with in-plane lattice constant decreases from the equilibrium a0=3.95 Å to 3.56 Å. The result was plot in Figure 5 and can be analyzed in the same way.
Figure 5. Polarization dispersions for BaTiO3 at different in-plane lattice constants.

3 Analytical methods based on Wannier Function
3.1 Wannier functions
Wannier functions are defined in the following forms
Eq5.png (5)
Or in a discretized form
Eq6.png (6)
where R runs over the whole real space lattice vectors. Combining Eq. 6 with Eq.2 and carrying out analytical integral over kparallel , for tetragonal perovskites, the polarization at each k can be derived as
Eq7.png (7)
Which can be rewritten as,
Eq8.png (8)
Where Φ0 is the phase contribution from the same unit cell. According to Eq. 8 , the polarization structure can be understood from three aspects:
i. dependence of the phase results from the overlap of the Wannier functions of different cells that are displaced by from each other within the plane that is perpendicular to the direction of polarization.
ii. Eq. 8 tells why the bandwidth of polarization dispersion is often smaller than 2π as only the second term in this equation is dependent and that the Wannier functions are generally well localized compared to the size of unit cell, one expects the overlap to be big.
iii. Third, since the dispersion in phase comes from the overlap of the Wanier functions between cells in the xy-in-plane directions, it explains why the polarization structure is very sensitive to in-plane strain, where by changing in-plane lattice constant, the distances between neighboring cells are effectively altered.
As Eq. (7) can be rewritten as
Eq9.png (9 )
Eq10.tif (10)
t(R) decays quickly with the increase of R , thus an approximate result of can be obtained with a cut off value of R by considering only several nearest neighbors, such as
Eq11.png (11)
Considering the symmetry of tetragonal, Eq (9) can be simplified as
Eq12.png (12)
At special K points, Γ, X1 and X2, phase as well as relative phase to the Γ can be easily calculated by Eq. (12), the value are in the form of t1 and t2, the value of which is listed as below for different in-plane lattice constant.

3.2 Comparison of PT and BT based on Wannier functions
It can be seen from the comparison between figure 4 and 5 that for both cases, the largest phase contribution comes from the zone boundary X2 instead of the zone center Γ. In addition, several differences between these two groups of calculations were also observed and they can be explained with Wannier functions.
In the calculation in BT, there is a larger phase at X1. This comes from different t1 and t2 values for these two structures since from Eq. (12) at X1, the phase difference from the Γ point can be approximated as
Eq13.png (13)
In the same way, as
Eq14.png (14)
Which is the reason that BT has a smaller band width comes from the smaller t1 value.

TABLE I. t1 and t2 parameters for PbTiO3 at different lattice constants.

Table 1 provides a comparison of t1 and t2 value for PbTiO3. The obtained curve is in good accordance agrees with the results from DFT. However, in some cases, such as, along the Γ-X1 curve, the small local maximum approximation beyond the second NNs would be necessary to investigate the subtle change in Figure 4. Table 1 also shows the ti(i denotes the ith NNS ) changes under different in-plane strains. t1 declines substantially as a decreases below 3.80 Å, while t2 shows less dependence on in-plane strain. This can be explained by that the main effect lies in altering the NNS interaction among the Wannier functions. For a=3.80 Å, t1 approximately equals 2t2, confirming the importance of the nearest-neighbor interaction. For large strains of a=3.72 Å, t1 and t2 become comparable, for which it is likely that higher orders of NNs are also needed.

Figure 6 compares how the phases at X1 and X2 point depend on the in plane lattice constant as a function of in-plane lattice constant for PT and BT. phases at both X1 and X2 are seen to be far greater in BaTiO3 than in PbTiO3 for a fixed a constant. The greater values of phase in BT could possibly due to the fact that Ba atom size is larger thus the Wannier functions in this material spread more.
Figure 6. Dependencies of phases at X1 point (left) and at X2 point (right) as a function of in-plane lattice constant for PT and BT.

4 Interfaces between thin film perovskites
Reference [3] provided a DFT calculation combining Berry-Phase theory and Wannier functions to see how an interface between metal and ferroelectric insulator. Reference [5] first suggested a dead layer effect in the metal-dielectric interfaces. Reference [4] used the method of [3] to investigate the phenomenon in [4] by calculating the response of a nanoscale capacitor under finite electric field. The finite electric field was applied using electric enthalpy by Eq. (15), to investigate the local dielectric response and capacitance of the whole capacitor.
Eq15.png (15)
The capacitor was modeled with periodic heterostructures consisting of alternating seven-unit-cell slabs of insulating SrTiO3 and metallic SrRuO3 . The in-plane cell parameter was fixed to the theoretical SrTiO3 equilibrium lattice constant of 3.85 Å, to simulate epitaxial growth on a SrTiO3 substrate. The ionic positions and out-of-plane lattice constant were relaxed first and until both the out-of-plane stress and the maximum force on each ion were smaller than 1 meV Å-1; Then the symmetry restricted force matrix, the Born effective charges and the linear variation of the charge density upon small displacements of the ions were calculated. Finally, the capacitance, the spatially resolved permittivity profile and the full dynamical properties of the device in the linear (small field 27.8mV/27.8Å) constraint was obtained.
Figure 7. Response of a SrRuO3 and SrTiO3 nanoscale capacitor under finite field. (a)Inverse relative dielectric constant at different position of the capacitor. (b) Comparison of the potential distribution between with and without the electronic relaxation.

The first figure 7(a) is the calculated inverse permittivity for the SrTiO3-SrRuO3 capacitor as compared to the classical, a clear decrease of dielectric constant can be observed at the interfaces, which is named as the dead layer effect for nanoscale capacitors and it caused a huge decrease of nanocapacitors capacitance as expected from the bulk. Figure (b) is the induced electrostatic potentials for an external potential difference 27.8mV. Red dotted curve is the case with electrons relaxed while keeping the atoms frozen in their zero-field positions. Blue solid curve is the case within the external bias which is in accordance with figure 7a. The difference between the two provides denoted by depolarizing field Ed (black dashed curve). Magnitude of the electric field is given by the slope.
Figure 8. Influence of the ionic contribution to the screening on the magnitude of the dead layer.

Figure 8 shows the work on separating the electronic contribution and ionic contribution to the screening at the interface. The fully relaxed result of Fig. 1 (solid curve) is compared to the frozen-electrode profiles, respectively allowing (dashed) or not allowing (dot-dashed) the interfacial SrO(I) layer to relax. The relaxation of SrO(I) provides the most important correction to the interfacial polarizability, while the rest of the SrRuO3 slab contributes to a lesser extent. Thus, the lattice polarization of SrRuO3 significantly enhances the dielectric response of the capacitor, but is only able to reduce the dead layer by half. In addition, the calculated profile for a Pt/SrTiO3 capacitor (shown as a thick red curve) demonstrates the superior properties that can be obtained by using elemental metals as electrodes.

5 Future work
Potential work concerning the polarization and minimize the dead layer effect are listed as follows:
i. With regarding to the polarization in perovskites, [2] pointed out that there is not much previous understanding in the literature, and that although we are able to compute precisely the polarization of individual K points. Thus, future to find an alternative way to understand the polarization structure and the computation results would be very useful. For instance, what determines the polarization at individual K points and why phase maximizes at the X2 point.
ii. For the application of perovskites in nanoscale capacitors, [4] indicated that elemental metals can provide better polarization than perovskites itself. Thus more work should be focused on the production of atomically sharp interfaces between elemental metals and high dielectric constant perovskites.

6 References
[1]. 1 R. D. King-Smith and D. Vanderbilt, Theory of polarization of crystalline solids, Phys. Rev. B 47, 1651 1993.
[2]. 2 Yanpeng Yao and Huaxiang Fu, Theory of the structure of electronic polarization and its strain dependence in ferroelectric perovskites, Phys. Rev. B 79, 014103 ,2009.
[3]. 3 Massimiliano Stengel and Nicola A. Spaldin, Ab initio theory of metal-insulator interfaces in a finite electric field, Phys. Rev. B 75, 205121 2007.
[4]. M. Stengel, and N. A. Spaldin, Origin of the dielectric dead layer in nanoscale Capacitors, Nature 443, 679, 2006.
[5]. Zhou, C. & Newns, D. M. Intrinsic dead layer effect and the performance of ferroelectric thin film capacitors. J. Appl. Phys. 82, 3081–-3088 1997.
[6]. Finnis, M. W. The theory of metal-ceramic interfaces. J. Phys. Condens. Matter. 8, 5811–-5836,1996.

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