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Phase transitions in ferroelectric-paraelectric superlattices
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10.1063/1.3662197
/content/aip/journal/jap/110/11/10.1063/1.3662197
http://aip.metastore.ingenta.com/content/aip/journal/jap/110/11/10.1063/1.3662197
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

(Color online) Schematic of the ferroelectric layer with thin dead layers having thickness d/2 between the ferroelectric and the electrode. This was the system that was investigated in Ref. 18.

Image of FIG. 2.
FIG. 2.

(Color online) ChT cell with thick dead layers where each paraelectric layer is l/2.

Image of FIG. 3.
FIG. 3.

(Color online) Comparison between the analytical (thin line) and the numerical (thick line) results of the transition temperature (in °C) in the ChT cell for ɛp  = 500. The thin curve reflect the small kl and the large kl limits as given in Eqs. (22) and (30). The material parameter values used in the calculations are TC  = 998 °C, Curie constant = 1.5 × 105 °C, g = 6.2 × 10-10 m3/F, = 50.

Image of FIG. 4.
FIG. 4.

(Color online) Bilayer cell with ferroelectric and paraelectric layers of equal thickness.

Image of FIG. 5.
FIG. 5.

(Color online) (a) Transition temperatures (in °C) as a function of layer thickness for the bilayer cell for ɛp  = 100 (hollow diamonds), ɛp  = 500 (dark thick line), and ɛp  = 1000 (gray triangles) and (b) Critical k as a function of layer thickness for the bilayer cell ɛp  = 100 (solid line), ɛp  = 500 (dashed line), and ɛp  = 1000 (line with the smallest k values). The material parameter values used in the calculations are TC  = 998 °C, Curie constant = 1.5 × 105 °C, g = 6.2 × 10−10 m3/F and  = 50.

Image of FIG. 6.
FIG. 6.

(Color online) Schematic of the non-symmetrical trilayer.

Image of FIG. 7.
FIG. 7.

(Color online) Comparison of (a) the numerical solutions for transition temperature for the bilayer cell (solid thick line), the non-symmetrical cell with l/4, 3l/4 paraelectric layer partitioning (hollow squares) and the ChT cell (hollow triangles); (b) the kc at the transition for the bilayer cell (thick solid line), the asymmetrical cell (line starting at 3.7 nm along the thickness axis), and the ChT cell (dashed line) for the BaTiO3—SrTiO3 system. The values used for BaTiO3 fully strained on SrTiO3 in the calculations are T C  = 998 °C (computed using the constants given in Ref. 25, Curie constant = 1.5 × 105 °C, g = 6.2 × 10−10 m3/F,  = 20, ɛp  = 300 for SrTiO3 and, for the sake of convenience, is assumed to be constant over the entire temperature range.

Image of FIG. 8.
FIG. 8.

(Color online) Schematic showing unit cells of the superstructure consisting of (a) bilayers and (b) ChT cells.

Image of FIG. 9.
FIG. 9.

(Color online) Two bilayer cell system mentioned in Sec. III A.

Image of FIG. 10.
FIG. 10.

(Color online) Polarization profile at the temperature of loss of stability of the paraelectric phase in the superstructure consisting of 4 bilayers (rapidly decaying curve from left to right with large period) and 8 bilayers (slowly decaying curve from left to right with small period) with 5 nm and 2.5 nm layer thickness, respectively. Note that the total thickness of the system in both cases is the same and fixed at 40 nm. The ferroelectric layers are BaTiO3 and the paraelectric ones are SrTiO3. Critical thickness for single domain state stabilization is 2.2 nm. The 5 nm layer has a much more rapidly decaying polarization along the thickness. The values used for BaTiO3 in the calculations are TC  = 998 °C (computed using the constants given in Ref. 25, Curie constant = 1.5 × 105 °C, g = 6.2 × 10−10 m3/F,  = 20, ɛp  = 300 for SrTiO3 and for the sake of convenience is assumed to be constant over the entire temperature range.

Image of FIG. 11.
FIG. 11.

(Color online) Two ChT cell system mentioned in Sec. III C.

Image of FIG. 12.
FIG. 12.

Comparison of the transition temperatures (in °C) of the two-bilayer cell (solid line), the two-ChT cell (hollow squares) and the secondary solution (hollow triangles) of the two-bilayer and the two-ChT cell as a function of layer thickness for ɛp  = 100 (a) and (b) ɛp  = 500. The material parameter values used in the calculations are TC  = 998 °C, Curie constant = 1.5 × 105 °C, g = 6.2 × 10−10 m3/F,  = 50.

Image of FIG. 13.
FIG. 13.

(Color online) Polarization wave profile at the temperature of loss of stability of the paraelectric phase in the superstructure consisting of 3 ChT cells, each layer having 8 nm thickness (curve with large period), and 4 ChT cells with each layer being 5 nm thick (curve with small period). Critical thickness for single domain state stabilization is 4.4 nm. The values used for BaTiO3 in the calculations are TC  = 998 °C, Curie constant = 1.5 × 105 °C, g = 6.2 × 10−10 m3/F, and  = 20, ɛp  = 300 for SrTiO3 and for the sake of convenience is assumed to be constant over the entire temperature range.

Image of FIG. 14.
FIG. 14.

(Color online) Polarization maps obtained in our numerical simulations 5 °C below the phase transition for the BaTiO3-SrTiO3 system strained on a thick electroded SrTiO3 substrate consisting of (a) 8 ChT cells and (b) 8 bilayers with each system having 80 nm total thickness. The system in (a) has a phase transition temperature around 300 °C and the one in (b) 440 °C, which agrees well with analytical results. The perpendicular colorbar scales are for normalized polarization. The values used for BaTiO3 in the calculations are TC  = 998 °C, Curie Constant = 1.5 × 105 °C, g = 6.2 × 10−10 m3/F,  = 20, ɛp  = 300 for SrTiO3 and for the sake of convenience is assumed to be constant over the entire temperature range.

Image of FIG. 15.
FIG. 15.

(Color online) Schematic of a superstructure with real electrodes (denoted by the presence of dead layers at the oxide-electrode interfaces). The electric field in the paraelectric (EP ) and in the ferroelectric (EF ) are in opposite directions to satisfy D = constant in the system.

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/content/aip/journal/jap/110/11/10.1063/1.3662197
2011-12-05
2014-04-19
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Phase transitions in ferroelectric-paraelectric superlattices
http://aip.metastore.ingenta.com/content/aip/journal/jap/110/11/10.1063/1.3662197
10.1063/1.3662197
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