^{1,2,3}and I. B. Misirlioglu

^{1,a)}

### Abstract

Within the phenomenological Landau–Ginzburg–Devonshire theory, we discuss the paraelectric-ferrolectric transition in superstructures consisting of ferroelectric and paraelectric layers of equal thickness. The polar axis of the ferroelectric is perpendicular to the layer plane as expected in fully strained BaTiO_{3}/SrTiO_{3} superstructures on SrTiO_{3} substrates with pseudomorphic electrodes. We concentrate on the electrostatic effects and do not take into account the boundary conditions other than the electrostatic ones. We find that when the ferroelectric phase transition in the superstructures is into a multidomain state, both its temperature and its character, i. e., the profile of the polarization appearing at the phase transition is strongly influenced by the nature of the near-electrode region. This is also the case for the layer thickness separating the single-and multidomain regimes of the transition. Such a finding makes us question the idea that these superstructures can be thought of as infinite systems, i.e., periodic superstructures similar to a crystal. The irrelevance of this idea in certain conditions is demonstrated by comparing the phase transitions in two different superstructures consisting of ferroelectric and paraelectric layers of the same thickness. In one of them, the ferroelectric layer is in immediate contact with an ideal metallic electrode, whereas at the other boundary, it is the paraelectric layer that is in contact with the electrode. In another superstructure, one paraelectric layer is split in two equal parts which are placed as the first and last layer between the electrodes and the ferroelectric layers which are closest to the electrodes. We show (with some formal reservations) that the phase transition temperature in the first superstructure can be over 100 °C more than in the second one if the material parameters of BaTiO_{3}/SrTiO_{3} are used for the estimations. Moreover, the profile of the polarization arising at the phase transition is inhomogeneous along the superstructure and has the maximum amplitude in the ferroelectric layer contacting the electrode. We argue that this situation is general and results in smearing of the phase transition anomalies for the layer thicknesses corresponding to multidomain transitions. The work is mainly analyical but numerical methods have been used to support some statements that have been put forward as hypotheses.

A.P.L. has been partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK) through the BİDEB Program and by the Ministry of Science and Education of Russian Federation (State Contract No. 02.740.11.5156). I.B.M. acknowledges the support of the Turkish Academy of Sciences (TÜBA) GEBİP Program.

I. INTRODUCTION

II. SMALL SYSTEMS

A. The Chensky and Tarasenko approach

B. The Chensky-Tarasenko cell with thick dead layer

C. Bilayer

D. Non-symmetrical trilayer

III. LARGE SYSTEMS

A. Two bilayers

B. Many bilayers

C. Two ChT cells

D. Many ChT cells

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Ferroelectric phase transitions
- 64.0
- Phase transitions
- 63.0
- Electrodes
- 49.0
- Polarization
- 44.0
- Multilayers
- 20.0

## Figures

(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.

(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.

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

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

(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 *ɛ _{p} * = 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

*T*= 998 °C, Curie constant = 1.5 × 10

_{C}^{5}°C,

*g*= 6.2 × 10

^{-10}m

^{3}/F, = 50.

(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 *ɛ _{p} * = 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

*T*= 998 °C, Curie constant = 1.5 × 10

_{C}^{5}°C,

*g*= 6.2 × 10

^{-10}m

^{3}/F, = 50.

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

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

(Color online) (a) Transition temperatures (in °C) as a function of layer thickness for the bilayer cell for *ɛ _{p} * = 100 (hollow diamonds),

*ɛ*= 500 (dark thick line), and

_{p}*ɛ*= 1000 (gray triangles) and (b) Critical

_{p}*k*as a function of layer thickness for the bilayer cell

*ɛ*= 100 (solid line),

_{p}*ɛ*= 500 (dashed line), and

_{p}*ɛ*= 1000 (line with the smallest k values). The material parameter values used in the calculations are

_{p}*T*= 998 °C, Curie constant = 1.5 × 10

_{C}^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F and = 50.

(Color online) (a) Transition temperatures (in °C) as a function of layer thickness for the bilayer cell for *ɛ _{p} * = 100 (hollow diamonds),

*ɛ*= 500 (dark thick line), and

_{p}*ɛ*= 1000 (gray triangles) and (b) Critical

_{p}*k*as a function of layer thickness for the bilayer cell

*ɛ*= 100 (solid line),

_{p}*ɛ*= 500 (dashed line), and

_{p}*ɛ*= 1000 (line with the smallest k values). The material parameter values used in the calculations are

_{p}*T*= 998 °C, Curie constant = 1.5 × 10

_{C}^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F and = 50.

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

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

(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, 3*l*/4 paraelectric layer partitioning (hollow squares) and the ChT cell (hollow triangles); (b) the *k _{c} * 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 BaTiO

_{3}—SrTiO

_{3}system. The values used for BaTiO

_{3}fully strained on SrTiO

_{3}in the calculations are T

_{ C }= 998 °C (computed using the constants given in Ref. 25, Curie constant = 1.5 × 10

^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, = 20,

*ɛ*= 300 for SrTiO

_{p}_{3}and, for the sake of convenience, is assumed to be constant over the entire temperature range.

(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, 3*l*/4 paraelectric layer partitioning (hollow squares) and the ChT cell (hollow triangles); (b) the *k _{c} * 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 BaTiO

_{3}—SrTiO

_{3}system. The values used for BaTiO

_{3}fully strained on SrTiO

_{3}in the calculations are T

_{ C }= 998 °C (computed using the constants given in Ref. 25, Curie constant = 1.5 × 10

^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, = 20,

*ɛ*= 300 for SrTiO

_{p}_{3}and, for the sake of convenience, is assumed to be constant over the entire temperature range.

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

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

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

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

(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 BaTiO_{3} and the paraelectric ones are SrTiO_{3}. 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 BaTiO_{3} in the calculations are *T _{C} * = 998 °C (computed using the constants given in Ref. 25, Curie constant = 1.5 × 10

^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, = 20,

*ɛ*= 300 for SrTiO

_{p}_{3}and for the sake of convenience is assumed to be constant over the entire temperature range.

(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 BaTiO_{3} and the paraelectric ones are SrTiO_{3}. 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 BaTiO_{3} in the calculations are *T _{C} * = 998 °C (computed using the constants given in Ref. 25, Curie constant = 1.5 × 10

^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, = 20,

*ɛ*= 300 for SrTiO

_{p}_{3}and for the sake of convenience is assumed to be constant over the entire temperature range.

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

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

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 *ɛ _{p} * = 100 (a) and (b)

*ɛ*= 500. The material parameter values used in the calculations are

_{p}*T*= 998 °C, Curie constant = 1.5 × 10

_{C}^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, = 50.

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 *ɛ _{p} * = 100 (a) and (b)

*ɛ*= 500. The material parameter values used in the calculations are

_{p}*T*= 998 °C, Curie constant = 1.5 × 10

_{C}^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, = 50.

(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 BaTiO_{3} in the calculations are *T _{C} * = 998 °C, Curie constant = 1.5 × 10

^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, and = 20,

*ɛ*= 300 for SrTiO

_{p}_{3}and for the sake of convenience is assumed to be constant over the entire temperature range.

(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 BaTiO_{3} in the calculations are *T _{C} * = 998 °C, Curie constant = 1.5 × 10

^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, and = 20,

*ɛ*= 300 for SrTiO

_{p}_{3}and for the sake of convenience is assumed to be constant over the entire temperature range.

(Color online) Polarization maps obtained in our numerical simulations 5 °C below the phase transition for the BaTiO_{3}-SrTiO_{3} system strained on a thick electroded SrTiO_{3} 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 BaTiO_{3} in the calculations are *T _{C} * = 998 °C, Curie Constant = 1.5 × 10

^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, = 20,

*ɛ*= 300 for SrTiO

_{p}_{3}and for the sake of convenience is assumed to be constant over the entire temperature range.

(Color online) Polarization maps obtained in our numerical simulations 5 °C below the phase transition for the BaTiO_{3}-SrTiO_{3} system strained on a thick electroded SrTiO_{3} 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 BaTiO_{3} in the calculations are *T _{C} * = 998 °C, Curie Constant = 1.5 × 10

^{5}°C,

*g*= 6.2 × 10

^{−10}m

^{3}/F, = 20,

*ɛ*= 300 for SrTiO

_{p}_{3}and for the sake of convenience is assumed to be constant over the entire temperature range.

(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 (*E _{P} *) and in the ferroelectric (

*E*) are in opposite directions to satisfy

_{F}*D*= constant in the system.

(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 (*E _{P} *) and in the ferroelectric (

*E*) are in opposite directions to satisfy

_{F}*D*= constant in the system.

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