Phase Transitions and Switching Dynamics of Topological Domains in Hafnium Oxide-Based Cylindrical Ferroelectrics from Three-Dimensional Phase Field Simulation

Abstract

The phase transitions and switching dynamics of topological polar textures in hafnium oxide (HfO₂)-based cylindrical-shell ferroelectrics are studied using a three-dimensional (3D) phase field model based on the self-consistent solution of the time-dependent Ginzburg–Landau model and Poisson equation. The comprehensive interplays of bulk free energy, gradient energy, depolarization energy, and elastic energy are taken into account. When a cylindrical ferroelectric device is biased under the in-plane radial electric field, there is a size-controlled phase transition between the ferroelectric (FE), antiferroelectric (AFE), and paraelectric (PE) phases, depending on ferroelectric film thickness and cylindrical shell radius. For in-plane polarization textures at the equilibriums, the FE phase has a Néel-like texture with a center-type four-quad domain, the AFE phase has a monodomain texture, and the PE phase has a Bloch-like texture with a vortex four-quad domain. These polarization domain textures are resultant from energy competition and topologically protected by the geometrical confinement. The polarization dynamics from polar states towards equilibriums are analyzed considering the separated contributions of x- and y-components of polarizations that are driven by x-y in-plane electric fields. The emergent topological domains and phase transitions provide guidelines for geometrical engineering of a novel nano-structured ferroelectric device that is different from the planar one, offering new possibilities for multi-functional high-density ferroelectric memory.

1. Introduction

As the information age rapidly advances, the von Neumann bottleneck problem stemming from the separation of storage and computing units has become more acute, leading to significant delays and power consumption [1,2,3]. Neuromorphic computing based on non-volatile memory (NVM) has gained significant attention due to its high energy efficiency and parallel-computing capability [4,5,6]. Hafnium oxide (HfO₂) offers good CMOS compatibility and exceptional scalability, making it highly advantageous for next-generation non-volatile memory (NVM) technologies [7,8,9], such as FeRAM [10,11], FeFET [12,13], and FTJ [14,15]. Moreover, antiferroelectric HfO₂ has been implemented in FTJ memory, highlighting its potential in NVM applications [16]. In recent years, the three-dimensional (3D) integration of these ferroelectric memories by vertically stacking memory devices has been successfully demonstrated, showing great potential to achieve ultrahigh memory density by maximizing cell area efficiency [17,18,19].

The polarization texture and ferroelectric dynamics can be theoretically explored by many simulation frameworks such as density function theory and phase field simulation [20,21]. For planar ferroelectric films, the application of vertical electric field perpendicular to the plane of planar ferroelectric films induces the formation of polar textures, including vortices [22], skyrmions [23], bubbles [24], and merons [25]. These polar textures are intricately related to a delicate balance between the bulk free energy, gradient energy, electrostatic energy, and elastic energy within the ferroelectric materials, depending on the device size and external excitation [26,27,28]. In recent years, the emergence of topological polar textures has been observed experimentally in perovskite ferroelectric films. Controlling these robust polar topological textures via electric fields holds promise for ultrahigh-density storage of ultrafine topological entities [29,30,31], paving the way for widespread applications in neuromorphic computing and broadening the scope of research in high-density, non-volatile memory.

A cylindrical-shell ferroelectric film is one of the key device structures to implement 3D memory integration using VNAND-like architecture, where the outer electrode is called plane electrode and the inner electrode is called pillar electrode [32]. However, compared with a planar ferroelectric film under a vertical electric field [22,23,24,25,26,27,28], polarization textures in such a cylindrical shell geometry-confined film that is biased via a radial electric field have barely been explored [33]. On the other hand, compared with that in perovskite ferroelectrics, topological polarization textures in HfO₂ ferroelectrics are more or less uncovered. Recently, it has been revealed that cylindrical ferroelectric capacitors (FeCAPs) show distinct switching behaviors from planar devices [34]. Therefore, elucidating the polarization textures and switching dynamics in cylindrical HfO₂ ferroelectrics is of great significance, to offer a deep understanding of ferroelectric memory devices such as FeRAM, FeFET, and FTJ, and also to provide theoretical guidance for emerging polar topological devices.

In this article, we investigate the polarization patterns and their switching dynamics within 3D nano cylindrical HfO₂ ferroelectric devices, employing 3D phase field modeling. Section 2 describes the simulation method and calibration, where the 3D time-dependent Ginzburg–Landau (TDGL) and Poisson equations are self-consistently solved using the finite element method. Section 3 studies the topological polar domains and their switching dynamics as a function of the inner radius of the capacitor and the thickness of the ferroelectric layer. It is revealed that 3D cylindrical-shell ferroelectric capacitors demonstrate a size-controlled phase transition between ferroelectric (FE), antiferroelectric (AFE), and paraelectric (PE) phases. The emergence of these unique topological textures is analyzed via the intricate energy competition. Section 4 gives the conclusions.

2. Materials and Methods

Figure 1a,b illustrate the metal–ferroelectric–metal (MFM) film capacitor device with a 3D nano cylindrical-shell structure, with a ground bias applied to the inner electrode and an external time-dependent bias voltage applied to the outer electrode. The inner radius is denoted as R_in; the thickness and height of the ferroelectric layer are T_FE and H_FE, respectively.

To study the spatial and temporal evolutions of polarization in ferroelectric films, the 3D TDGL and Poisson equations are self-consistently calculated within the ferroelectric region [35,36,37,38,39,40]. The total energy F in TDGL comprises bulk free energy f_bulk, gradient energy f_grad, elastic energy f_elas, and electrostatic energy f_elec,

$$-\frac{1}{\Gamma }\frac{\partial P_i}{\partial t}=\frac{\partial F}{\partial P_i}(i=1,2,3)$$

(1)

$$F={\displaystyle {\int }_V(f_{bulk}+f_{grad}+f_{elas}+f_{elec})dV}$$

(2)

where the local polarizations are denoted as P_i (i = 1, 2, 3), along with the x axis (i = 1), y axis (i = 2), and z axis (i = 3) in Cartesian coordinates. t represents the time, and Γ is a dynamic coefficient that describes the polarization reversal speed relative to the applied voltage [38].

The bulk free energy of the ferroelectric material is described by the LGD model [38],

$$\begin{array}{cc}\hfill f_{bulk}& ={\alpha }_1(P_1^2+P_2^2+P_3^2)\hfill \\ & +{\alpha }_{11}(P_1^4+P_2^4+P_3^4)\hfill \\ & +{\alpha }_{12}(P_1^2{P}_2^2+P_2^2{P}_3^2+P_1^2{P}_3^2)\hfill \\ & +{\alpha }_{111}(P_1^6+P_2^6+P_3^6)+{\alpha }_{112}[P_1^4(P_2^2+P_3^2)\hfill \\ & +P_2^4(P_1^2+P_3^2)+P_3^4(P_1^2+P_2^2)]+{\alpha }_{123}(P_1^2{P}_2^2{P}_3^2)\hfill \end{array}$$

(3)

where α₁, α₁₁, α₁₂, α₁₁₁, α₁₁₂, and α₁₂₃ are Landau coefficients used to describe the types of stable ferroelectric phases. Note that α₁ is positive above the Curie temperature and negative below it.

The gradient energy, represented by the polarization gradient, characterizes the energy of dipole–dipole interaction arising from spatially non-uniform polarization [38],

$$\begin{array}{cc}\hfill f_G& ={\displaystyle \frac{1}{2}}{G}_{11}(P_{1,1}^2+P_{2,2}^2+P_{3,3}^2)+G_{12}(P_{1,1}{P}_{2,2}+P_{2,2}{P}_{3,3}+P_{1,1}{P}_{3,3})\hfill \\ & +{\displaystyle \frac{1}{2}}{G}_{44}[{(P_{1,2}+P_{2.1})}^2+{(P_{2,3}+P_{3,2})}^2+{(P_{1.3}+P_{3,1})}^2]\hfill \\ & +{\displaystyle \frac{1}{2}}{G^{\prime }}_{44}[{(P_{1,2}-P_{2.1})}^2+{(P_{2,3}-P_{3,2})}^2+{(P_{1.3}-P_{3,1})}^2]\hfill \end{array},$$

(4)

where G₁₁, G₁₂, G₄₄, and $G_{44}^{\prime }$ are the gradient energy coefficients and $P_{i,j}=\partial P_i/\partial x_j$ [40].

The elastic strain energy density can be expressed as [38]

$$\begin{array}{l}{f}_{elastic}={\displaystyle \frac{1}{2}}{C}_{ijkl}({\epsilon }_{ij}-{\epsilon }_{ij}^0)({\epsilon }_{kl}-{\epsilon }_{kl}^0)\end{array},$$

(5)

where C_ijkl are the elastic stiffness tensor, ε_ij are the total strain, and ${\epsilon }_{ij}^0$ are the eigenstrain. The strain solution satisfies the boundary conditions and mechanical equilibrium as follows [38],

$${\displaystyle \frac{\partial {\sigma }_{ij}}{\partial x_j}}=0,$$

(6)

The elastic energy is given by the multiplication of the elastic stress [38],

$${\sigma }_{ij}=C_{ijkl}({\epsilon }_{ij}-{\epsilon }_{ij}^0),$$

(7)

where C₁₁, C₁₂, and C₄₄ are the independent elastic constants. The stress-free strain induced by the polarization field is described by the equation below [38],

$$\begin{array}{c}{\epsilon }_{11}^0=Q_{11}{P}_1^2+Q_{12}(P_2^2+P_3^2),{\epsilon }_{23}^0=Q_{44}{P}_2{P}_3\\ {\epsilon }_{22}^0=Q_{11}{P}_2^2+Q_{12}(P_3^2+P_1^2),{\epsilon }_{13}^0=Q_{44}{P}_1{P}_3\\ {\epsilon }_{22}^0=Q_{11}{P}_3^2+Q_{12}(P_1^2+P_2^2),{\epsilon }_{12}^0=Q_{44}{P}_1{P}_2\end{array},$$

(8)

where Q_ij are the electrostrictive coefficients.

Electrostatic energy, also called depolarization energy, can be expressed as [39,40,41]

$$f_{elec}=-E_i{P}_i(i=1,2,3),$$

(9)

where the electric field E_i within a ferroelectric is solved by the Poisson equation as follows [39,40,41]:

$${\epsilon }_0{\epsilon }_{FE}{\nabla }^2\phi =\nabla \cdot \mathit{P}$$

(10)

where ε₀ and ε_FE are the vacuum and the relative permittivity, respectively, and φ is the electric potential. The inner and outer boundaries of the ferroelectric film are set as Dirichlet boundary conditions with φ_in = 0 V and φ_out = V_a, where V_a is the external applied voltage, while the top and bottom boundaries are set as Neumann boundary conditions.

The 3D TDGL and Poisson equations are self-consistently solved using the finite element method. The calibration of the P-V characteristic of the HfO₂-based capacitor is shown in Figure 1c, where the simulated results are consistent with the measured data [42]. The good agreement between simulation and experiment verifies the coercive field and remnant polarization, with E_c ~ 1 MV/cm and P_r ~ 25 μC/cm², which are typical values for HfO₂ ferroelectrics, ensuring an accurate prediction of polar texture in HfO₂ ferroelectrics. In the simulation, the mesh grid resolution is set as 1–2 nm for the three-dimensional cylindrical-shell structure, depending on the inner radius and ferroelectric thickness. The V_a is set as a time-dependent ramp bias with triangular waveform, where the applied voltage amplitude and frequency are 3 V and 10 kHz, which are consistent with the measurement setup in [42]. The time step is set as T/100 with T = 1/f, where T and f are the period and frequency of the triangular waveform. The total simulation time is selected as 2T, and the simulation results in this work are derived from the second period. To start the self-consistent calculation of the TDGL and Poisson equations, the initial polarizations are set to zero in the whole ferroelectric region. In addition, the convergence criterion to end the self-consistent calculation is that the potential update satisfies |φⁿ⁺¹ − φⁿ| < ε, where the convergence criterion ε is ~10⁻⁴. The structure parameter H_FE is set as 10 nm, and T_FE and R_in are varied. The phase field parameters, which are listed in

Table 1. Parameters used in the phase field simulation.

Parameters	Values	Units
α1	−2.5 × 109	V·m/C
α11	−2 × 108	V·m5/C3
α12	5 × 108	V·m5/C3
α111	3.2 × 1011	V·m9/C5
α112	3.5 × 1011	V·m9/C5
α123	3.5 × 1011	V·m9/C5
G11	1 × 10−9	V·m3/C
G12	0	V·m3/C
G44	1 × 10−9	V·m3/C
G′44	1 × 10−9	V·m3/C
C11	450.129	GPa
C12	124.003	GPa
C44	5.469	GPa
Q11	0.0056	m4/C2
Q12	−0.0059	m4/C2
Q44	0.0385	m4/C2
Γ	0.01	S/m
εFE	30	1
f	10	kHz
HFE	10	nm

, are derived following the method in [42]. The temperature is set as 273 K in the simulation.

3. Results and Discussion

3.1. Phase Transitions Along with Topological Domain Patterns

Figure 2 shows the polarization characteristics of cylindrical-shell HfO₂ ferroelectric devices. It is observed that there a phase transition between FE, AFE, and PE phases with varied ferroelectric thickness and cylinder radius. A similar thickness-controlled phase transition in a planar ferroelectric thin film biased under vertical electric field is discussed in [40]. From Figure 2a, with T_FE = 10  nm and R_in = 24 nm, the P-V curve of the FE phase shows a single hysteresis loop and two stable remnant polarization states, which are two typical features of ferroelectrics. From Figure 2b, with T_FE = 13 nm and R_in = 1 nm, the P-V curve of the AFE phase shows a double hysteresis loop and zero remnant polarization. From Figure 2c, with T_FE = 14 nm and R_in = 1 nm, the P-V curve of the PE phase shows a slim hysteresis and nearly linear characteristics. Note that there is a very small and non-zero net remnant polarization in PE phase, which are consistent with the P-V curve of the experimental PE-phased HfO₂ ferroelectrics [43,44,45].

To reveal the microscopic domain pattern under an in-plane radial electric field E = (E_x, E_y, 0), namely, E = E_xy, Figure 3 shows the topological domain patterns of in-plane polarization P_xy in cylindrical-shell HfO₂ ferroelectric devices at the equilibrium state of V_a = 0 V, as the in-plane P_xy is the dominant component of total polarization. From Figure 3a, P_xy in the FE phase appears as a Néel-like texture. In particular, a center-type four-quad domain forms, and a quadruple domain wall in the FE case. From Figure 3b, P_xy in AFE phases exhibit a monodomain-like pattern. The overall P_xy gives rise to zero net polarization at equilibrium, as this circular-shape monodomain can be regarded as a double semicircle that has opposite polarization direction with respect to the center, one of which points towards the center <P_xy,in>, and the other outwards from the center <P_xy,out>. Therefore, zero net polarization in the AFE phase results from <P_xy,in> = −<P_xy,out>. This is analogous to the anti-parallel strip domains in planar AFE devices as discussed in [39,40], i.e., <P_z,up> = −<P_z,down>, thereby giving rise to zero net polarization. From Figure 3c, P_xy in the PE phase exhibits another topological texture, i.e., a Bloch-like texture with the vortex four-quad domain pattern. The overall net P_xy is very small compared with the FE phase, because P_xy significantly deviates from the radial electric field direction.

3.2. Switching Dynamics of Ferroelectric Phase

To further understand the formation of center-type four-quad Néel-like domain in FE-phase cylindrical ferroelectrics, as shown in Figure 3a, Figure 3 describes the switching dynamics of in-plane polarization components including P_x, P_y, and P_xy. When switched from negative to positive remnant polarization <P_R>, the evolution of polarization distribution driven by in-plane E_xy is clearly illustrated.

Figure 4a shows the local P_x distribution, namely, P_x(x,y), where positive P_x means that it points right and negative that it points left. In the negative <P_R> case at equilibrium (point A), a double domain pattern within left and right semicircles forms, each of which direction points towards the outside of the cylindrical ferroelectric device. At the same time, a double 180° tail-to-tail DWx forms at the x = 0 interfaces. The domain wall width W_DW is about 1.78 nm. When V_a increases (point B), the reversed domains with inward direction nucleate at the outer edges at y = 0, which is marked with a dashed box. Based on the phase field simulation, the activation energy of ferroelectric reversal is calculated to be 0.3 eV. This value is within the reasonable range for HfO₂ ferroelectrics [46]. Then, the reversed domains rapidly grow, going through the conditions when the total inward polarization <P_x,in> is equal to the total outward polarization <P_x,out>, namely, <P_x,in> = <P_x,out>, reaching zero net polarization at coercive voltage (point C). After this, the total polarization becomes positive, as the total inward polarization exceeds the total outward polarization. The growth of reversed domains stops until P_x is fully switched inwards (point D). From point D to point E, the polarization intensity is strengthened by the response of background dielectrics. When we reach the positive <P_R> case at equilibrium (point F), a double domain pattern with P_x pointing inwards is stabilized, along with a double 180° head-to-head DW_x.

Similarly, Figure 4b shows the local P_y distribution, namely, P_y(x,y), where positive P_y means that it points upwards and negative that it points downwards. A double domain pattern within the top and bottom semicircles is stabilized at equilibriums, and a double 180° DW_y forms at the y = 0 interfaces. There is an outward-directed polarization and tail-to-tail DW_y in the negative <P_R> case and an inward-directed polarization and head-to-head DW_y in the positive <P_R> case, in analogy to that of P_x and DW_x. The polarization reversal of P_y proceeds similarly to the reversed domain nucleation and growth processes.

Figure 4c shows the local P_xy distribution, namely, P_xy(x,y) = (P_x(x,y), P_y(x,y)). As a combination of P_x and P_y, P_xy exhibits a center-type four-quad domain pattern. Meanwhile, a quadruple Néel-type 90° DW_xy forms, due to a combined effect of a double 180° DW_x and a double 180° DW_y. The polarizations are inhomogeneous around the DW regions. This implies that the enhancement of gradient energy can alleviate the enhancement of electrostatic energy that is due to the strong depolarization field, thereby effectively suppressing the enhancement of total energy. This is the reason why the center-type four-quad domain pattern can be stabilized in cylindrical-shell ferroelectrics. Moreover, in-plane polarization P_xy can be switched back and forth between convergent and divergent states driven by in-plane E_xy, driven by the reversed domain nucleation and growth processes. To be specific, under V_a with a frequency of 10 kHz, the switching time t_s (see Figure 2a) is estimated to be 7 μs, and the corresponding domain wall velocity during polarization switching is about 0.067 m/s for the FE phase. Furthermore, the four-quad domain pattern of P_xy switched between convergent and divergent states is consistent with the experimental observation of the planar cylindrical film biased under a vertical electric field by vector piezoresponse force microscopy (PFM) in [27]. In addition, the projection of two-semicircle P_x and P_y, obtained by our phase field simulation, is also consistent with the PFM measurements in [27]. This good agreement between experiments and simulation ensures that our results are reasonable.

3.3. Switching Dynamics of Antiferroelectric Phase

To further explore the emergent domain patterns in the AFE phase as shown in Figure 3b, Figure 5 describes the switching dynamics of in-plane polarization components including P_x, P_y, and P_xy as labeled in Figure 2b.

Figure 5a shows the local P_x distribution in the AFE phase. At equilibrium (point A), P_x is uniformly distributed with identical directions parallel to the x-axis, implying that net P_x is zero because overall <P_x,in> = −<P_x,out>, where P_x in the left semicircle pointing right means inward P_x (namely, P_x,in), and P_x in the right semicircle pointing right means outward P_x (namely, P_x,out). When positive voltage is applied (point B), P_x begins to flip from the center on the left semicircle. The flipped area gradually expands as bias voltage increases (point C), leading to a reduction in P_x,out and an increase in P_x,in, and thereby an increase in net <P_x,in>. Also note that the P_x intensity close to the center is enhanced. Therefore, a dual-center domain forms with respect to P_x, namely, a P_x,in center and P_x,out center, respectively. As the voltage decreases (point D), the flipped region progressively returns to its initial state and ultimately recovers to be identical to the original configuration at 0 V. A mirror-symmetry switching process takes place when negative voltage is applied (D-E-F-A).

Figure 5b shows the local P_y distribution in the AFE phase. The P_y distribution and switching dynamics are very similar to those of P_x. At equilibriums, P_y is uniformly distributed with identical directions parallel to the y-axis in response to E_y. This results in zero net P_y because overall <P_y,in> = −<P_y,out>, where P_y in the upper semicircle pointing downwards means inward P_y (namely, P_y,in), and P_y in the lower semicircle pointing downwards means outward P_y (namely, P_y,out). When positive or negative voltage is applied, P_y in one of the upper and lower semicircles begins to flip from the center and gradually expands as bias voltage increases, while the P_y intensity close to the center of the two semicircles is enhanced. Therefore, a dual-center domain forms with respect to P_y, namely, a P_y,in center and P_y,out center, respectively. In addition, the domain wall width W_DW is about 1.14 nm.

Figure 5c shows the local P_xy distribution in the AFE phase, resulting from a combination of P_x and P_y. At equilibriums, in-plane P_xy is almost uniform in terms of both intensity and direction. This is regarded as <P_xy,in> = −<P_xy,out>, where P_xy,in is contributed by the upper-left semicircle and P_xy,out by the lower-right semicircle. As the external voltage is biased, a dual-center emerges, where the P_xy intensity of one of the two centers is enhanced, and the other one is weakened. At the same time, the weakened one radially moves away from the center to the edge, accompanied by an increase in the flipped area as the bias voltage increases. The domain pattern is switched back and forth between monodomain and dual-center domain depending on the equilibrium or non-equilibrium bias conditions. In particular, when V_a varies with a frequency of 10 kHz, the switching time t_s (see Figure 2b) is estimated to be 2 μs for forward switching (t_sf) and 5 μs for backward switching (t_sf)) in one of the double hysteresis loops in the AFE phase. The corresponding domain wall velocity during polarization switching is about 0.825 m/s for the AFE phase, which is about one order of magnitude higher than that of the FE phase.

3.4. Switching Dynamics of Paraelectric Phase

To deeply understand the Bloch-like patterns in the PE phase as shown in Figure 3c, Figure 6 describes the switching dynamics of in-plane polarization components including P_x, P_y, and P_xy, under varied applied voltage as labeled in Figure 2c.

Figure 6a shows the local P_x distribution in the PE phase. At an initial 0 V (point A), P_x in the upper semicircle is polarized right, while P_x in the lower semicircle is polarized left. The overall circle is formed by two anti-parallel domains, thereby leading to nearly zero remnant polarization. At the same time, there is an Ising-type DW_x parallel to the x-axis. The domain wall width W_DW is about 0.84 nm in the PE phase. Note that the two DWs located left and right are not fully aligned, and the slim misaligned regions around the DW_x give rise to a very small amount of net <P_x,out>. When a positive voltage is applied (points B and C), the two Ising-DW start to move anticlockwise. Consequently, the net polarization changes from net <P_x,out> to net <P_x,in>. When the voltage returns to 0 V (point D), the DW_x move clockwise, and polarization recovers to a state similar to the initial one with a small amount of net <P_x,in>. A mirror-symmetry switching process takes place when negative voltage is applied.

Figure 6b shows the local P_y distribution in the PE phase. The two vertical DW_y move anticlockwise or clockwise, depending on the voltage polarity. The polarization is switched between a small amount of net <P_y,out> and net <P_y,in>. As the P_y distribution and switching dynamics are very similar to those of P_x with a rotation of 90°, they are not described to avoid repetition.

Figure 6c shows the local P_xy distribution in the PE phase. At 0 V, the remnant polarization is weak, because the vortex-like four-quad domains are polarized far from the radial direction, and are nearly tangent to the radial direction. This means that the polarization direction is tangent to the electric field direction. As bias voltage is increasingly applied, the vortex-like domain moves as a whole and rotates slightly off the tangent direction. Similarly to that of FE and AFE phases, the switching time t_s (see Figure 2c) and the domain wall velocity in the PE phase are derived to be 6 μs and 0.2 m/s, respectively.

3.5. Energy Competition in Different Phases

The polarization characteristics of the FE, AFE, and PE phase transitions highlight the complex dynamic behaviors in cylindrical-shell ferroelectrics under external radial electric fields. These distinct polarization patterns, including Néel-like, monodomain, and Bloch-like configurations, are deeply rooted in the intrinsic energy competition. Hence, Figure 7 gives the energy variation in the abovementioned FE, AFE, and PE phases, including f_bulk, f_grad, f_elec, and f_elas, at equilibriums, the corresponding domain textures of which are shown in Figure 3.

Furthermore, Figure 8a gives the size-controlled R_in-T_FE phase diagram for the cylindrical MFM ferroelectric capacitor to fully capture the phase transition between FE, AFE, and PE. From Figure 7, f_grad between the three phases changes most. Therefore, the related energy diagram of f_grad as a function of R_in-T_FE is presented in Figure 8b. The relative value of f_bulk between FE, AFE, and PE in Figure 7 is consistent with [47], where the AFE phase has relatively low f_bulk, followed by the FE and PE phases. The energy competition between FE, AFE, and PE based on Figure 7 and Figure 8 is discussed in the following paragraphs.

In the FE phase, where R_in is comparably larger than T_FE, the cylindrical ferroelectric capacitor behaves like the planar thin film one, and the formation of a domain wall can effectively relieve the strong depolarization effect. Therefore, the gradient energy increases, but the electrostatic energy decreases. Also noteworthy are the vertical 180° strip domains that exist in planar thin film ferroelectrics; their domain size changes as ferroelectric thickness varies [39,40]. In contrast, the center-type four-quad domain pattern of the topological polar texture is stabilized in cylindrical structures, which are topologically protected due to geometry confinement, meaning that they are independent of R_in and T_FE as long as they are in the FE phase. Such topological polar textures are energetically favorable as the bulk free energy remains much lower in the relatively deep double wall with the appearance of remnant polarization.

In the AFE phase, where R_in is smaller than T_FE, the curvature asymmetry between inner and out boundaries is greatly enhanced. The evolution of the domain wall vanishes towards equilibrium; otherwise, the gradient energy would be much higher and the domain pattern would become unstable. The formation of a quasi-monodomain without DW is actually a bidomain that has anti-parallel opposite directions inwards and outwards from the center, giving rise to a zero net polarization. The bidomain pattern is attributed to the increased electrostatic energy that is determined by anti-parallel inhomogeneous polarization [48]. In other words, the decrease in gradient energy is dominant compared with the increase in electrostatic energy, making the AFE phase stable.

In the PE phase, where T_FE is large and R_in is small, polarization with tangent directions suggests that the gradient energy is significantly high in such ultra-nonuniform conditions. This explains the formation of a vortex-like four-quad domain in the PE phase, with a very thick domain wall compared with the FE phase. Additionally, the great thickness of the ferroelectric layer reduces the overall electric field, making the electrostatic energy remarkedly lower.

4. Conclusions

We have systematically investigated the phase transitions, polarization topologies, and switching dynamics within three-dimensional cylindrical nano-ferroelectric devices using a time-dependent phase field model. The interplay of bulk free energy, gradient energy, depolarization energy, and elastic energy is critical in determining the phase behavior and stability of these devices. We have demonstrated that by controlling key size parameters such as inner radius and ferroelectric layer thickness, it is possible to induce phase transitions between ferroelectric, antiferroelectric, and paraelectric phases, each of which exhibits distinct topological patterns including Néel-like, monodomain, and Bloch-like textures. These findings underscore the potential of cylindrical ferroelectric devices to be used in high-density non-volatile memory and neural computing devices, and the ability to manipulate topological domains via geometry engineering offers exciting possibilities for device design and functionality. Future research should focus on further observation of these topological polar textures by advanced measurement techniques and of the origins of these FE, AFE, and PE phases by density function theory, as well as optimizing these devices for practical applications, exploring their integration into larger memory architectures, and investigating the dynamic control of their topological states for enhanced performance in real-world environments.