W. M. Xiongab,
G. L. Jiangab,
J. Y. Liuab,
Qiang Shengab,
W. J. Chenabc,
B. Wangac and
Yue Zheng*ab
aState Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China. E-mail: zhengy35@mail.sysu.edu.cn; Tel: +86-20-8411-3231
bMicro & Nano Physics and Mechanics Research Laboratory, School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, China. E-mail: chenweijin@mail.sysu.edu.cn; Tel: +86-20-8411-3370
cSino-French Institute of Nuclear Engineering and Technology, Sun Yat-sen University, Guangzhou 510275, China. E-mail: wangbiao@mail.sysu.edu.cn; Tel: +86-20-8411-5692
First published on 11th March 2016
Here we propose a method to detect the degenerated tetragonal vortex states (i.e., the toroidal axis is along the x-, y- or z-axis) in ferroelectric nanodots by applying a controllable surface charge (CSC) condition. Electrodes are placed at two parallel surfaces of the nanodot to form a short circuit. Surface charges with a controllable density are then applied to another two parallel surfaces of the nanodot. Under this CSC condition, a characteristic short-circuit current vs. time (I–t) curve related with the evolution of domain structure in a nanodot can be detected. The evolution paths and the characteristic short-circuit I–t curves of the degenerated vortex states in ferroelectric nanodots have been systematically revealed by our phase field simulations by solving the time-dependent Ginzburg–Landau (TDGL) equations. It is found that the degenerated vortex states exhibit distinct evolution features under the CSC condition. In the stages of placing electrodes and increasing surface charges, one, two, and zero short-circuit I–t peak(s) are observed in the nanodots with 〈100〉, 〈010〉 and 〈001〉 vortex states, respectively. Therefore, the unknown vortex states of a nanodot can be distinguished. We further investigate the effects of temperature and nanodot size on the characteristic I–t curves of the vortex states. The results show that the vortex states can be nondestructively distinguished by applying the CSC condition if the nanodot size is within a moderate range (i.e., 8–12 nm). Our study provides an alternative way of detecting the degenerated tetragonal vortex states in ferroelectric nanodots without the use of a scanning probe microscope, and also sheds light on the application of ferroelectric vortex domain structures in novel devices such as memories, sensors and actuators.
In the literature, using a first-principles-derived effective Hamiltonian approach, Naumov et al.13 predicted that low-dimensional ferroelectric systems under open-circuit condition form VDS to decrease the depolarization energy. The smallest vortex size was found about 3.2 nm, indicating that by using VDS to store information one can reach an ultimate storage density of 60 Tbit per inch2. A phase transition from a polar state to a vortex state by varying the screening condition of the depolarization field has also been further revealed.14 Based on molecular dynamics simulation, it has been found that ferroelectricity with a toroidal ordering can be sustained in nanoparticles with a few lattice constants.15 Meanwhile, quite a lot of researches have been taken on the controllability of VDS in ferroelectric nanostructures through electrical and mechanical means.17–23 Particularly, Prosandeev et al.18 showed that the single vortex state in asymmetric ferroelectric nanorings can be switched by a homogeneous electric field. Chen et al.19–22 studied the effects of mechanical loads on VDS, and found that mechanical loads can not only effectively control the vortex size and orientation but also lead to a novel vortex pining effect as well as the feasibility of vortex switching by a homogeneous electric field.
In contrast with the theoretical progresses, it remains a challenge to characterize and control the nanoscale ferroelectric VDS in experiments. Nevertheless, one can note that much experimental attempts have been made.25–29 For example, Hong et al.27 used focused ion beam technique to prepare well-shaped ferroelectric nanodots and observed VDS in the nanodots via scanning transmission electron microscope. Combining pulse laser deposition and piezoresponse force microscopy, Gao et al.28 prepared well-ordered ferroelectric nanodot arrays, and images of their measurements indicated the existence of bubble domain patterns in the nanodots. Balke et al.26 succeeded to control the formation of VDS in BiFeO3 film, and found an enhanced conductivity at the vortex core.29 With the further improvement of fabrication and characterization techniques in the near future, one can expect that more and more theoretically predicted features of the VDS could be revealed experimentally.
In another aspect, it is well known that a depolarization field would arise inside the ferroelectric if the surface polarization charges have not been completely screened. As a consequence, the domain structure of a ferroelectric can be controlled by varying the screening extent of surface polarization charges.14,30 For a typical capacitor structure, the surface polarization charges of the ferroelectric are compensated by the electron gas from the electrodes. The screening extent can be controlled by choosing proper electrodes. Screening of surface polarization charges can be also achieved by surface ionic charges, e.g., by those from atomic and molecular adsorbates.9,31,32 Interestingly, when placing a ferroelectric in a chemical environment, e.g., oxygen environment, surface redox reaction can occur, and then an electric field across the sample could be induced by the net charges generated during the reaction. In the literature, it was found that VDS of ferroelectric nanostructures are sensitive to the surface charge conditions.14 More recently, Wu et al.23 have predicted that fruitful domain patterns and evolution paths of VDS can be realized in the ferroelectric nanodots via applying a controllable surface charge (CSC) conditions to the nanodots. The sensitivity of VDS to the surface charge condition indicates potential applications of ferroelectrics on charge, gas or other nano-sensors.
For a regularly shaped nanostructure, its vortex states are not only degenerated in chirality but also degenerated in orientation. The typical examples include nanodots (e.g., PbTiO3) with degenerated tetragonal vortex states and nanodots (e.g., BaTiO3) with degenerated orthorhombic vortex states. It is natural to ask if we can distinguish these similar vortex states according to their signals during the same detection. While previous works on ferroelectric VDS focused on the formation and controllability of VDS,12–24,33 there were hardly any works on how to detect the vortex states except those using high resolution scanning probe microscopes to directly probe the local dipole states.27–29 In this paper, we propose a novel method to detect the degenerated tetragonal vortex states of cubic ferroelectric nanodots, and study the size and temperature effects on the domain evolution of the ferroelectric nanodots under a controllable surface charge (CSC) conditions. The distinct evolution behaviors of the degenerated vortex states and consequently the characteristic current vs. time (I–t) curves of the vortex states have been clearly revealed and completely summarized.
The temporal evolution of the spontaneous polarization in ferroelectric nanodot can be simulated by solving the time-dependent Ginzburg–Landau (TDGL) equations, i.e., ∂Pi/∂t = −MδF/δPi (i = 1, 2, 3), where M is the kinetic coefficient related to the domain wall mobility, F is the total free energy, and t is time. In order to simulate the domain structure of ferroelectric materials in nanoscale, it is necessary to consider effects of inhomogeneous electromechanical fields as well as that of surface/interface on the total free energy of the system. In our simulations, the total free energy can be expressed as , where V and S are the volume and surface of the nanodot, respectively. fLand is the so-called Landau free energy density. fgrad stands for the density of gradient energy, which describes the free energy contributed by the spatial polarization variation. The mechanical stress field and its coupling with polarization contribute to the elastic energy density felas. felec is the density of electric energy, including the energy contributed by the depolarization field and external electric field. Due to the truncation of polarization near surfaces, an additional surface energy should be considered, and the surface energy density is fsurf.
A polynomial of Landau energy density fLand is written as37–39
According to previous works,13,33 three degenerated tetragonal-like vortex domain patterns favor to form in the free-standing ferroelectric nanodot under the ideal open-circuit boundary condition as shown in Fig. 1a. Specifically, these three vortex states are defined as 〈100〉 (i.e., |gx| ≫ |gy| and |gz|), 〈010〉 (i.e., |gy| ≫ |gx| and |gz|) and 〈001〉 (i.e., |gz| ≫ |gx| and |gy|) vortex states, according to the direction of toroidal moment, which can be along 〈100〉, 〈010〉 and 〈001〉 axis direction, respectively. Generally, nanodots with different initial vortex orientations have different characteristics, e.g., different current–voltage signals under the same applied voltage. This is our starting point to propose a method to distinguish these three degenerated vortex states through detecting the I–t curves under a CSC condition.
The detection can be divided into two main stages, as shown in Fig. 1b. In stage 1, the short-circuit boundary condition is applied at the two parallel surfaces along 〈100〉 direction. This can be obtained by placing two parallel electrodes with ideal screening capability. Then, in stage 2, surface charges with a controllable density are applied to the upper and lower surfaces of the nanodot. The controllable negative or positive surface charges are produced by the adsorption reaction at the two surfaces. In general, the reaction and thus the amount of surface charges should be a strong function of the adsorption gas pressure. To simplify the calculation, we directly set the amount of the surface charge density to be σ to mimic the CSC condition, but ignore the detailed chemical equilibrium process.
As the short-circuit boundary condition is along 〈100〉 direction, a short-circuit current is generated related to the movement of screening charges in the electrodes. This charge movement is governed by the net polarization along 〈100〉 direction, i.e., 〈P1〉. The short-circuit current is given by I = dQe/dt = Sedqe/dt,47 where Qe is the total charge in the electrodes, Se is the area of the electrode, and dqe/dt is the current density. The above formula shows the positive correlation between the short-circuit current and the current density. dqe/dt is proportional to dP1/dt. For numerical stability of simulations, the following dimensionless variables are employed as , t* = |α0|Mt, P* = P/P0, ε*0 = |α0|ε0, α*1 = α1/|α0|, α*11 = α11P02/|α0|, α*12 = α12P02/|α0|, α*111 = α111P04/|α0|, α*112 = α112P04/|α0|, α*123 = α123P04/|α0|, Q*11 = Q11P02, Q*12 = Q12P02, Q*44 = Q44P02, s*11 = s11(|α0|P02), s*12 = s12(|α0|P02), s*44 = s44(|α0|P02), G*11 = G11/G110, G*12 = G12/G110, G*44 = G44/G110, G′*44 = G44/G110, where α0 is the value of α1 at room temperature, P0 is the magnitude of the spontaneous polarization at room temperature, and G110 is a reference value of the gradient energy coefficients. To simplify the calculation, dP*1/dt* will be calculated instead of calculating the exact short-circuit current.
In Fig. 3c, dependence of dP*1/dt* on σ is plotted, and the corresponding domain patterns of selected points are also inset. During the detection, a current peak appears right after placing the electrodes. It could be explained by the simulated results shown in Fig. 3b. It is found that changes to a nonzero value when placing electrodes. Moreover, the vortex state also maintains in the nanodot. This coexistence of both net polarization and toroidal moment in the nanodot indicates that the domain state is actually a polar-vortex multi-order state. Then, comparing the corresponding domain patterns of labeled B and C points, we could find the vortex core shifts to the side surface of the nanodot with the increase of σ. This is because that an electric field along z-axis positive direction is formed by the surface charges to tilt the dipoles along the z-axis. When σ increases to ∼0.2 C m−2, the toroidal order disappears and a single c-domain state forms. Specially, changes from nonzero to null at this moment, but there is no obvious current change. That is because the change process is so long that the change of for each time step is very little. Then, the single c-domain is stable with the further increase of σ and there is no current peak.
As shown in Fig. 4a and b, for the nanodot initially with 〈010〉 vortex state, a sudden decrease of |gy| from ∼1.2 to 1.0 e Å−1 is observed after placing two parallel electrodes. As the surface charges is applied on the upper and lower surfaces, a net is induced and linearly increases with the increase of σ. Meanwhile, |gy| firstly increases a little bit and then gradually deceases to zero, indicating the vanishing of the toroidal ordering. When σ increases to 0.16 C m−2, drops from 0.38 to 0.11 C m−2. Then, with a further increase of σ, linearly increases with σ. As σ further increases to 0.36 C m−2, changes to zero and only keeps nonzero.
From the corresponding domain patterns and dependence of dP*1/dt* on σ, a 180° domain pattern with a domain wall of ∼2 nm forms after placing the electrodes. As shown in Fig. 4c, this domain pattern is similar to that found in the BaTiO3 nanodot under the high charge screening condition.23 During the process of placing electrodes, keeps zero and no short-circuit current peak is observed. When σ increases to 0.16 C m−2, a current peak appears due to the change of domain structure from the 180° domain pattern to a single ac-domain pattern with the disappearance of domain wall. Then, with a further increase of σ, c-domain state forms, meanwhile, drops to ∼0 and another reverse current peak is observed.
Similar to the previous case, for the nanodot initially with 〈001〉 vortex state, |gz| suddenly decreases from ∼1.2 to 1.0 e Å−1 after placing two parallel electrodes (see Fig. 5a and b). With the increase of σ, |gz| decreases slowly and keeps increasing, while and keeps ∼0. When σ increases to ∼0.26 C m−2, |gz| drops to zero, and has a sudden increase from 0.1 C m−2 to 0.5 C m−2. Then, linearly increases with a further increase of σ. As shown in Fig. 5c, the short-circuit current peak has not been observed during the whole process of increasing σ. According to the domain patterns shown in Fig. 5c, a transformation from vortex state to 180° domain pattern with a domain wall of ∼2 nm is induced by placing electrodes. Then, domain wall tilts and gradually vanishes with the increase of σ. When σ increases to ∼0.26 C m−2, the 180° domain pattern disappears and a single c-domain pattern forms in the nanodot. For this nanodot initially with 〈001〉 vortex state, keeps ∼0 and no current peak can be detected during the detection.
Summarizing the results in Fig. 3c, 4c and 5c, one can see that ferroelectric nanodots with degenerated tetragonal vortex states exhibit quite different behaviors during the detection. In order to clearly illustrate these behaviors, we plot the characteristic I–t curves of the 8 × 8 × 8 nm3 nanodots with degenerated vortex states under different temperatures as shown in Fig. 6. The domain patterns can be given as a function of temperature and σ, and represented by the phase diagrams in Fig. S1 in the ESI.† In the detection, one, two and zero peak(s) in the I–t curves can be observed for the nanodots initially with 〈100〉, 〈010〉 and 〈001〉 vortex states, respectively. These interesting results indicate that an unknown vortex state in a ferroelectric nanodot could be distinguished by observing its characteristic short-circuit I–t curve, and the features of the I–t curves can't be affected by the temperature.
Fig. 6 Dependence of dP*1/dt* on σ in nanodots (8 × 8 × 8 nm3) with 〈100〉, 〈010〉, and 〈100〉 vortex states under (a) T = 0 K, (b) T = 300 K, and (c) T = 400 K. |
As shown in Fig. 7b for the nanodot with initial 〈010〉 vortex state, lineally decreases until σ decreases to 0.25 C m−2. When σ decreases from 0.24 to 0.2 C m−2, increases from zero to 0.5 C m−2, and decreases from 0.44 to 0.13 C m−2. This indicates that an ac-domain state forms. Then, slowly increases with the decrease of σ, and lineally decreases. An a-domain state forms when the ionic surface charges are all removed. After the electrodes are removed, |gy| increases abruptly from zero to 1.2 e Å−1, indicating that the initial 〈010〉 vortex state re-forms. In the nanodot with 〈001〉 vortex state, the domain evolution is similar to the evolution of the nanodot with 〈010〉 vortex state as shown in Fig. 7c. The phase diagrams depicting the domain pattern are summarized in Fig. S2 in the ESI.†
Note that c-domain state forms for all the three vortex states when the nanodot is under a high σ (Fig. 7). Generally, the single c-domain patterns evolved from the three vortex states have no obvious difference, and the re-formed vortex state could be along any one of the directions after the CSC condition is turned off. However, the nanodot could keep the initial vortex orientation after the processes of increasing and decreasing σ. Comparing the Fig. 7a–c, similar single domain patterns are obtained under the condition of σ = 0.6 C m−2, and small differences of polarization components can be observed. Here we consider that the single domain state under a high σ is related to the initial state, and these small differences can affect the domain evolution and determine the final domain structure. In a real measurement experiment consisting of several steps, it is not likely for a system to reach a totally stable state. Therefore, information related to the initial state will exist to affect the evolution of domain structure. Nevertheless, in practical application, we suggest using a lower upper limit of σ in the detection to avoid forming single domain.
Interestingly, the evolution behaviors of polarization and toroidal moment exhibit huge differences in the 14 × 14 × 14 nm3 nanodots with initial 〈100〉, 〈010〉 and 〈001〉 vortex states at room temperature. As shown in Fig. 9a, in the nanodot with initial 〈100〉 vortex state, |gx| drops from 2.06 e Å−1 to zero and increases from zero to 0.57 C m−2 when placing the electrodes on the 〈100〉 surfaces. Then, with the increase of σ, keeps decreasing and keeps increasing. After σ reaches 0.52 C m−2, drops to zero and has a large increase from 0.39 C m−2 to 0.62 C m−2. As shown in Fig. 9b, for the nanodot with initial 〈010〉 vortex state, |gy| decreases from 2.06 e Å−1 to 1.40 e Å−1 after placing the electrodes. Meanwhile, the components of polarization keep zero. With σ increasing, |gy| keeps unchanged and keeps increasing when σ is less than 0.1 C m−2. When σ increases to 0.12 C m−2, |gy| drops to zero and increases from 0.12 C m−2 to 0.41 C m−2. Then, linearly increases with the increase of σ. In the nanodot with initial 〈001〉 vortex state (see Fig. 9c), |gz| keeps a value of 2.06 e Å−1 when electrodes are placed on 〈100〉 surfaces. Meanwhile, another two components of toroidal moment and components of polarization always keep zero. |gz| gradually decreases and linearly increases with the increase of σ. When σ increases to 0.36 C m−2, |gz| drops to zero, has a sudden increase from zero to 0.53 C m−2, and has a small decrease. Then, with a further increase of σ, and linearly decreases and increases, respectively. suddenly decreases from 0.27 C m−2 to ∼0 and increases from 0.33 C m−2 to 0.57 C m−2 when σ increases to 0.54 C m−2.
The dependence of dP*1/dt* on σ in the 14 × 14 × 14 nm3 nanodots with initial 〈100〉, 〈010〉 and 〈001〉 vortex states is shown in Fig. 9d. A current peak appears with the placement of electrodes on the nanodot with initial 〈100〉 vortex state, while no current peaks are observed for the nanodot with initial 〈010〉 or 〈001〉 vortex state at this moment. With the increase of σ, a small current peak appears for the nanodot with initial 〈100〉 vortex state. Different with the results in 8 × 8 × 8 nm3 nanodot, there are no obvious current peaks for the nanodot with initial 〈010〉 vortex state throughout the whole process of increasing σ. For the nanodot with initial 〈001〉 vortex state, two weak opposite current peaks can be observed. The first peak appears with a sudden increase of and another peak appears because of the decrease of from a large value to zero. From the characteristic I–t curves, an unknown vortex state can be detected through this method. The corresponding domain patterns in the nanodots with initial 〈100〉, 〈010〉 and 〈001〉 vortex states are depicted in Fig. 9e–g, respectively. From the evolution of domain patterns during the process of decreasing σ, the evolution behaviors are similar for the nanodots with three degenerated vortex states. 90° domain forms when ionic surface charges are completely removed and a domain pattern with two 〈010〉 vortices forms in the nanodot after removing the electrodes. This indicates that the initial domain pattern has been destroyed after the detection. After the detection, the domain pattern transforms from a tetragonal-like vortex domain pattern to a vortex domain pattern with two 〈010〉 vortices.
To be more systematic, the dependence of dP*1/dt* on σ in the nanodots with their sizes ranging from 6 nm to 14 nm is summarized in Fig. 10. For the 6 × 6 × 6 nm3 nanodot with one of the degenerated vortex states, one current peak appears with the placement of electrodes. During the process of increasing σ, another reversed current peak can be observed. It is found that characteristic I–t curves for the nanodots with initial 〈100〉, 〈010〉 and 〈001〉 vortex states are similar. In addition, the similar results can be calculated in the nanodots with their sizes smaller than 6 nm, e.g., 4 nm (not given). For the nanodots with their sizes ranging from 8 nm to 12 nm, a current peak appears for the nanodots with initial 〈100〉 vortex state when placing two parallel electrodes on 〈100〉 direction, while no current peak is observed for nanodots with another two vortex states. With σ increasing, two opposite current peaks generate for the 8 × 8 × 8 nm3 and 10 × 10 × 10 nm3 nanodots with initial 〈010〉 vortex state, and one current peak is observed for the 12 × 12 × 12 nm3 nanodot with initial 〈010〉 vortex state. Thus, the degenerated vortex states in the nanodot could be distinguished by their characteristic I–t curves. And the final domain patterns are the same with the initial domain patterns, indicating that the detection is nondestructive. In the 14 × 14 × 14 nm3 nanodot, two opposite current peaks generate for the nanodots with initial 〈100〉 and 〈001〉 vortex states while current peaks cannot be detected in the nanodot with initial 〈010〉 vortex state. Differently, in the nanodot with initial 〈100〉 vortex state, one strong current peak is induced by the placement of electrodes and another weak current peak appears during the process of increasing σ. While both of the opposite current peaks for the nanodot with initial 〈001〉 vortex state are detected in the process of increasing σ. Three degenerated vortex states exhibit distinct evolution behaviors. But the final domain patterns shown in Fig. 9e and f indicate that the initial vortex domain states transform from a tetragonal-like vortex domain state to a vortex domain state with two 〈010〉 vortices. The initial domain pattern is thus destroyed during the detection. Moreover, the initial domain pattern in a larger nanodot, i.e., 16 nm, also is destroyed according to the simulated results.
Fig. 10 A summary of the dependence of dP*1/dt* on σ in the nanodots with their dimensions ranging from 6 nm to 14 nm. |
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c5ra25193a |
This journal is © The Royal Society of Chemistry 2016 |