Frustration of Negative Capacitance in Al2O3/BaTiO3 Bilayer Structure

Enhancement of capacitance by negative capacitance (NC) effect in a dielectric/ferroelectric (DE/FE) stacked film is gaining a greater interest. While the previous theory on NC effect was based on the Landau-Ginzburg-Devonshire theory, this work adopted a modified formalism to incorporate the depolarization effect to describe the energy of the general DE/FE system. The model predicted that the SrTiO3/BaTiO3 system will show a capacitance boost effect. It was also predicted that the 5 nm-thick Al2O3/150 nm-thick BaTiO3 system shows the capacitance boost effect with no FE-like hysteresis behavior, which was inconsistent with the experimental results; the amorphous-Al2O3/epitaxial-BaTiO3 system showed a typical FE-like hysteresis loop in the polarization – voltage test. This was due to the involvement of the trapped charges at the DE/FE interface, originating from the very high field across the thin Al2O3 layer when the BaTiO3 layer played a role as the NC layer. Therefore, the NC effect in the Al2O3/BaTiO3 system was frustrated by the involvement of reversible interface charge; the highly stored charge by the NC effect of the BaTiO3 during the charging period could not be retrieved during the discharging process because integral part of the polarization charge was retained within the system as a remanent polarization.

Scientific RepoRts | 6:19039 | DOI: 10.1038/srep19039 equation of the displacement continuity at the DE/FE interface under the short circuit condition with or without external bias voltage should be written as equation (1), where ε 0 represents the vacuum permittivity; E f (E d ) is the electric field inside the FE (DE) layer; ε b is the background dielectric constant of the FE layer; and ε d is the dielectric constant of the DE layer. While the presence of ε b is widely accepted in electrostatic calculations of critical phenomenon 23,24 , depolarization [14][15][16][17][18] , and dielectric response of the DE/FE superlattice structures [25][26][27] , the precise definition and its value are controversial. From literatures, various ε b values, such as optical dielectric constant (~5) 14,15,25,28 , ~10 18,29 , ~50 22,30 and > 100 17,31 , could be found for various perovskite FE materials. In this work, 50 were taken for the ε b of the c-axis oriented BTO epi-layer to calculate the thermodynamic states and the dielectric response of the FE single layer as well as the DE/ FE stacked layer. Actually, ε b of BTO varies according to the electric field because it could vary with variation of P s along the applied field direction. However, this is generally the case for randomly oriented material, where its P s state is heavily dependent on the applied field. In this work, where the epitaxial BTO film is c-axis oriented and its c-axis lattice parameter is even elongated along the surface-normal direction, the P s is always aligned along the out-of-plane direction. This makes the ε b quite invariant throughout the most part of voltage application. There could be bias conditions where the permittivity increases when the material is depolarized near the coercive voltage region. In fact, the depolarized state variation is quite small in one case (Fig. S3 of on-line supplementary information of ref. 32). but could be as three times high as the polarized state 33 . However, such voltage region is very narrow compared with the entire tested voltage region. Hence, a constant permittivity assumption in the calculation induced minimum error. In many theoretical cases 14,15,17,18,20,22,25,[28][29][30][31] this value has been taken as constant.
As can be understood from the equation (1), the displacement in the DE layer could be induced by only the ε 0 ε d E d term. Therefore, if ε 0 ε d of the DE layer is much smaller than the capacitive contribution from ε 0 ε b and P s , E d becomes very high. When the DE layer is very thin, interface charge (σ i ) can be formed at the DE/FE interface by carrier injection across the DE layer. Under this circumstance E dep in the FE layer decreases, which can stabilize P s . The electric field generated by the presence of P s and σ i at the FE and DE layer under the short circuit condition can be represented as follows, where E int d is the internal electric field across the DE layer, and l f (l d ) is the thickness of the FE (DE) layer. Figure 1a shows the distribution of E dep and E int d within the Al 2 O 3 (AO)/BTO stacked layer, where the BTO layer possesses P s and the AO/BTO interface contains σ i . The free energy (or thermodynamic potential) of an order parameter (P s ) in a FE layer can be described by LGD equation as shown in equation (4).  Then, the electrostatic field acting on P s , can be expressed as where α, β and γ are Landau coefficients of a FE material. Figure 1b,c show the electric force-polarization diagrams for the cases of σ i = 0 and σ i < 0, respectively. Here, the electric force can be easily calculated by multiplying the electric field with charge. The reason why a electric force is invoked can be easily understood from Fig. 1d Here, the E ext f is the portion of E ext applied over the FE layer. The coefficient of P s 2 term ) is determined by the relative magnitude of E dep and E pol . If E pol > E dep , α′ has a negative value and the FE layer is in the FE state. If E pol < E dep , α′ becomes positive, and the FE layer becomes paraelectric-like. This state is critical for the emergence of the NC effect from the DE/FE structure. Detailed material parameters in equation (6) for the AO/BTO system are summarized in table 1 (Note: the ferroelastic energy had been also taken into account in energy calculations, see the equation (8).). It should be noted that the equation (6) represents the thermodynamic energy function for a given σ i . If σ i varies, the function needs to be rewritten for new σ i , and the transition states between different values of σ i cannot be thermodynam i cally described by this method. Therefore, only the thermodynamic states before and after the σ i change are described analytically in this work, and the transition between them is only empirically described.
Then the capacitance of the paraelectric-like DE/FE system could be achieved from the general definition of capacitance. In fact, for DE/FE system, there are two distinctive layers, so the capacitance can be calculated from either layers. The derivation processes are described in detail in method section based on displacement-continuitiy at the interface between DE and FE layers and minimization of electrostatic energy. Capacitance of the DE/FE system can be represented by equation (7).
Based on these formalisms, the electrical behavior of the AO/BTO bilayer structure was examined. First, the case with σ i = 0 is considered in Fig. 2. Figure 2a shows the free energy diagram (U-P diagram) of a 5 nm-thick AO/150 nm-thick BTO bilayer structure calculated using the equation (6) at room temperature. For reference, the U-P curves of a single layer AO and BTO were also plotted. With this geometry, the U-P curve shows a single minimum at P = 0, suggesting that the ferroelectricity of the FE layer is totally destabilized due to the influence by the large depolarization field. This behavior is influenced by the relative thicknesses of the DE and FE layers. Figure 2b shows the graph of (d 2 U/dP 2 ) −1 at P = 0, which corresponds to α′ , as a function of AO thickness for the given BTO thickness of 150 nm calculated by the equation (6). The capacitance showed critical variations at 3.5 nm, which is called the critical thickness (l cr ). When the AO film is thinner than the l cr , C has a negative value and diverges to − ∞ as l cr is approached. This corresponds to the unstable state of the DE/FE system near P = 0, so such overall negative capacitance cannot be experimentally achieved. By contrast, when the AO film is thicker Material T c (K) ε b /ε d α(10 5 C −2 ·m 2 ·N) β(10 8 C −4 ·m 6 ·N) γ(10 9 C −6 ·m 10 ·N) Ref. than the l cr , C has a positive value and diverges to ∞ as l cr is approached. This is a very useful stable state of the DE/ FE for the capacitance boost near l cr , which is due to the involvement of the NC state of a FE layer. Nevertheless, there are two limitations on the use of such increased capacitance, which can be found from Fig. 2c,d. Figure 2c shows the capacitance -voltage (C-V) curve of the 5 nm-thick AO/150 nm-thick BTO bilayer. Here, V was calculated by multiplying the field and thickness of each layer and summing them. The C-V curve of single layer AO is also plotted within the same graph and shows a constant value which corresponds to a dielectric constant of 8.9. Within the voltage range from ~− 5 V to ~5 V, the C value of the bilayer is higher than that of the AO single layer, suggesting the emergence of the NC state within the FE layer. However, at voltages outside this range, the C value decreases to lower values than the value of a single AO layer, which is due to the fact that the capacitance of the BTO layer changes from negative to positive at certain high voltages. Figure 2d shows the calculated P-V curves of the bilayer. For reference, P-V curves of single layer AO and BTO are also plotted. The P-V curve of the bilayer does not contain any negative slope region, suggesting that AO/BTO is overall in the PC state, whereas BTO layer shows the NC state. Usefulness (and limitation too) of the AO/BTO bilayer as the charge storage capacitor can be understood from these figures. For example, when the bilayer capacitor was biased from − 6 V to + 6 V, ~0.4 C m −2 is stored within the capacitor. In contrast, the single layer AO capacitor can store only ~0.2 C m −2 . For the single BTO layer capacitor, it can store a much higher value of ~0.7 C m −2 . However, when the capacitor voltage was released to 0 V, only ~0.05 C m −2 could be extracted from the single BTO layer capacitor because it is now with FE state, so ~0.65 C m −2 remains within the capacitor as a P r . For the single layer AO and bilayer capacitor, half of the stored charges is released by the same operation. As the applied voltage range increases, the stored (so released) charge density increases linearly for the AO capacitor and non-linearly for the bilayer capacitor, and finally the charge density values become almost identical for − 20-20 V range. This can be understood from the decrease of capacitance at higher voltages for the case of bilayer capacitor while that of AO capacitor is constant over a whole range of voltage in Fig. 2c. The maximum storable charge density in the bilayer cannot be higher than the 2P s of the BTO layer, which is the ultimate limitation of DE/FE systems as high capacitance capacitors.
One may be curious about what would happen when the thickness of the AO layer is near l cr ? According to Fig. 2b, the capacitance can be infinite, whereas Fig. 2c shows that the voltage range for such enhanced capacitance becomes infinitesimal. Therefore, there is an upper bound for the drivable charge density, which is ~2P s . Meanwhile, there is another critical side effect that mitigates the emergence of the capacitance boosting effect as shown in the next section. Formation and influence of σ i on DE/FE system. Figure 3a shows the variations in the electric field (See the equation (22) in method section) over the AO and BTO layers as a function of E ext when the field was applied over the 5 nm-thick AO/150 nm-thick BTO structure with σ i = 0. In this graph, E ext was simply calculated by dividing the applied voltage (V app ) by a total film thickness (155 nm). For the actual field calculation in each layer, the E ext was divided into two parts: E ext f and E ext d , which are inversely proportional to the dielectric constant of each layer (50 for BTO and 8.9 for AO), and E dep , calculated from the equation (2), was added to estimate the net electric field over the BTO layer. Similarly, E int d was calculated from the equation (3), which is determined by the field exerted by the spontaneous polarization from the FE layer and interface charge density. E int d was added to the component of E ext over the AO layer. It is quite notable that the internal field across the BTO layer decreases when E ext increases within − 2.5 MV cm −1 < E ext < + 2.5 MV cm −1 , meaning that the BTO layer works as a NC layer under this circumstance. Such decrease in the internal field of the BTO layer is compensated by the increase of the internal field across the AO layer, meaning that the capacitance boosting might be acquired. This outcome is obvious from the NC operation of the BTO layers. It is also notable that the field over the AO layer is extremely high even for the quite small E ext , which significantly influences the charge distribution as will be discussed below. Figure 3b shows a schematic energy band diagram of the Pt/5 nm-thick AO/150 nm-thick BTO/Pt capacitor when E ext = 100 kV cm −1 was applied to total structure (V app = 1.55 V). Due to the NC effect of BTO under this bias condition, the BTO band is tilted in the opposite way to the applied bias direction which is compensated by the very high tilting of the AO band in accordance with the applied bias direction. The opposite tilting of the BTO band within the NC region occurs because E ext f is overcompensated by E dep . Under this circumstance, a deep potential well is formed at the AO/BTO interface of which the depth is deeper than the conduction band offset at the Pt/AO interface. Due to a very high band tilting of the AO layer, in addition to its small thickness (5 nm), there must be a very high chance of carrier injection (by most probably tunneling) as represented by the lateral arrows in Fig. 3b. When such carrier injection occurs, the σ i can compensate for the polarization charge within the BTO layer totally or partially at the interface, and E dep will be diminished. This means that the NC effect could be also diminished under this circumstance. Nevertheless, it has to be noted that the influence of σ i on the NC operation of the BTO layer is dependent on the bias application and carrier transport across the AO layer as discussed below. An approximate time estimation of the charge transport across the 5 nm-thick AO layer by the Fowler-Nordheim tunneling to compensate about 0.2 C m −2 varies from several μs to several tens μs depending on the bias voltage magnitude and other interface conditions. The influence of such carrier injection effect will be discussed in detail with Figs 4 and 5.
By contrast, it could be quite different for the case of the STO/BTO as shown in Fig. 3c. The band diagram was calculated for SrRuO 3 (SRO)/25 nm-thick STO/50 nm-thick BTO/SRO structure. Due to a non-linear dielectric response of the paraelectric STO, the calculation was performed using a self-consistence method based on the LGD equation of STO. When E ext = 100 kV cm −1 , the calculated E int d of the STO layer was 364 kV cm −1 and relative dielectric constants of STO layer were calculated as 215. Due to the relatively high dielectric constant of the STO, the internal band tilting of the STO layer (~0.9 eV) was much lower than that of the AO layer under the identical E ext condition. The BTO band also tilts in the opposite direction to the bias voltage suggesting that the BTO layer is in NC mode. Under this band configuration, the electron tunneling from the SRO into the DE/FE interface is not expected to be active, and thus, the chance for observing the NC effect from this structure would be high 9 .
This carrier injection does not necessarily correspond to the total elimination of the NC effect as long as the value of σ i is invariant during the subsequent bias application. Here, the U-P and P-V curves of the previously mentioned AO/BTO structure were formulated again based on the equation (6) assuming σ i , present at the AO/ BTO interface, to be − 0.2, − 0.1, 0, 0.1, and 0.2 C m −2 , which stayed invariant throughout the entire voltage application. The results are shown in Fig. 4a,b. As can be understood from Fig. 4a, the structure shows certain monostable polarization values corresponding to a minimum energy, with different σ i values. Figure 4b shows that the P-V curves are non-hysteretic due to the monostable configuration of polarization under these circumstances, and the capacitance enhancement could be achieved for all cases. However, the voltage region to observe the NC operation (region with steep slope in the P-V curve) varies according to the σ i values. The presence of σ i induced an invariant internal field inside the structure and shifted the P-V response along the voltage direction. Therefore, the NC effect is achieved at a shifted V (or E ext ) without a hysteretic P-V switching behavior according to this internal field effect. This can be qualitatively understood as that the invariant σ i stabilizes only one of the two possible P s 's of the BTO layer, and this stabilized P s decreases uniformly making the BTO layer be within the NC region when a bias, whose polarity is opposite to this stabilized P s , is applied to the AO/BTO structure. Nonetheless, as discussed previously, the change of σ i along with the change in the bias polarity largely decreases the amount of retrievable charge at the discharging step, hindering the NC effect. Experimental proof of such NC frustration is demonstrated in the next section.   well-defined interface structure. From this HRTEM image, the lattice-parameter along the c-axis (normal to the film surface) was determined to be ~0.408 nm which corresponds to an in-plane misfit strain of ~− 1.2% with SRO bottom layer. More detailed structural characterization of this bilayer structure can be observed in on-line SI. Figure 5b shows the experimental P-V curves of a single layer BTO film with a Pt top electrode (diamond symbol) and AO/BTO bilayer film with two different AO thickness (circle symbol for 5 nm and square symbol for 9 nm). The severe shift of the P-V curve of the BTO single layer into a positive bias direction is due to the work function mismatch between Pt and SRO electrodes with possible contribution of epitaxial strain. Being different from calculation results shown in Fig. 4b, the P-V curve of the AO/BTO bilayer shows clear emergence of (distorted) hysteresis curve for the two AO thicknesses. This phenomenon can be understood from the following calculation. From equation (6), P-V curve of this system can be simulated. Because of epitaxial strain, modified LGD equation was adopted to correctly describe the U-P relationship as equation (8), which takes into account the ferroelastic energy terms 34 ,  Fig. 5b. The simulation fits the experimental P-V curve quite well in the voltage region of ~− 1 V-10 V for upper branch, and of − 10 V-1 V for lower branch, which means that the σ i barely changed within each voltage region. The significant mismatch outside these voltage regions and transition between the two branches can be understood from the variations in σ i according to the bias voltage application. For example, the P-V curve of the lower branch corresponds to the case where the negative σ i (− 0.120 C m −2 ) is stabilized mostly in the negative bias region. However, as the voltage increased into the positive bias region, σ i changes to positive value most probably by tunneling through the thin AO layer (indicated by an upward green arrow in the figure). As a result, when the bias voltage reaches to + 10 V it becomes 0.080 C m −2 . With decreasing bias voltage, this positive interface charge appears to be retained down to ~− 1 V but changes back to the negative value also by another tunneling process (indicated by downward green arrows in the figure). Being compared with the P-V curve of a single layer AO (indicated by a black line in the figure), the P-V curves of the 5 nm-AO/BTO bilayer showed a higher slope especially in the P region with yellow background, suggesting that the capacitance of the double layer is higher than that of the single layer AO. This corresponds to the NC operation of a BTO layer.

Experimental study on NC effects in
Nevertheless, the consecutive change in σ i occurring in this region makes it impossible to observe the desired NC effect in the AO/BTO structure. The capacitance enhancement could have been achieved if charging-discharging process follows the trajectory of (non-hysteretic) P-V curve within NC region. However, the change of σ i during the high voltage application shifts the P-V curve from one position to another making the overall P-V curve shape to be ferroelectric-like hysteretic one. This means that an integral part of the stored charges during the charging step remained in the bilayer capacitor as remanent polarization as well as the interfacial charge with the opposite sign during the discharging step. It has to be noted that the largely stored charge must be discharged spontaneously with voltage decrease when the capacitance was enhanced by the NC effect of the FE layer within the DE/FE bilayer, which was not the case in this AO/BTO sample. The tunneling through the AO layer can be easily anticipated from the very high field when BTO layer operates in NC mode (Fig. 4b).
The time range for the P estimation at each V value during achieving the P-V loop was 100 μ s. This time constant is long enough to induce the sufficient charge transport across the AO layer which partly compensates for the polarization switching. It could be further noted that the trapped charge density was still lower than the P s value of the BTO single layer, suggesting that induced charges on the metal electrodes were partly responsible for the stabilization of the ferroelectric bound charges.
In conclusion, the capacitance boost effect in a DE/FE system by the NC effect of a FE layer, which was originally suggested by Khan 8 , could be realized under certain limited conditions, such as no FE poly-domain formation and well-balance between the thickness and material parameters of the DE and FE layers as long as the total capacitance is in the positive regime. However, the original studies had taken some problematic assumptions, which induced a self-contradictory outcome from the calculation of the DE/FE system (See on-line SI for details). In addition, when a low dielectric DE layer is adopted, its LGD formula is not generally well known making the application of the previous formalism (Landau-Khalatnikov model) 8 to calculate the total free energy to such cases improbable. Therefore, an alternative model was suggested in this work that could calculate the capacitance at the DE/FE system based on the general theory on the depolarization effect 14-18 of the FE layer when the FE bound charge is not compensated well. This approach explains the experimental results more accurately. The model was also adopted to the case where the DE/FE interface had trapped charges which could (partly) compensate for the P s of the FE layer. The charge trapping could be induced by tunneling through the thin DE layer during the NC operation of the FE layer, which augmented the potential applied over the DE layer. The interfacial charging appeared to be almost inevitable when a low permittivity DE layer, such as AO, was adopted of which the thickness must be very thin to match the absolute capacitance values of the DE and FE layers. The trapped charges stabilize one of the two P s 's of the FE layer, which could have induced the emergence of the NC effect from the BTO layer in the AO/BTO bilayer during the subsequent voltage application with opposite bias polarity. However, when the FE layer falls within the NC region, a significant change in the trapped charge occurs making the opposite P s stabilized. Therefore, a FE-like hysteretic P-V loop is achieved, major portion of the accumulated charges during a voltage application is retained as the remanent polarization and injected charges in the capacitor during the subsequent voltage are released. This is detrimental to use the AO/BTO capacitor as an extremely high capacitance capacitor. It is also possible that some other factors that have not been considered in this work could contribute to the detailed P-V behavior of the AO/BTO structure. However, the dynamically varying interfacial charge model can provide a reasonable explanation to the experimental results. Pulse-type measurement will provide another details on the switching kinetics, which will be reported elsewhere. It was also elucidated that even when the positively infinite capacitance is realized by the perfect match between the PC of DE and the NC of FE, the overall driven charge density cannot be higher than 2P s of the FE layer. This is because as the capacitance increases, the voltage range for the enhanced capacitance decreases inversely proportional to the capacitance.

Methods
Experimental setups. The BTO layer was epitaxially grown on a SRO/DyScO 3 single crystal substrate by a PLD, and the AO layer was deposited by an ALD, as shown in Fig. 5a. Details for the PLD processes of the BTO and SRO layers are reported elsewhere 34 , and the ALD of Al 2 O 3 was performed using trimethylaluminum and O 3 (with a concentration of 250 gm −3 ) as the Al-precursor and oxygen source, respectively, at a sample temperature of 250 °C. Pt top electrode with an area of 6000 μ m 2 was fabricated by a lift-off lithographic process followed by an electron-beam evaporation of 70 nm-thick Pt layer. The P-V measurements were performed using Aixacct TF-2000 ferroelectric tester with an AC frequency of 1 kHz in a virtual ground mode. It is assumed that the polarization (P s ) of the FE layer is homogeneous and that the trapped charges existing at the interface between the DE and FE layers fully or partly compensate the P s . It is further assumed that the displacements in the dielectric and ferroelectric layers ( ) absence of any interfacial trapped charges, continuity of displacement at the DE/FE interface must be maintained. Under this circumstance, the accumulated charge, Q, can be described either at M/D or F/M interface as follows; where E tot f and E tot d are total electric field in ferroelectric and dielectric respectively, and ε d is a permittivity of an insulator. Here, the E tot f and E tot d encompass both electric field components from the externally applied voltage and internal charge mismatch. The capacitance of the MIFM capacitor, therefore, is