## Metastable ferroelectricity driven by depolarization fields in ultrathin Hf0.5Zr0.5O2

We first present the scaling trends of HZO capacitors and the main switching characteristics of the thinnest (5 nm) ferroelectric HZO obtained. Thickness scaling and proper choice of electrode materials are both important for operation of single layer FTJ. Semiconductor bottom electrodes such as Ge or Nb-doped SrTiO_{3} (NSTO) are used because of their lower carrier densities compared to metals which allow for uncompensated polarization charges, the latter being very important for the FTJ functionality. Ge produces sharp, crystalline interfaces with HZO^{17} therefore, it is preferred over Si which typically forms low-k interfacial oxides or silicates. On the other hand, NSTO has been used successfully for the fabrication of BaTiO_{3} (BTO)/NSTO FTJs^{4}, where the giant tunneling electroresistance effect was obtained by modulating the Schottky barrier thickness at the interface, for various Nb concentration^{18}, therefore it could be a good candidate also for HZO/NSTO FTJs. HZO is deposited by plasma assisted molecular beam deposition^{17,19,20} and crystallization annealing is performed via Rapid Thermal Annealing (RTA)^{20} or Flash Lamp Annealing (FLA)^{17} as described elsewhere^{17,19,20,21} and in Methods.

The main structural characteristics of the thinnest TiN/HZO/Ge capacitor structures are shown in Fig. 1 and Supplementary Note 1. Grazing Incidence X-ray Diffraction (GIXRD) scans before (gray line, amorphous) and after annealing (blue line) are shown in Fig. 1a. The latter shows no evidence of the non-polar monoclinic phase (green lines at 28.4 and 31.7 deg^{22}) while the main peak at 30.7 deg (Fig. 1a) is attributed to a combination of the ferroelectric orthorhombic phase Pca2_{1}^{23}) and the tetragonal phase P42/nmc^{22}.

From the Scherrer fit of the peak at 30.7 deg, with k-factor ranging from 0.7 to 1, the average crystalline grain size along the direction ω−2θ/2 = 15 deg (approximately in-plane) is estimated to be between 4 and 5.6 nm. A HZO thickness of 5 nm is accurately measured by X ray reflectivity (Fig. 1b) and cross-sectional Transmission Electron Microscopy (TEM) (Fig. 1c and Supplementary Fig. 1). No interfacial layers at the top and bottom surface of HZO are detected by TEM (Fig. 1c and Supplementary Fig. 1), although a non-discernible amorphous TiO_{x}N_{y} layer at the top interface cannot be excluded. Based on filtering of the Fast Fourier Transform (FFT) pattern (Fig. 1d)), the grain size of the HZO polycrystalline film is estimated between 5 nm and 20 nm (Supplementary Fig. 1), noting that the lower limit is compatible with the GIXRD results discussed above. Based on FFT (Fig. 1d, e) and real image (Fig. 1g) simulations, the observed nanocrystallites (Fig. 1f) are compatible with the ferroelectric orthorhombic phase Pac2_{1}. However, the distinction between orthorhombic and tetragonal at the macroscale based on GIXRD (Fig. 1a) is difficult. It is anticipated that possible tetragonal phase will be present in rather small quantities since, after wake-up, the remanent polarization *P*_{r} is high (>20 μC cm^{−2}) (see Supplementary Note 2 and discussion below) indicating the dominance of the ferroelectric orthorhombic phase.

Figure 2 shows the evolution of the switching characteristics as a function of decreasing HZO thickness for Ge and NSTO Metal Ferroelectric Semiconductor (MFS) capacitors. The general trend is that thick HZO capacitors show full ferroelectric hysteresis loops in the pristine state, while as the thickness is reduced, the hysteresis loops become distorted and at 5 nm, they are pinched (AFE-like) yielding low remanent polarization. Full hysteresis loops are recovered (“waked-up”) after field cycling while the two switching peaks in the displacement current nearly merge.

For thick HZO films (Fig. 2a and b), only few cycles are needed but for thinner films, larger number of cycles are required to wake-up the devices. In some cases (as in Fig. 2c, g), the devices are not affected by cycling at all and remain pinched. It should be noted that for 4 nm HZO and thinner (Fig. 2d, h, l, p), ferroelectricity becomes obsolete and HZO behaves like a normal dielectric.

The waked up devices down to the thinnest 5 nm HZO show high remanent polarization *P*_{r} > 20 μC cm^{−2} after correction, with only small parasitic (non-switchable) polarization (see PUND measurements in Supplementary Fig. 2) and a low coercive voltage *V*_{c} ~ 1 V. These performance characteristics are very promising for low voltage/low power FTJ memory devices. Moreover, the high *P*_{r} indicates the predominance of the ferroelectric orthorhombic phase as already mentioned above.

The capacitance-voltage measurements on p-Ge/HZO/TiN capacitors at 1 kHz are shown in Fig. 3 featuring the characteristic “butterfly” structure due to ferroelectricity in HZO. Due to the “butterfly” structure which masks the capacitance variations with bias, and the low energy gap of Ge (consequently the high minority carrier concentration) the C-V at room temperature looks more like an MFM capacitor rather than MFS. Note however, that at lower temperatures where the minority carrier concentration is dramatically reduced, a more typical MOS-like behavior is recovered with a clear distinction between accumulation and depletion^{17}. The accumulation capacitance *C*_{acc} (measured at > −2V) of the 5 and 15 nm capacitors are measured to be 3.1 and 1.7 μF cm^{−2}, respectively. If a single layer HZO is assumed, the measured *C*_{acc} yields a value of HZO relative dielectric constant *ε*_{F} = 18 which is not compatible with the orthorhombic ferroelectric phase, since the latter has an expected value of *ε*_{F} in the range between 30 and 40 [refs. ^{20,24,25,26,27,28}]. It is therefore necessary to consider an interfacial capacitance *C*_{int} in series with the FE capacitance *C*_{F} in order to account for the measured *C*_{acc}. Taking into account both 5 nm and 15 nm capacitors and by assuming the same *C*_{int} and *ε*_{F}, values of *C*_{int} ~ 6 μF cm^{−2} and *ε*_{F} ~ 39 are extracted. This value of *C*_{int} is considerably smaller than the values of 30–110 μF cm^{−2} reported for Metal-Ferroelectric-Metal (MFM) capacitors^{27,29} with very thin or no interfacial layers.

In our case, *C*_{int} could be dominated from the semiconductor region near the interface due to the finite screening length *l*. Using an expression^{3} for \(l=\frac{{{{{{\rm{h}}}}}}}{q}* \sqrt{\frac{{\varepsilon }_{0}{\varepsilon }_{{Ge}}}{3{m}^{* }}}* {\left(\frac{3}{8\pi }\right)}^{\frac{1}{3}}* {\left(\frac{1}{{n}_{h}}\right)}^{\frac{1}{6}}\) (*h* is the Planck constant and *q* is the electron charge), derived for metallic systems and assuming a hole concentration *n*_{h} = 1.6*10^{17} cm^{−3}, effective mass *m** = 0.29*m*_{e}^{30} and *ε*_{s} (Ge) = 16, a value of *l* ~ 3.5 nm and a corresponding capacitance *C*_{s}~*ε*_{0}*ε*_{s}/*l~*3.6 μF cm^{−2} are estimated. Although *C*_{s} is only a rough estimate of the semiconductor capacitance, it does not differ much from the *C*_{int} derived from capacitance measurements. In fact, a capacitance *C*_{it} due to interface defect charging could be added in parallel to *C*_{s} so the total capacitance *C*_{int} = *C*_{s} + *C*_{it} could better match the measured value of 6 μF cm^{−2} (see above). In conclusion, the semiconductor capacitance, including possible contribution from interface defects, could account for the rather small value of measured *C*_{int} even in the absence of interfacial (dead) dielectric layers.

### Metastable ferroelectricity and pinched pristine hysteresis curves

To analyze the switching characteristics of the thin HZO MFS capacitors,

we assume a 1st order ferroelectric with Gibbs free energy [Eq. 1].

$$G=\frac{1}{2}a{P}^{2}+\frac{1}{4}\beta {P}^{4}+\frac{1}{6}\gamma {P}^{6}-{E}_{F}P$$

(1)

where *P* is the polarization order parameter and *α* < 0, *β* < 0 and *γ* > 0 are the Landau coefficients. From simple electrostatics (see Supplementary Note 3, for details and refs. ^{31,32}), the electric field *E*_{F} in the ferroelectric is given by

$${E}_{F}=\frac{V}{{t}_{F}}{\left[1+\frac{{C}_{F}}{{C}_{{int}}}\right]}^{-1}-\frac{P+\sigma }{{\varepsilon }_{0}{\varepsilon }_{F}}{\left[1+\frac{{C}_{{int}}}{{C}_{F}}\right]}^{-1}$$

(2)

Where *V* is the applied bias on the MFS, and *C*_{F}, *C*_{int} are the capacitances of the ferroelectric and interfacial layers, respectively and *t*_{F} is the FE thickness. The first term in the right-hand side of the equation accounts for the reduction of the volage in the ferroelectric by a factor \({\left[1+\frac{{C}_{F}}{{C}_{{int}}}\right]}^{-1}\), since part of the voltage drops on interfacial (dead layers), or, on the semiconductor near the interface. The second term in the right-hand side of Eq. (2) is the electric field built as a result of uncompensated charge. The surface charge *σ* is the charge trapped at pre-existing defects at the interface; it has a sign opposite to that of *P* and assists in the compensation of polarization charges thus stabilizing *P*. Charge *σ* becomes important only during field cycling (wake-up) when charge injection and trapping occur and it will be discussed in subsequent sections. Here in this section, we study the pristine case where *σ* ≈ 0. It should be noted that when *V* = 0, then *P* = *P*_{s} (spontaneous polarization) and *E*_{F} *=* *E*_{dep}= \(-\frac{{P}_{s}}{{\varepsilon }_{0}{\varepsilon }_{F}}{\left[1+\frac{{C}_{{int}}}{{C}_{F}}\right]}^{-1}\), that is the DF.

Using (1) and (2) with *σ* ≈ 0, the FE thickness dependent expression for *G* becomes

$$G= \frac{1}{2}a{P}^{2}+\frac{1}{4}\beta {P}^{4}+\frac{1}{6}\gamma {P}^{6}+\,\frac{1}{{\varepsilon }_{0}{\varepsilon }_{F}}{\left[1+\frac{{t}_{F}{C}_{{int}}}{{\varepsilon }_{0}{\varepsilon }_{F}}\right]}^{-1}{P}^{2}\\ -\frac{V}{{t}_{F}}{\left[1+\frac{{\varepsilon }_{0}{\varepsilon }_{F}}{{t}_{F}{C}_{{int}}}\right]}^{-1}P$$

(3)

In the last equation, *C*_{F} has been expressed as *C*_{F} *=* *ε*_{0}*ε*_{F}*/t*_{F}, where *t*_{F} is the FE thickness. A corresponding expression for *C*_{int} cannot be given due to the lack of knowledge about the interfacial (dead) layer characteristics. Instead, experimental *C*_{int} values are deduced from *C*–*V* analysis (see Fig. 3 and related discussion above) which yield *C*_{int}.

The polarization switching dynamics under the influence of a time varying bias *V* are described by the Landau-Khalatnikov (L-K) equation \(\delta \frac{{dP}}{{dt}}=-\frac{{dG}}{{dP}}\) ^{33,34} where *δ* is the kinetic (or viscosity) coefficient. Using Gibbs free energy from Eq. (3), the L-K equation becomes:

$$\delta \frac{{dP}}{{dt}}=\,-\left[\left(a+\,\frac{2}{{\varepsilon }_{0}{\varepsilon }_{F}}{\left[1+\frac{{t}_{F}{C}_{{int}}}{{\varepsilon }_{0}{\varepsilon }_{F}}\right]}^{-1}\right)P+\beta {P}^{3}+\gamma {P}^{5}-\frac{V}{{t}_{F}}{\left[1+\frac{{\varepsilon }_{0}{\varepsilon }_{F}}{{t}_{F}{C}_{{int}}}\right]}^{-1}\right]$$

(4)

The L-K Eq. (4) is solved by applying time-varying, triangular waveforms for *V* (see Supplementary Note 4, Supplementary Note 5) to simulate the experimental *P-V* curve while equation \(J=\frac{{dD}}{{dt}}={\varepsilon }_{0}\frac{d{\rm E}}{{dt}}+\frac{{dP}}{{dt}}\) is complementarily used to calculate the displacement current *J*–*V* curves. The simulated curves (Fig. 4a and b) using a kinetic coefficient *δ* = 2.5*10^{6} s cm F^{−1} and Landau parameters *α* = −23*10^{10} cm F^{−1}, *β* = −21.4*10^{19} cm^{5} F^{−1} Cb^{−2} and *γ* = 15.5*10^{28} cm^{9} F^{−1} Cb^{−4}, are in semiquantitative agreement with the experiment.

The deviations are partly due to non-idealities of the experimental curves. A small offset in the voltage (~0.15 V) was set in the calculations in order to account for rigid shifts to higher bias due to small workfuncion differences between top TiN and bottom Ge electrodes. The simplified phenomenological LGD model which ignores extrinsic factors affecting switching dynamics (see discussion section) also accounts for the discrepancies between for the experimental and simulated curves.

At a fixed FE thickness of *t*_{F} = 5 nm, *C*_{int} and *ε*_{F} are the two parameters which control the depolarization ~*P*^{2} term in Eq. (3), hence the pinching of the hysteresis curve. It should be emphasized that the pinching characteristics (reduced *P*_{r}) are best simulated with *C*_{int} = 5.5 μF cm^{−2} and *ε*_{F} = 39 which compare well with the values of *C*_{int} and *ε*_{F} obtained from the analysis of the *C-V* measurements (Fig. 3). Our modeling indicates that the positive depolarization term (~*P*^{2}) in the Gibbs free energy counteracts the negative ¼ *βP*^{4} (and the negative ½ *αP*^{2}) term and results in the triple potential well (Fig. 4c) which describes a metastable HZO ferroelectric. The stable paraelectric phase (*P* = 0) is then responsible for the pinching of the *P-V* hysteresis reducing the remanent polarization to near zero values. More insights are given in the discussion and conclusions section below.

To better illustrate the influence of *C*_{int} on the ferroelectric stability and the switching characteristics, the Gibbs free energy landscapes and the *P-V* curves for two different *C*_{int} values of 10 and 20 μF/cm^{2} are given in Fig. 5. When *C*_{int} is has an intermediate value ~10 μF cm^{−2}, the 6 nm film shows metastable ferroelectricity (pink curve) and a weakly pinched loop with reduced *P*_{r}. The 4 and 2 nm HZO show very small remanent polarization and an ordinary dielectric behavior. When *C*_{int} becomes larger (20 μF cm^{−2}), the situation is improved since the 6 nm HZO has stable ferroelectricity and a fully open hysteresis loop with high *P*_{r}.

The 4 nm HZO is marginally stable ferroelectric (blue curve), with a nearly open ferroelectric loop and only the very thin 2 nm film shows metastable ferroelectricity resembling an ordinary dielectric behavior. Therefore, it can be concluded that by increasing the *C*_{int} above 20 μF cm^{−2} it is possible to drive the system to a stable ferroelectric state even for very thin films <4 nm.

In MFM capacitors, an infinitely high *C*_{int} corresponds to perfect metal electrodes forming sharp interfaces^{27,29} with the HZO (without dead layers). However as already briefly mentioned in the introduction, at least one of the electrodes should be less than perfect to allow for incomplete compensation of the polarization charge, which is essential for the operation of ultrathin FTJs. Therefore, in most cases of interest related to FTJs, *C*_{int} is expected to be small resulting in loop pinching. Nevertheless, severely pinched loops at small thickness <6 nm can be recovered by injecting charge during field cycling, as described in the following section.

### Field cycling (“wake-up”) in the framework of Landau-Ginzburg-Devonshire theory

Charge injection^{35} can occur during field cycling since pre-existing defect states are filled with trapped electrons which could tunnel through interfacial layer barriers, if such barriers exist. This mechanism (Fig. 6) provides efficient screening of polarization charges mitigating *E*_{dep} thus stabilizing polarization. Both, polarization *P* and applied bias *V*, control the states available for trapping and consequently the total trapped charge.

Although the analysis presented below is made for the case of identical I and M-layers on either side (Fig. 6a), with appropriate adaptations it could be valid more generally for a structure with different insulator (I) and metal (M) layers on the opposite sides including the cases where one I-layer is missing or one M-layer is a semiconductor. As a first approximation, a constant energy density of states is considered here.

Once established (Fig. 6b), the polarization lowers the potential for electrons at the interface by an amount *Φ* proportional to *P* according to *Φ* = (*t*_{F}/2)|*E*_{dep} | = *t*_{F}*P*/2*ε*_{0}*ε*_{F}(1 + *C*_{int}/*C*_{F}). As a consequence, a portion of interface states within an energy *ΔE*_{Φ} = *eΦ* moves below *E*_{F} and becomes accessible to electrons for trapping.

An applied external bias *V* further lowers the potential (Fig. 6c) at the interface by an amount *V*_{int} = *V*/2(1 + *C*_{int}/*C*_{F}) making more states accessible to electrons in an energy interval *ΔE*_{V} = e*V*_{int}. The total states accessible for charge trapping are within the energy interval *ΔΕ* = e(*Φ* + *V*_{int}) therefore, the total trapped charge, in absolute value, is

$$\left|\sigma \right|=e{n}_{{tr}}\triangle E=\frac{1}{2}e{n}_{{tr}}{\left(1+\frac{{C}_{{int}}}{{C}_{F}}\right)}^{-1}\left(\frac{{t}_{F}}{{\varepsilon }_{0}{\varepsilon }_{F}}P+V\right)$$

(5)

In Eq. (5), *n*_{tr} is the trapped electrons density per unit energy which is determined by solving the Shockley Read Hall^{36,37} rate equation ∂*n*_{tr}/∂t = *c*_{n}(*D*_{it} −*n*_{tr}) −*e*_{n}*n*_{tr}. *D*_{it} is the density of interface states per unit energy and *c*_{n}, *e*_{n} are the capture and emission rates^{38}.

In Fig. 7a the trapped carrier density per unit energy is plotted as a function of field cycles and for two different *D*_{it}, while the details near the origin are shown in Fig. 7b. The carriers are captured in the first half of the cycle, while a smaller portion is emitted back to the electrodes during the second half of the cycle. A positive balance of carriers remains captured after each complete cycle so gradually the trapped density increases and finally saturates at a value that depends on the *D*_{it}: a larger *D*_{it} allows for more trapped charge density as can be seen from the comparison between the green and red curves in Fig. 7a.

The corresponding total charge density *σ* captured in all energy levels made available by *P* and *V* is calculated using Eq. (5) and is plotted in Fig. 7c. The detail at the origin in Fig. 7d shows the different contributions from the polarization *P* (red) and the bias *V* (blue) according to Eq. (5) as well as the total *σ* (black). It should be noted that *σ* has always opposite sign to both, the polarization charges and the bias therefore it changes sign in each half period when *P* and *V* also change sign. The two interfaces have charges of opposite sign which alternates in each half period. Setting a relatively high value of *D*_{it} ~ 1.4 × 10^{13} eV^{−1} cm^{−2}, the calculated charge is accumulated to both interfaces increasing in amplitude and saturating after about 160 cycles at a value of about 10 μC cm^{−2} which is a substantial fraction of the polarization charge (~20 μC cm^{−2}) (Fig. 7c).

This indicates that after cycling, the polarization charge is sufficiently screened, thus minimizing the DF and stabilizing ferroelectricity.

A large *D*_{it} determines the screening efficiency of the polarization charges and ensures the opening of the ferroelectric loop. On the other hand, *e*_{n} (consequently *c*_{n}) determine how fast the traps are filled up to the saturation value. Higher *e*_{n} (and *c*_{n}) ensures faster filling so wake-up and loop opening is easier occurring only after a small number of cycles.

Introducing the contribution of trapped charge *σ* (Eq. (5) and Fig. 7c)) in the expression for the electric field *E*_{F} (Eq. (2)), additional *P*^{2} and *P* terms appear in the Gibbs free energy (Eq. (3)) and L-K equations (Eq. (4), and Supplementary Fig. 5)), respectively, that counteract the depolarization terms and make ferroelectricity stable again alleviating the pinched characteristics after cycling is completed. The results are summarized in Fig. 8 above.

Although a direct fitting of the experimental curves is not possible due to non-idealities in the experimental data, a qualitative agreement between experimental data (Fig. 8a) and simulation (Fig. 8b) is observed. Using *D*_{it} = 1.4 × 10^{13} eV^{−1} cm^{−2} and *e*_{n} = 20 sec^{−1} the simulation reproduces the trend of the P-V curves which shows a progressive opening of the hysteresis loop as the number of cycles increases. The loop opening is accompanied by a corresponding transition from a metastable ferroelectric state (pristine, black curve) to a stable ferroelectric state (160 cycles, pink curve) as seen from the Gibbs free energy plots in Fig. 8c.

It should be noted that both *D*_{it} and *e*_{n} are compatible with experimental observations in the literature and in the present work. Briefly, we refer to our electrical admittance spectroscopy (Supplementary Note 6): the *D*_{it} value at Ge/HZO interface is estimated from ac conductance losses to be 0.93–3.5 × 10^{13} eV^{−1} cm^{−2} (Table S1), which is very similar to the *D*_{it} value deduced from the simulation results in Fig. 8. It should be noted that a *D*_{it} which is an order of magnitude smaller (see Supplementary Note 7), is insufficient to open the loop inducing only marginal changes to the pristine curves. The low *D*_{it} could explain the behavior of some of the thinnest samples where the pristine, pinched characteristics do not recover after extensive cycling (See Fig. 2, and Supplementary Fig. 7).