Introduction

Random numbers are essential in everyday life. While small amounts of random numbers can be generated using mechanical methods such as throwing a dice or spinning a roulette wheel, larger quantities required for fields like statistics, computing, cryptography, and other branches of science and engineering typically rely on electronic random number generators. Various entropy sources are used in hardware-based electronic methods, including thermal noise from diodes1, charge variations in capacitors1,2, and current fluctuations in memristors3,4. Recently, random telegraph noise (RTN) in semiconductor devices has emerged as a promising candidate for entropy sources5,6,7. The stochastic transition between two discrete current levels in RTN offers the potential to generate random bit-strings without additional software and costly circuitry. However, the transient and unstable nature of RTN limits the development of reliable random number generators8.

Two-level systems (TLS) are well-established building blocks for RTN9,10,11,12,13. In electronic devices, an individual point defect in dielectric oxides functions as a TLS at the microscopic level. The point defects typically create defect states in the bandgap of the oxides. If one of the defect states is shallow, a conduction electron can occupy the state altering its dynamic behavior. Through a combination of quantum tunneling and external coupling to the environment, the point defect incoherently flips between the electron-occupied and -unoccupied states, resulting in the intrinsic RTN. At the mesoscopic or macroscopic scale, numerous point defects distributed over a certain volume contribute to charge trapping phenomena. Previous studies have shown that such collective point defects can function as a quasi-TLS and generate RTN in dynamic equilibrium14. However, it is evident that the TLS based on the collective point defects is inherently unstable at room temperature. Various factors, such as the spatial migration of point defects, the structural relaxation of charged point defects, and the presence of metastable states contribute to the evolution of TLS15,16. The complex interplay between these factors increases the number of accessible electronic states, often resulting in typical 1/f noise rather than RTN17,18,19,20. Additionally, frequency-independent noise may dominate the current signal due to excessive thermal energy21. Therefore, it is desirable to establish a novel TLS that can secure the entropy source along with a simple two-state scheme at room temperature for developing advanced random number generators.

Herein, we introduce a two-level quantum system (TLQS) based on SrRuO3/LaAlO3/Nb-doped SrTiO3 (SRO/LAO/Nb:STO) heterostructure, incorporating two different types of point defects, the oxygen vacancies (VO) and TiAl defects. Temporal localization of electrons at the TiAl defects shifts the energy level of nearby VO by instantaneous Coulomb interaction, resulting in the two-level current fluctuation across the interface. The RTN-like current signals from the TLQS exhibit high stability at room temperature. The two-level fluctuation could be successfully maintained for over 169 s without intermittent vanishing. We demonstrate that this highly stable two-level current fluctuation in the TLQS can be utilized to generate random number bit-strings, which are suitable for practical stochastic machine learning algorithms. This study provides a fundamental approach to designing reliable entropy sources using the complementary interactions between cation and anion point defects in oxide heterostructures, which are crucial for hardware-based random number generators.

Results

Two-level quantum system

Normal RTN is based on the presence of two energetically different but related states in an electronic system (Fig. 1a). The most well-known mechanism involves the TLS based on the charge trapping and detrapping of VO in oxides. Regardless of the microscopic mechanism at play, the temporary localization of electrons at the VO sites results in two discrete states with slightly different energy levels22,23,24. In these classical TLS, or fluctuators, RTN is driven only when the thermal energy kBT of the system is not larger than the height of the energy barrier between two energy wells. The barrier height is primarily determined by the spatial distribution and energy level of the defects18,25. Since VO defect levels are typically shallow in oxides, the barrier height is quite low. Additionally, other factors such as interstitial defects, interfacial magnetism, and ferroelectric dipoles can influence the barrier shape and create metastable states15,26,27. Therefore, at room temperature, normal RTN is difficult to control and often exhibits unstable and transient behaviors, including the iterative disappearance, temporary change in stochastic properties, and the multi-level fluctuations8,28. Addressing this fundamental instability is crucial for practical applications of RTN signals.

Fig. 1: Two-level quantum system (TLQS) designed with two types of point defects in oxide heterostructures.
figure 1

a Schematic coordination diagram for a classical two-level systems (TLS). Ee represents the energy of electrons. The normal RTN involves the random transition between the state A and the state A’. Since shallow defects act as major charge traps with a small energy barrier (\({\phi }_{b}\)) the state transition is fast and unstable. b TLQS employs the shallow (Type-1) and the deep (Type-2) defects to control the state transition behavior. The fast and unstable transitions of the Type-1 defects are suppressed, as depicted by the gray-colored dashed lines, by controlling the internal potential profile. Instead, the complementary interaction between the different types of defects creates state A and B. The state transition depends on the temporal localization of electrons at Type-2 defects, resulting in the highly stable current fluctuation. c Schematic illustration of electron transport in TLQS. (Top) At a Schottky interface, Type-1 defects work as electron hopping sites, resulting in a specific electron transfer rate \({\varGamma }_{A}\). (Bottom) When an electron occupies the type 2 defects, the energy level of Type-1 defects upshifts and leads to the increased electron transfer rate \({\varGamma }_{B}\). Thus, depending on the charge state of the Type-2 defects, the tunneling current fluctuates between discrete two states. d TLQS device structure based on SrRuO3/LaAlO3/Nb-doped SrTiO3 heterostructure.

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To overcome the limitations of classical fluctuators, we design a TLQS of the oxide-based Schottky junction incorporating two different types of point defects (Fig. 1b). The first type, referred to as Type-1 defects, are shallow defects utilized primarily for electron transport through the Schottky junction. Unlike in the classical TLS, the Type-1 defects do not directly function as an entropy source (see the dotted and bold lines for state A). By controlling the internal potential profile, we can minimize the fast and unstable transition between the occupied and empty states of the shallow defects at room temperature (Supplementary Fig. 1). Instead, we employ a second type of point defects, so-called Type-2 defects, which possess a deeper energy level. The Type-2 defects stochastically capture and release electrons with a comparatively low trapping frequency. The deep-level traps do not significantly contribute to the electron transport process but do influence the energy levels of nearby Type-1 defects (i.e., the shallow defects) through Coulomb interaction. Consequently, depending on the charge state of the Type-2 defects, the energy level of the Type-1 defects consistently transitions between two discrete values (Fig. 1c), resulting in the two-level current fluctuation.

Based on to this hypothetical model, we build a TLQS using SRO/LAO/Nb:STO heterostructures (Fig. 1d). The band offset between the metallic SRO and the semiconductor Nb:STO induces a Schottky barrier at the interface29. The barrier height varies with the thickness of the LAO insertion layer. Previous studies have shown that, with a monolayer of LAO, the barrier height is approximately 0.5 eV30. Despite the significant Schottky barrier height, the inevitably present VO defects form shallow defect states (~0.28 eV below the conduction band edge)23, enabling defect-mediated tunneling. We utilize the VO defects as Type-1 defects. Additionally, we use an Al-deficient LAO as the insertion layer to create interface-confined cation defects31,32,33. The interdiffusion of Ti ions from Nb:STO into the Al-deficient LAO layer forms TiAl defects, which have deeper mid-gap states34. We exploit the TiAl defects as Type-2 defects. In this study, we focus on the interaction between the Type-1 and Type-2 defects and the resultant two-level current fluctuation in SRO/LAO/Nb:STO heterostructures.

Preparation of oxide heterostructures for TLQS

The SRO/LAO/Nb:STO heterostructure is epitaxially synthesized by a pulsed laser deposition technique. After growing the SRO and LAO thin films, the top SRO layer is partially etched with an aqueous solution of NaIO4 to fabricate circular pads (Supplementary Fig. 2)17,19. Fig. 2a shows an annular dark field scanning transmission electron microscopy (ADF-STEM) image of an as-grown SRO/LAO/Nb:STO heterostructure. It is seen that the LAO monolayer is coherently grown on the Nb:STO(001) substrate, forming an atomically sharp interface with the top SRO layer. The structural coherence is confirmed by X-ray diffraction reciprocal space mapping in Supplementary Fig. 3. The presence of the high-quality LAO monolayer is further confirmed by atomic-resolution energy-dispersive X-ray spectroscopy (EDS) investigations (Fig. 2b). The EDS maps clearly show that the LAO monolayer has full coverage with minimal interdiffusion into the top SRO or the bottom Nb:STO region. To form TiAl defects at the LAO/Nb:STO interface, we controlled the cation stoichiometry of the LAO layer. By growing a slightly Al-deficient LAO film on the Nb:STO substrate, we induce ionic diffusion of Ti to the LAO layer, thereby forming interface-confined TiAl defects (Fig. 2c)33. The La:Al ratio can be reproducibly controlled by adjusting the energy fluence of excimer laser during pulse laser deposition (PLD) process (see Methods)31. The cation stoichiometry of the Al-deficient LAO thin films was investigated using X-ray photoelectron spectroscopy (XPS). Figure 2d, e show the La 3d5/2 and Al 2p core level spectra of the as-grown Al-deficient LAO film, respectively. The cation stoichiometry was evaluated based on the XPS peak area and the relative sensitivity factors for the individual atoms31. The La/Al ratio was estimated as 1:0.95 (Supplementary Note 1), where the TiAl defects can readily be formed at the interface. Our density functional theory (DFT) calculations indicate that TiAl defects can be formed within the LAO layer between SRO and Nb:STO (Supplementary Note 2). The calculated formation energy of the TiAl defects is −0.77 eV with respect to elemental Ti and Al solids under standard temperature and pressure, suggesting that Ti atoms can spontaneously replace Al in the LAO layer. Although direct observation of individual TiAl defects is highly challenging, both our XPS and DFT analyses consistently confirm the presence of Type-2 defects (i.e., the TiAl defects) in the SRO/LAO/Nb:STO heterostructures. In the case of VO defects, unlike TiAl defects, we intended to achieve a uniform distribution of a small amount of VO around the interface rather than confining their spatial distribution to a specific location (Supplementary Fig. 4). In this way, we could facilitate intimate interactions between the two types of point defects at the interface.

Fig. 2: TLQS in SrRuO3/LaAlO3/Nb-doped SrTiO3 (SRO/LAO/Nb:STO) heterostructure.
figure 2

a ADF-STEM image and b the atomic-resolution EDS elemental mapping of the epitaxially-grown SRO/LAO/Nb:STO heterostructure. The scale bars represent 2 nm. c Schematics illustrating the formation of point defects in the LAO monolayer. The Al-deficient LAO film spontaneously forms TiAl defects, which are expected to intimately interact with the nearby oxygen vacancies. d, e X-ray photoelectron spectroscopy (XPS) (d) La 3 d and (e) Al 2p core level spectra for an as-grown SRO/LAO/Nb:STO heterostructure. The blue curve represents the main La 3d5/2 peak. The violet and green curves correspond to satellite peaks associated with charge transfer from La ions. f, g XPS Ti 2p spectra (f) before and (g) after the electric refresh process. The green and violet peaks represent the main Ti 2p signals. The dark green and blue sub-peaks correspond to Ti3+-related components. h,i Atomic force microscopy (AFM) images measured on the surface of the SRO/LAO/Nb:STO heterostructure (h) before and (i) after the refresh process. j, k Surface potential maps (j) before and (k) after the refresh process measured by Kelvin probe force microscopy. Note that the surface potential is increased by the refresh process, while the surface topography is preserved. The bottom panels quantitatively show the contact potential difference VCPD profile. Source data are provided as a Source Data file.

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In the SRO/LAO/Nb:STO heterostructure, the VO defects are expected to act as electron hopping sites and facilitate the electron transfer across the Schottky barrier at the interface. However, VO defects can capture some electrons, forming polarons35, and do not always work as hopping sites. Therefore, we employ a simple refresh process to remove interface-localized electrons. By applying a positive bias voltage of +3 V across the SRO/LAO/Nb:STO heterostructure for 60 s, we effectively remove the interface-confined electrons. The removal of the excess electrons is confirmed by comparing the XPS spectra of Ti 2p core levels obtained before and after the refresh process (Fig. 2f, g). The Ti3+ peak centered at 462.9 eV, which reflects the presence of excessive electrons in the Nb:STO, is significantly reduced after the refresh process. The relative area of the Ti3+ peak is decreased from 22.7% to 14.1%. Importantly, this refresh process does not drive the migration or creation of VO defects (Supplementary Fig. 5), which are often associated with resistive switching in dielectric oxides36,37,38. To further understand the effect of the refresh process, we examined changes in surface potential using Kelvin probe force microscopy (KPFM). We scanned the atomic force microscopy (AFM) probe tip over a \(500\times 500\) nm2 region on the sample surface with a constant voltage of +3 V to visualize the effect of the refresh process. The surface topography was not changed by the refresh process (Fig. 2h, i). Previous studies have shown that the accumulation of VO in perovskite oxides results in local volume expansion, which can be identified by AFM imaging39,40. Thus, our topography images indicate that the spatial distribution of VO is not significantly changed by the refresh process. Additionally, the surface potential of the refreshed region increases, as shown by the potential maps taken before and after the refresh process (Fig. 2j, k). If the electropositive VO had moved downward, the potential would have decreased. Furthermore, the change in current due to the refresh process is only ~109% (Supplementary Fig. 6), which is orders of magnitude lower than that observed in typical resistive switching devices41. Thus, taken all together, these results confirm that the refresh process removes the localized electrons from the interface with minimal change in the VO distribution.

Two-level current fluctuations in TLQS

Figure 3 shows the electrical characteristics of the TLQS device based on the SRO/LAO/Nb:STO heterostructure. The inset shows the circular SRO patterns on the TLQS devices, which serve as top electrodes. After a refresh process with +3 V for 60 sec, the current trace over time I(t) was measured at a constant bias voltage Vread at room temperature. The TLQS device clearly exhibits two-level current fluctuation at a Vread of −0.03 V (Fig. 3a). The current fluctuates between two discrete levels of ~−0.377 μA (State A) and ~−0.365 μA (State B) for a long period of time. Considering the transient properties of conventional RTN, the two-level fluctuation in TLQS is remarkably stable at room temperature. We observed that the two-level fluctuation was maintained for over 169 s without intermittent vanishing. In measurements lasting over 1800 s, only a small number of intermittent vanishings were observed. Additional two-level fluctuation data, including extended current traces and evidence of reproducibility, are provided in Supplementary Fig. 7. This high stability reveals that Type-2 defects, such as TiAl defects, can work as a high-quality entropy source. Obviously, the charge trapping by deep defects, involving the transition of their charge-state, is less influenced by environmental conditions, such as a small change in temperature, low-energy photon absorptions, and external electromagnetic fields, as compared to that by shallow defects (Supplementary Note 3). Thus, given a reasonable electronic accessibility, the cation-related deep defects (i.e., Type-2 defects) can serve as a better entropy source for the TLQS than the anion-related shallow defects (i.e., Type-1 defects) in terms of stability. Indeed, the TLQS device exhibited stable operation over an exceptionally long period exceeding one year, demonstrating robustness against visible light exposure and temperature variations (Supplementary Note 4). In addition, we confirmed that these two-level current fluctuations are significantly influenced by the thickness of the LAO layer (Supplementary Fig. 11) but not by the size of the SRO electrode (Supplementary Fig. 12). This indicates that the observed two-level current fluctuations arise from the point defects, uniformly distributed at the LAO/Nb:STO interface, as designed.

Fig. 3: Two-level current fluctuation in TLQS.
figure 3

a Current trace over time I(t) measured at Vread of −0.03 V. The insets are optical images of TLQS devices. The scale bars represent 2 mm and 200 μm, respectively. b Current power spectral density (PSD) SI(f) spectrum measured at the same voltage. The fitting curves (blue) clearly indicate the presence a distinct Lorentzian component (red) along with a typical 1/f-type noise (green). c Histogram of 3578 analog current values representing discrete two levels. d Time lag plot obtained from the identical current data set. The state transition between the two levels is seen. e–g Histograms of (e) uniformity U, (f) mean entropy, and (g) hamming distance evaluated from the generated 209 bit-strings of 32 bit each. h Autocorrelation function with error bars at different lag numbers. The error bars represent the standard deviation at each number of lags. Source data are provided as a Source Data file.

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The role of Type-2 defects can be further clarified by electrical noise analysis. Figure 3b shows a current power spectral density (PSD) SI spectrum measured with the TLQS device at the same Vread. The overall PSD spectrum clearly consists of two different noise components: a typical 1/f-type (\({S}_{I}\left(f\right)\propto 1/f\): green curve) and a 1/f2-type or Lorentzian noise (\({S}_{I}\left(f\right)\propto \tau /(1+{(2\pi f\tau )}^{2})\): red curve), where f and τ represent frequency and a characteristic time constant. Given the broad and uniform distribution of VO in the SRO/LAO/Nb:STO heterostructure, the 1/f-type noise is attributed to electron trapping and de-trapping processes by VO around the interface25. On the other hand, the Lorentzian component indicates the presence of another type of point defects which serve as major charge traps. Note that the characteristic time constant reflects the defect level and defect position20. Therefore, the appearance of a single Lorentzian spectrum indicates that Type-2 defects exist and are spatially confined to a specific location (i.e., within the LAO monolayer). For comparison, we suppressed the formation of TiAl defects in a SRO/LAO/Nb:STO heterostructure and confirmed that the two-level current fluctuations, along with the corresponding Lorentzian spectrum, disappeared (Supplementary Note 5).

The electrical characterizations unambiguously show the presence of two distinct current levels (Fig. 3c) with transition occuring randomly. Based on this two-level fluctuation behavior, now we focus on the randomness characteristics of the current data generated by the TLQS. Figure 3d shows a time lag plot (TLP) of the output current data taken from the TLQS. The TLP, which plots the current value at a certain moment Ii+1 versus the previous value Ii, directly visualizes the state transition behavior of the device6. The TLP of the SRO/LAO/Nb:STO heterostructure shows that, although the thermal fluctuation at room temperature makes it less sharp, the transition clearly occurs between the two states of A and B. To further evaluate the randomness of these transitions, we assess 4 representative parameters, the uniformity, Shannon entropy, hamming distance, and correlation42. For these randomness tests, analog current values are first binarized as “1” or “0” using the mean value as a threshold (the blue plot in Fig. 3a). Note that the sampling rate can influence the random characteristics of the original data when converting analog data into a digitized bit-string. Therefore, we examined various sampling rates below the data acquisition rate of 39 Hz and selected 8 Hz, as it showed the best randomness characteristics after the sampling process (Supplementary Fig. 14). The binarized 6688 bits are divided into 209 bit-strings of 32 bit each. The total 209 bit-strings generated by the TLQS are then subjected to the randomness tests.

Figure 3e shows the histogram of uniformity evaluated from the 209 bit-strings. We define the uniformity U as the proportion of “1” out of 32 bits for each bit-string. The mean value of U is estimated as ~0.492, which is quite close to the ideal value of 0.5. This indicates that the TLQS generates “1” or “0” with almost the same probability. This is one of the important properties of random number generators, because it guarantees the generation of unbiased random numbers. Based on the U values, the uncertainty of the bit-strings generated by the TLQS is quantified by the Shannon entropy. The Shannon entropy ES is defined as \({E}_{S}=-\left(U{\log }_{2}U+\left(1-U\right){\log }_{2}\left(1-U\right)\right)\). Notably, the maximum entropy value of 1 is achieved when the U is exactly 0.5. Figure 3f shows the histogram of ES values evaluated from the same 209 bit-strings. The averaged ES is evaluated as ~0.98 with a standard deviation of 0.01, which implies that the randomness of the bit-strings is comparable to that of the already-established random number generators42,43.

Another important parameter that can evaluate the strength of the randomness is the hamming distance. The intra hamming distance quantifies the difference between a pair of bit-strings generated by the TLQS. For security purposes in cryptography, the 50% of the string length is considered to be an ideal hamming distance. Figure 3g shows the histogram of the mean intra hamming distance calculated from the same datasets. Remarkably, the intra hamming distance values exhibit a Gaussian distribution with the mean value of 16.48, that is ~51.5% of the 32-bit bit-strings. This performance indicates that the random bits generated by the TLQS are weakly biased and thus difficult to decipher by brute force trials.

We also evaluate the autocorrelation function (ACF) of the current datasets. The ACF values can reveal the existence of short-range periodicity in bit-strings. The autocorrelation of +1 and −1 indicate the maximum correlation and the anticorrelation, respectively. When the autocorrelation is 0, the bits in the bit-strings are considered to have no correlation at all. Figure 3h shows the plot of autocorrelation versus lag obtained from the total 209 bit-strings. The existence of the short-range periodicity in the consecutively-generated bits can be identified by high magnitude of autocorrelation at low lags. Thus, the small and flat ACF values in Fig. 3h indicate that the short-range periodicity in the bit-strings is negligible, and the data generated by the TLQS is highly random.

Lastly, we verify the randomness of the TLQS bit-strings using the National Institute of Standards and Technology (NIST) randomness test suite (NIST 800-22)43,44. A total of 500,000 bits obtained from the TLQS device were analyzed by the NIST 800-22 test suite with the standard protocols. The random bit-strings from the TLQS device successfully pass all the 15 NIST statistical tests (Supplementary Note 6), confirming their high randomness quality.

Evaluation of TLQS-generated randomness through machine learning

Next, as a proof of concept, we examine the randomness of the bit-strings generated by the TLQS by using them in a stochastic machine learning algorithm, the stochastic gradient descent (SGD)-based neural network training. Figure 4a shows the SGD-based image super-resolution process (Supplementary Note 7). Given pairs of high-resolution and low-resolution images, our Very Deep Super Resolution (VDSR) model learns how to recover the high-resolution images from the corresponding low-resolution images45. Note that the SGD repeatedly calculates gradients of the objective function with respect to the model parameters using randomly-sampled training data (Fig. 4b)46. That is, independent and identically-distributed (IID) random variables are one of the important prerequisites for fast and accurate training (Supplementary Note 8).

Fig. 4: Stochastic gradient descent (SGD)-based neural network training based on bit-strings generated by TLQS.
figure 4

a Simplified illustration of image super-resolution with VDSR. Note that, in this image super-resolution process, random number sequences are necessarily used for the random sampling of VDSR image dataset. b Schematics explaining the effect of random shuffling for SGD-based model training. The figure shows a two-dimensional parameter space depicting the loss landscape. The deeper (i.e., darker) the landscape, the lower the loss value. c, d The random shuffling enables faster and more accurate training. (c) Training loss and (d) validation accuracy curves for the network training with different shuffling methods. e–g The high-resolution images recovered from an identical low-resolution image with (e) no-shuffling, (f) Python’s NumPy shuffling, and (g) TLQS-based shuffling, respectively. The smooth and clear image recovered with the TLQS-based shuffling indicates the high quality of the random number sequences generated by the TLQS. Source data are provided as a Source Data file.

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We perform the model training with different data shuffling schemes: the conventional Python-based shuffling and the TLQS-based shuffling (Supplementary Note 9). For comparison, we perform the model training with no shuffling as well. Figure 4c, d show the training loss and validation accuracy curves obtained with the different data shuffling methods. We first find that the TLQS-based shuffling increases the speed of model training as compared to the training with no shuffling. This result verifies the TLQS bit-strings consist of random numbers with uniform distribution. Additionally, the TLQS-based shuffling provides a learning curve comparable to that of the Python’s numpy random number generator47. Considering the already-proven performance of the numpy random number generator based on PCG-64 algorithm48, these results imply that the TLQS generates random numbers with practically strong randomness.

Figure 4e–g show the high-resolution images recovered from the low-resolution images using no-shuffling, the Python-based shuffling, and the TLQS-based shuffling, respectively. Figure 4e shows that there exists a blocking symptom on the image, caused by JPEG compression algorithm49. The compression artifacts remaining on the image indicate that the training without shuffling does not make sufficient progress due to the biased gradient approximations. In contrast, the other two images show similarly smooth borderlines, indicating that the SGD runs well with either the TLQS or the numpy’s random number generator. It should be noted that the hardware-based TLQS ensures the high quality of random number sequences, similarly to the already-established software random number generator. This implies that the TLQS offers a promising possibility to develop a hardware-based true random number generator with physically unclonable functions.

Before ending this paper, we emphasize that the TLQS holds great promise for practical device applications44. The key to TLQS lies in forming a heterointerface uniformly incorporating two types of point defects. Therefore, if the interface is properly designed, the TLQS can operate effectively even at thicknesses below a few nanometers, which is a significant advantage in terms of device integration. Additionally, previous studies have demonstrated the feasibility of epitaxially growing STO on commercial Si substrates50,51, suggesting that the proposed TLQS is already compatible with existing Si-based CMOS systems. Furthermore, by incorporating recently proposed free-standing membrane structures52,53, the TLQS device could be fabricated in a crossbar array configuration, paving the way for integration into real electronic systems. Although the data throughput of the TLQS device is lower than that of conventional ultrafast fluctuators, it can still be applied to practical applications when integrated with a pseudo-random number generator, such as a nonlinear feedback shift register, similar to existing low-throughput devices5,44.

Discussion

In summary, we report a TLQS based on SRO/LAO/Nb:STO heterostructure that drives highly stable two-level current fluctuation at room temperature. The TLQS employes two different types of point defects, the VO as Type-1 (electron hopping sites) and the TiAl defects as Type-2 (electron trapping sites). The VO defects facilitate electron tunneling across Schottky barrier at the interface, while the TiAl defects trigger the two-level fluctuation by modulating the energy level of VO. The TLQS maintained the two-level fluctuation for at least 169 s at room temperature without intermittent vanishing, revealing remarkable stability. We utilized this stable two-level current fluctuation to generate random bit-strings and confirmed their use in practical stochastic machine learning algorithms for image super-resolution. This study presents a fundamental approach to designing reliable entropy sources based on point defects in oxides. Notably, we emphasize that the complementary interaction between cation and anion defects is key to achieving stable two-level current fluctuation in oxide heterostructures. Further understanding of the complex interactions among polar point defects will provide a way to tune the random behavior of current signals from oxide-based electronic systems.

Methods

Preparation of SRO/LAO/Nb:STO heterostructures

The SRO and LAO thin films were epitaxially grown on (0.05 wt%) Nb:STO (001) substrates by pulsed laser deposition (PLD). First, the as-received Nb:STO substrates were etched by a buffered hydrofluoric acid and annealed at 900 °C for 6 h. Then, a LAO monolayer was grown on the thermally-treated Nb:STO substrate. The deposition rates for SRO and LAO were calibrated by XRR measurements. To grow Al-deficient LAO films, a LAO ceramic target was ablated by a KrF excimer laser (\(\lambda \sim\) 248 nm) with a repetition rate of 3 Hz and an energy fluence of ~1.1 J/cm2. Note that the standard stoichiometric LAO films can be grown by a slightly higher fluence of ~1.3 J/cm2. During the growth of LAO thin films, the temperature of the substrates and oxygen partial pressure was kept as 550 °C and 0.75 mTorr, respectively. Subsequently, the SRO top layer was deposited at 610 °C. The oxygen partial pressure was set to be 250 mTorr. The repetition rate and the laser fluence were, respectively 3 Hz and ~1.1 J/cm2. Lastly, the as-grown SRO/LAO/Nb:STO heterostructures were cooled down to room temperature in oxygen atmosphere. All the electrical contacts were fabricated with a commercial indium without conventional lift-off processes.

Scanning transmission electron microscopy (STEM) measurements

A cross-section specimen for STEM analysis was fabricated using a focused ion beam (FIB) machine (Helios G4, FEI) to achieve sub 30 nm thickness. The atomic resolution annular dark field STEM (ADF-STEM) image and EDS spectra of the SRO/LAO/Nb:STO sample were measured using a double Cs-corrected transmission electron microscope (Spectra Ultra, Thermo Fisher) operated at a 300 kV acceleration voltage with a beam convergence semi-angle of 21.4 mrad. Both ADF-STEM and EDS images were obtained with a screen current of 150 pA, an image size of 1024 \(\times\) 1024 pixels, a pixel size of 12.2 pm, and a dwell time of 10 \({{\rm{\mu }}}{{\rm{s}}}\). For ADF-STEM imaging, detector inner and outer angles were set to 39 mrad and 200 mrad, respectively. EDS data were collected using an Ultra-X EDS detector, and each elemental image was constructed by integrating the charactersitic x-ray of Ru-\({L}_{\alpha }\), La-\({L}_{\alpha }\), Sr-\({K}_{\alpha }\), Ti-\({K}_{\alpha }\), Al-\({K}_{\alpha }\), and O-\({K}_{\alpha }\), respectively.

Surface characterization by Kelvin probe force microscopy (KPFM)

The surface potential before and after the electrical refresh process was characterized using KPFM mode of an atomic force microscopy (AFM) system (NX10, Park Systems). We performed the amplitude-modulated KPFM measurements with the AC modulation voltage of 1 V at 17 kHz. A commercial Pt/Cr-coated tip (ElectriMulti75-G, Budget) was used for the KPFM measurements.

Electrical charaterizations

The current trace over time was measured by a semiconductor analyzer, Keithley 4200A-SCS. Before the electrical characterizations, we applied a constant bias voltage of +3 V to the top SRO pattern for 60 s to remove the interface-localized electrons. This electrical refresh process was performed only one time before the first measurement. Unless otherwise noted, the I(t) curves were measured at a constant bias voltage of −0.03 V. The current PSD spectra were measured using a function waveform generator (33220 A, Agilent Technologies), a low-noise pre-amplifier (SR570, Stanford Research Systems), and a dynamic signal analyzer (SR785, Stanford Research Systems). All the electrical measurements were performed in a dark and shielded box.

Bit-string generation

The analog data sets of current values were digitized and binarized. First, we treated a small number of current values that deviated by more than 100% from the average current values in State A and State B as outliers and excluded them from the digitization process. Next, we calculated the mean value of the analog current data and lastly performed the binarization by classifying values greater than the mean as “1” and values smaller than the mean as “0”. We recalibrated the mean value every 30 min to avoid inadvertently caused outliers.