Resonant tunneling in a quantum oxide superlattice

Resonant tunnelling is a quantum mechanical process that has long been attracting both scientific and technological attention owing to its intriguing underlying physics and unique applications for high-speed electronics. The materials system exhibiting resonant tunnelling, however, has been largely limited to the conventional semiconductors, partially due to their excellent crystalline quality. Here we show that a deliberately designed transition metal oxide superlattice exhibits a resonant tunnelling behaviour with a clear negative differential resistance. The tunnelling occurred through an atomically thin, lanthanum {\delta}-doped SrTiO3 layer, and the negative differential resistance was realized on top of the bipolar resistance switching typically observed for perovskite oxide junctions. This combined process resulted in an extremely large resistance ratio (~10^5) between the high and low-resistance states. The unprecedentedly large control found in atomically thin {\delta}-doped oxide superlattices can open a door to novel oxide-based high-frequency logic devices.

also, more importantly, the interfacial electronic phase of the electrode layers 13 . Unfortunately, however, a clear RT behaviour has not been experimentally realized in TMO heterostructures.
Although a clear RT behaviour has not been widely explored with TMOs, the NDR behaviour itself has been reported in various complex oxide junction structures [15][16][17][18] , In particular, nonvolatile resistance switching (RS) in TMO-based heterostructures has attracted scientific attention for developing next-generation memory devices based on memristors [19][20][21] . In typical RS devices, the electric resistance of a junction can be modulated by applying external bias, and the device can have two (or more) resistance states, i.e., a high resistance state (HRS) and a low resistance state (LRS).
The switching from a LRS to a HRS usually accompanies a NDR behaviour, as the amount of current decreases upon increasing the bias above the switching voltage.
In the following, we investigate the junction transport property of a QW superlattice (SL) precisely designed by inserting atomically thin LaTiO 3

(LTO) between SrTiO 3 (STO) barrier layers. A clear RT
behaviour with NDR is observed. Moreover, as oxide heterostructures can reveal RS, we further propose that the combination of RS with NDR is an efficient way to maximize the resistance ratio between the HRS and LRS in oxide QW heterostructures.

Results
Precision design of oxide quantum heterostructures by pulsed laser epitaxy. In order to realize a TMO QW heterostructure, a La δ-doped STO SL with atomically sharp interfaces was fabricated 22 .
We believe that a sample with an excellent crystalline quality is highly necessary for the observation of the quantum resonant behaviour in TMOs. Therefore, we epitaxially grew a high quality [LTO 1 /STO 6 ] 10 SL with minimized defect states using our atomic scale synthesis capability 6,8 .
Detailed information on the sample growth by PLE and experimental detail, as well as the excellent crystallinity of our SL is demonstrated in the Methods and Supplementary Figure 1.
A schematic drawing of the SL structure is shown in Fig. 1(a). It should be noted that La δ-doping creates a two dimensional electron gas (2DEG) with high-density conducting carriers at the interface (0.5 electrons per unit cell), due to the leakage of the 3d 1 25,27,28 , little is known about the out-of-plane junction properties 29 . In addition, unlike typical RT devices composed of tens of nanometres-thick semiconductor QWs, the tunnelling in our devices occurs through only an atomically thin layer, the fundamental lattice unit, providing a promising outlook for an ultra-fast transit time.
Nonlinear current-voltage characteristics and negative differential resistance. Figure 2 shows the junction current-voltage characteristics of the TMO QWs at various temperatures (T). At room T, a weak hysteresis curve, which manifests the typical bipolar RS behaviour, was observed. The two different resistance states could be achieved by switching the polarity of the bias voltage. Indeed, the memory characteristics have been confirmed to show a typical RS behaviour. In general, understanding the switching mechanisms is a major challenge in achieving better nonvolatile memory performance. In order to explain the bipolar RS phenomenon, various electric field polarity dependent models, including ion migration 19 , Mott transition 30 , and formation of Schottky barrier [31][32][33] , have been suggested. In particular, the Schottky barrier formed at the interface has been identified to trigger the RS behaviour for Nb:STO based junctions 31,33 . More specifically, changes in the Schottky barrier by either a charge trap at the defect states or oxygen migration due to applied bias voltage has been frequently attributed for the bipolar RS behaviour 31,32,34 . Similar mechanism, if not the same, seems to play a major role in our SL as well. As a qualitatively similar bipolar RS behaviour was also observed for a Pt/Nb:STO junction (inset of Fig. 2a), we believe that the RS behaviour originates mainly from the interface between the SL and Nb:STO.
As T decreases, the overall current level increases for both voltage polarities, and the RS becomes more distinct with a larger discrepancy between the LRS and HRS. Among these parameters, ϵ r of an oxide heterostructure with 2DEGs has not been exactly understood.
However, it is expected that the creation of interfacial charges would drastically reduce the ϵ r value of the highly dielectric STO. As this change leads to a significant modification in the dielectric screening of the conducting carriers, we focus here on the effect of ϵ r . Figure  As ϵ r increases, the well becomes shallower, weakening the quantum confinement -i.e., both the energy difference between the states and the number of confined states are reduced upon the increase of ϵ r . For ϵ r ≤ 100, the well is deep enough to accommodate three confined states (ground, first and second excited states) within the QW, while only two confined states are possible for ϵ r > 100 (Fig.   3a). Based on the energy level separation and the QW geometry, we can calculate V R as a function of ϵ r (Fig. 3b). As ϵ r increases, a smaller V R is expected because of the smaller energy difference between the ground and first excited states. For ϵ r = 100, V R = 1.23 V is needed to induce RT between the ground and first excited states in the deliberately designed oxide SL, as shown in Fig. 1c. The theoretical calculation implies that it would be rather difficult to observe a well-defined higher order peak in the tunnelling current. The confinement of the second excited state is only possible for ϵ r < Practically, the overall high current level makes it impossible to observe clear high voltage features.
Especially, the current substantially increased to a very high level at low T and the current compliance set to prevent thermal damage of the oxide QW obscures the peak feature. (We noted some T fluctuation near the lowest T due to the Joule heating, but the I(V) characteristic was highly stable and reproducible.)

Discussion
It should be noted that the RT behaviour is observed for the positive bias only, which can be attributed to the asymmetric sample geometry and resultant current profile. The current level for the negative bias is too large (for both HRS and LRS), possibly hindering the observation of the NDR behaviour.
In addition, the current reaches to the compliance limit much faster, which makes it even more difficult to observe the NDR behaviour for the negative bias. Finally, the I(V) characteristic of our oxide device is remarkably similar to that from a SiO 2 /Si resonant tunnelling diode, 4 strongly supporting that the NDR behaviour is based on RT in our TMO heterostructure.
We note that a few previous studies on TMO junctions claimed an observation of resonant states 16,35 .
These reports indicated that resonant states could be formed with V R on the order of 0.  36 . Due to the broad feature of the resonance peak (~1 eV) for our TMO QW, we obtain a rather small τ RT value (~0.7 fs), which is orders of magnitude smaller than that obtained in semiconductor QWs 36 . This value is also rather small compared to the traverse lifetime across the tunnelling barrier, and therefore, strong coherence is not expected in the complex oxide QWs. In fact, the small τ RT in our QW SL is expected as such a heterostructure has a rather small relaxation time (τ μ = ~42.4 fs) 24 of the charge carriers, which is also orders of magnitude smaller than those obtained from conventional semiconductors, possibly due to strong correlation 37 . In addition, the peak-to-valley current ratio (PVCR) and the peak current density (PCD) of our TMO-based RT diode are about 1.3 and 120 A/cm 2 , respectively, at the lowest T (4 K).
PVCR is rather small compared to that of conventional semiconductor RT diodes where the typical value is larger than 3 at room T. On the other hand, PCD of semiconductor RT diodes span seven orders of magnitude from tens of mA/cm 2 to hundreds of kA/cm 2 . Therefore, the PCD value we have measured from our TMO RT device is within the range of the semiconductor-based RT diodes.
The RT behaviour in our oxide QW SL enhances yet another advantageous functionality, which is unique to the TMOs. Indeed, the resistance ratio between the HRS and LRS is observed to be largely enhanced due to the RT feature. Figure 4 shows the junction resistance of the HRS and LRS measured at 0.5 and -0.5 V, respectively, as a function of T. The inset shows the resistance plot as a function of applied bias voltage. The difference between the HRS and LRS resistance increased with decreasing T.
Below 150 K, we observed the ON/OFF ratio to be larger than 10 5 , while it was slightly larger than 10 at room T. Typically, the ON/OFF ratio for bipolar RS is about 10 2 , much smaller than that for unipolar RS. The completely different T-dependent behaviour of the resistance (HRS shows insulating behaviour while LRS shows metallic T-dependent behaviour.) mainly due to the RT feature greatly enhances the ON/OFF ratio. Below 120 K, the compliance current played a role, somewhat decreasing the HRS resistance value (empty symbols in red). Nevertheless, we could still observe a large resistance ratio (~10 5 ) with an increasing trend towards the lowest T we employed. Undoubtedly, such a large enhancement in the resistance ratio stems from the RT behaviour of the QW SL. The drastically enhanced tunnelling probability near the resonant bias voltage substantially decreased the resistance of the LRS near V R . The dip feature for the resistance value at ~1 V for the LRS shown in the inset of Fig. 4 clearly signifies the enhanced tunnelling probability.
In summary, we have observed an intriguing NDR feature in a SrTiO 3 /LaTiO 3 /SrTiO 3 QW superlattice at low temperatures. The NDR behaviour is attributed to a resonant tunnelling occurring through the deliberately designed oxide QWs. In particular, because of the existence of the resonant tunnelling, a largely enhanced ON/OFF ratio has been achieved in the bipolar resistance switching, which occurs at the interface between the heterostructure and metallic substrate. Our study also demonstrates the potential of oxide heterostructures for a quantum mechanical behaviour that has been thought to be possible only in conventional semiconductor heterostructures. Thus, we believe that the discovery of resonant tunnelling through oxide-based QWs can lay down a stepping stone to oxide electronics. Theoretical calculation. The shape of electrostatic potential (Fig. 1b) induced by La δ-doping in STO was obtained by solving the Poisson and the Schrödinger equation self-consistently, without any electric field applied. The self-consistent calculations were performed iteratively using Broyden's second method until the convergence of the electrostatic potential is reached. 38 An external electric field was then applied to obtain envelope wavefunctions and energy levels of electrons (Fig. 1c). We examined ϵ r values of STO in the range of 10-10 3 , (Note, while ϵ r of bulk STO at low T is known to be very large, ϵ r strongly varies with the electric field, temperature, and sample geometry.) and effective mass of 4.4 (out-of-plane band effective mass of STO) was used for the calculations. 39     superlattice. Therefore, the overlapping of the wavefunction and hence the tunnelling probability drastically decreases. Indeed the sample becomes more insulating.

Supplementary Note 3: Discussion on the phase transitions of the constituent layers
Perovskite SrTiO 3 and LaTiO 3 undergo important phase transitions with lowering the temperature.
SrTiO 3 undergoes a structural transition from a cubic to a tetragonal phase at around 105 K and LaTiO 3 undergoes a phase transition from a nonmagnetic insulator to an antiferromagnetic insulator near 146 K. 3,4 While these phase transitions might influence the RT behaviour of the TMO superlattices, we did not find any anomaly across the temperatures within the error bar of our experiment. It should be noted that more detailed structural and magnetic studies across the temperatures might elucidate the subtle correlation between the phase transition in oxide perovskites and quantum tunneling behaviour.