We report a gate-tunable dimensional crossover in sub-micrometer-scale channels created at the LaAlO3/SrTiO3 interface. Conducting channels of widths 10 nm and 200 nm are created using conducting atomic force microscope lithography. Under sufficient negative back-gate tuning, the orbital magnetoconductance of the 200 nm channel is strongly quenched, and residual signatures of low-field weak-antilocalization become strikingly similar to that of the 10 nm channel. The dimensional crossover for the 200 nm channel takes place near the conductance quantum G = 2e2/h. The ability to tune the dimensionality of narrow LaAlO3/SrTiO3 channels has implications for interpreting transport in a variety of gate-tunable oxide-heterostructure devices.

Over the past fifteen years, the development of complex-oxide heterostructures has enabled gate-tunable control over superconductivity,1 spin-orbit interactions,2 and magnetism,3 greatly extending the motif of ordinary semiconductor-based heterointerfaces. The most extensively investigated complex-oxide heterointerface, which can confine electrons in a two-dimensional layer at the interface, consists of a thin layer of LaAlO3 (LAO) grown on TiO2-terminated SrTiO3 (STO).4 Most investigations utilize electrostatic gating from the back of the STO substrate to modulate the carrier density at the interface.5 The interplay of dimensional confinement with a rich palette of physical behavior can lead to new physics and potentially new applications.

A variety of experiments involving local probes have revealed the existence of naturally forming quasi-1D channels in macroscopic LAO/STO devices, associated with ferroelastic domain patterns that intersect the 2D interface.6,7 Low-dimensional LAO/STO nanostructures can be artificially created using a variety of methods, such as optical lithography,8,9 electron-beam lithography,10,11 and electrostatic top gating.12 Here we utilize conductive atomic force microscope (c-AFM) lithography13 to control locally the interfacial conductivity of critical-thickness (3 u.c.) LAO/STO heterostructures. The mechanism for conduction is believed to be controlled by surface protons that are distributed on the LAO surface,14,15 enabling reconfigurable nanoscale control of the electron gas at the interface with a precision which is typically in the 5-10 nm range (measured at room temperature), but which can be as small as 2 nm.16 Quantum devices created with this c-AFM lithography technique include single electron transistors (SETs),17 quantum dots,18 and Fabry-Perot interferometer cavities,19 which rely on the ability to control dimensionality into quasi-1D and 0D regimes.

We investigate magnetotransport properties of quasi-1D and quasi-2D nanostructures formed at the LAO/STO interface, with specific attention paid to the effects of electrostatic gating. Conducting channels with varying widths wi are “written” using c-AFM lithography13 on critical-thickness (3.4 unit cell) LAO/STO heterostructures grown by pulsed-laser deposition. The STO substrate is etched in buffered hydrofluoric acid for 60 s and annealed at 1000°C for 6 h to achieve an atomically smooth TiO2-terminated surface. LAO is subsequently grown by pulsed laser deposition at a temperature of 550°C and oxygen pressure of 1 × 10−3 mbar, and gradually cooled to room temperature.20–23 Electrical contact to the LAO/STO interface is made by Ar+ etching (25 nm) followed by sputter deposition of Ti/Au (5 nm/20 nm). To minimize possible systematic variations associated with, e.g., growth or processing conditions, experiments are compared between pairs of channels connected in series on a single 25 μm × 25 μm canvas. Here we focus on one particular pair of channels, and note that the results are highly reproducible in a qualitative sense (i.e., specific values of resistances and gate ranges may change but the overall qualitative features are reproducible).

A typical device [Fig. 1(a)] consists of a main channel connected by two leads (labeled 1 and 7) and voltage probes (labeled 2-6) that enable simultaneous measurement of both a narrow channel with width w1 = 10 nm and a wider channel with width w2 = 200 nm, connected in series. The width of the 10-nm channel is estimated from erasure experiments13 on similar devices written over the same area under identical writing conditions, at room temperature. The 200-nm channel is created by writing a series of closely spaced lines (10 nm apart), so that the overall potential profile has a specified effective width w2. A back gate (voltage Vbg) contacts the bottom of the STO substrate (using silver epoxy), which enables overall tuning of the chemical potential of the entire device. The two channel lengths are identical (l1 = l2 = 5 μm). All measurements are performed at T = 2 K and under vacuum, where the conducting channels created with c-AFM are stable over time scales much longer than the experimental time scale.16 

FIG. 1.

Device schematic and typical conductance behavior. (a) Schematic of a typical device consisting of two Hall bar structures of widths w1 and w2 in series between leads 1 and 7, with leads 2-6 acting as voltage sensing leads. (b) Conductance G from a lockin measurement at T = 2 K as a function of back-gate voltage Vbg during a forward Vbg sweep for both a w1 = 10 nm channel (blue) and w2 = 200 nm channel (red).

FIG. 1.

Device schematic and typical conductance behavior. (a) Schematic of a typical device consisting of two Hall bar structures of widths w1 and w2 in series between leads 1 and 7, with leads 2-6 acting as voltage sensing leads. (b) Conductance G from a lockin measurement at T = 2 K as a function of back-gate voltage Vbg during a forward Vbg sweep for both a w1 = 10 nm channel (blue) and w2 = 200 nm channel (red).

Close modal

The conductances G1 = dI/dV23 and G2 = dI/dV34 are measured for both the narrow (w1 = 10 nm) and wide (w2 = 200 nm) channels as a function of back-gate voltage [Fig. 1(b)]. The conductance of both channels is hysteretic with back-gating, as is commonly observed in SrTiO3-based heterostructures and has been attributed to movement of ferroelastic domain walls,6,7 and saturates at sufficiently positive and negative back gate voltages. For simplicity, we focus on data acquired for the forward sweep direction; similar results are obtained for the reverse sweep direction. At the highest back gate voltage used (Vbg = 7 V), the w1 = 10 nm channel conductance saturates around 2e2/h, in agreement with previous transport measurements of LAO/STO nanoscale channels of comparable dimensions.24 Note that lower voltages are required to tune the electronic properties compared with larger devices,25 due to focusing of the electric field from the conducting back plane of the STO substrate.

Signatures of a dimensional crossover from 2D to 1D behavior appear in the magnetoconductance G(B). For positive and zero back-gate bias [Fig. 2(b)], the wide channel exhibits a negative, primarily quadratic Kohler magnetoconductance that is similar to other reports in the literature.5,26 The quadratic contribution to the magnetoconductance is strongly suppressed in the 10 nm channel [Fig. 2(a)], similar to reports on a 50-nm-wide channel by Chang et al.11 

FIG. 2.

Magnetoconductance at selected back-gate voltages for the (a) w1 = 10 nm channel and (b) w2 = 200 nm channel (blue), and quadratic fits (red). At Vbg = 7 V and Vbg = −2 V, the negative quadratic magnetoconductance observed in the wide channel (b) is suppressed in the w1 = 10 nm channel (a). At Vbg = −7 V, the magnetoconductance of the w1 = 10 nm channel and w2 = 200 nm channel look nearly identical as the quadratic magnetoconductance becomes suppressed in the wide channel.

FIG. 2.

Magnetoconductance at selected back-gate voltages for the (a) w1 = 10 nm channel and (b) w2 = 200 nm channel (blue), and quadratic fits (red). At Vbg = 7 V and Vbg = −2 V, the negative quadratic magnetoconductance observed in the wide channel (b) is suppressed in the w1 = 10 nm channel (a). At Vbg = −7 V, the magnetoconductance of the w1 = 10 nm channel and w2 = 200 nm channel look nearly identical as the quadratic magnetoconductance becomes suppressed in the wide channel.

Close modal

The effect of back-gate voltage Vbg on orbital magnetoconductance can be quantified by fitting the high-field conductance (|B|>2 T) to a second-order polynomial, G(B) = aB2 + bB + c. The quadratic coefficient a [Fig. 3(a)] for the 10-nm channel remains close to zero for the entire gate-voltage range, indicative of quenching of orbital magnetoconductance. For the 200-nm channel, a is relatively large near Vbg = 0 V, saturating at slightly higher values upon positive gating, and strongly suppressed under increasingly negative back-gating. At the lowest back-gate voltage Vbg = −7 V, the a coefficients are close to zero for both channels. Plotting the quadratic coefficients as a function of conductance, instead of Vbg, we find that a scales quadratically for the 200-nm channel, and overlaps in the narrow range of conductance for the 10-nm channel. The universality of the relationship between the magnitude of the conductance and the orbital magnetoconductance behavior is revealed in Fig. 3(b), where a is plotted versus G(B = 0) for six channels of varying widths. Re-scaling the conductance by a coefficient C1, which incorporates differences in the relative transmissions, results in all curves collapsing onto a single curve that is itself a quadratic function of G. This unexplained scaling is expected to saturate as the channel approaches bulk behavior.

FIG. 3.

Suppression of quadratic magnetoconductance in the wide channel at negative back-gate voltages. (a) By fitting the high field data to G(B) = aB2 + bB + c, the suppression of the quadratic magnetoconductance can be clearly seen by plotting the B2 coefficient a as a function of Vbg. While a remains close to zero for the w1 = 10 nm channel for all Vbg, a transition occurs in the wide channel from a0 at low Vbg to a saturating at a large positive value at high Vbg. (b) Quadratic coefficient a versus zero-field conductance G(B = 0) at each back-gate voltage for three devices, each with two sections of wire of varying widths, shows the universal relationship between a and G(B = 0). The black line is a parabolic fit of the points.

FIG. 3.

Suppression of quadratic magnetoconductance in the wide channel at negative back-gate voltages. (a) By fitting the high field data to G(B) = aB2 + bB + c, the suppression of the quadratic magnetoconductance can be clearly seen by plotting the B2 coefficient a as a function of Vbg. While a remains close to zero for the w1 = 10 nm channel for all Vbg, a transition occurs in the wide channel from a0 at low Vbg to a saturating at a large positive value at high Vbg. (b) Quadratic coefficient a versus zero-field conductance G(B = 0) at each back-gate voltage for three devices, each with two sections of wire of varying widths, shows the universal relationship between a and G(B = 0). The black line is a parabolic fit of the points.

Close modal

Superimposed on the quadratic magnetoconductance is a weak anti-localization (WAL) signature at low magnetic fields, indicative of spin-orbit coupling. The WAL conductance peak, which is observed in both channels for all values of Vbg, is highlighted in Fig. 4 by plotting the difference between the magnetoconductance data and the quadratic fits versus magnetic field. Despite the very different high-field behavior discussed in Figs. 2 and 3, the size of the low-field WAL peak remains similar for both channels over a wide range of conductance values G(B = 0). At the most negative gate voltage for the w1 = 10 nm channel, the channel approaches an insulating phase [G(B=0)0.25e2/h], with an associated suppression of WAL. In the w2 = 200 nm channel, the peak changes shape with backgating but remains approximately the same size; notably, the shape of the WAL feature in the w2 = 200 nm channel at negative back-gate voltages is similar to the w1 = 10 nm channel. The independence of the size of the WAL feature over a broad range of back-gate voltage and conductance suggests that the underlying spin-orbit coupling has a one-dimensional character.

FIG. 4.

Weak anti-localization signature. Subtracting the high-field quadratic fit from the magnetoconductance results in a sharp peak near zero-field. Curves are offset by 0.03 e2/h. (a) In the w1 = 10 nm channel, this feature is mostly constant except at the lowest back-gate voltage. (b) In the w2 = 200 nm channel, although the shape changes, the size of the peak remains nearly constant across all back-gate voltages.

FIG. 4.

Weak anti-localization signature. Subtracting the high-field quadratic fit from the magnetoconductance results in a sharp peak near zero-field. Curves are offset by 0.03 e2/h. (a) In the w1 = 10 nm channel, this feature is mostly constant except at the lowest back-gate voltage. (b) In the w2 = 200 nm channel, although the shape changes, the size of the peak remains nearly constant across all back-gate voltages.

Close modal

The suppression of orbital magnetoconductance under negative gate-bias at the conductance quantum (G = 2e2/h), and the striking resemblance of the magnetoconductance signature to an artificially constructed 10-nm channel, provides strong circumstantial evidence for a gate-tunable dimensional crossover between 2D transport and quasi-1D transport. In this work, we focus on high-aspect-ratio (L/w) devices, which have previously been observed to have increased mobility relative to both 20 nm × 20 nm and 1 μm × 1 μm devices.27 We predict that the dimensional crossover behavior would likely also appear in wide wires with smaller aspect ratios approaching 1 (e.g., L>200 nm); however, as the length is further decreased, the approach of the ballistic regime could have a significant effect on the magnetoconductance.

While the magnetoconductance is distinctly different than that observed at the LAO/STO metal-insulator transition,25 the precise manner in which the 2D region shrinks to a 1D channel cannot be directly inferred from transport measurements alone. Further insight into the precise nature of the 2D to 1D crossover could come from cryogenic scanning probe techniques (e.g., piezoforce microscopy,28 scanning tunneling microscopy, and scanning gate microscopy). One possibility is that ferroelastic domain boundaries, located at both edges of the channel, are driven inward under increasingly negative back-gate bias, causing the channel to shrink in width. This type of motion has already been observed in naturally occurring ferroelastic domain boundaries by scanning single electron transistors (SETs) measurements at the LAO/STO interface7 and by scanning SQUID microscopy.6 Another possible scenario is that insulating regions form within the interior of the conducting channel and expand outward. This topology would be qualitative distinct from the prior scenario, possibly leading to quantum interference effects. These scenarios could be distinguished by low-temperature piezoforce microscopy measurements28 to spatially map charge depletion in the w2 = 200 nm channel. While reported scanning SET resolution is likely too low to observe the domain wall motion over channel widths 200 nm, scanning SET would still be useful to identify whether ferroelastic domain walls are indeed pinned at the c-AFM-written nanowires.

The ability to electrically tune the dimensionality of mesoscopic LAO/STO channels using modest back-gate voltages provides a new and useful parameter with which one can investigate how spin-orbit coupling, superconductivity, and electron-electron interactions vary with dimensionality. These results also provide further evidence that electrostatic gating—especially via back-gating—affects the electron density at the LAO/STO interface in a non-uniform and non-trivial fashion.

We are grateful to David Pekker and Anthony Tylan-Tyler for helpful discussions. J.L. acknowledges support from the Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through Grant No. N00014-15-1-2847. Work at the University of Wisconsin was supported by funding from the DOE Office of Basic Energy Sciences under Award No. DE-FG02-06ER46327.

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