Thermal and electrostatic tuning of surface phonon-polaritons in LaAlO3/SrTiO3 heterostructures

Phonon polaritons are promising for infrared applications due to a strong light-matter coupling and subwavelength energy confinement they offer. Yet, the spectral narrowness of the phonon bands and difficulty to tune the phonon polariton properties hinder further progress in this field. SrTiO3 – a prototype perovskite oxide - has recently attracted attention due to two prominent far-infrared phonon polaritons bands, albeit without any tuning reported so far. Here we show, using cryogenic infrared near-field microscopy, that long-propagating surface phonon polaritons are present both in bare SrTiO3 and in LaAlO3/SrTiO3 heterostructures hosting a two-dimensional electron gas. The presence of the two-dimensional electron gas increases dramatically the thermal variation of the upper limit of the surface phonon polariton band due to temperature dependent polaronic screening of the surface charge carriers. Furthermore, we demonstrate a tunability of the upper surface phonon polariton frequency in LaAlO3/SrTiO3 via electrostatic gating. Our results suggest that oxide interfaces are a new platform bridging unconventional electronics and long-wavelength nanophotonics.


Introduction
Phonon polaritons (PhPs) 1,2,3 -hybrid modes involving light and ionic vibrations -allow spatial squeezing of the electromagnetic energy 4,5,6,7,8 thus facilitating applications in molecular sensing 9,10 , thermal management 11,12 , sub-diffraction imaging 13,14,15,16,17 and other technologies.However, PhPs only emerge within narrow spectral windows -Reststrahlen bands -above the transverse optical phonon frequency  , where the real part of the material permittivity () is negative 1 .In the case of propagating surface phonon-polaritons (SPhPs), the upper limit is set by the surface optical frequency  , where Re[( )] =-1.As compared to theory, practically relevant SPhP windows are significantly more narrow when reasonable figures of merit are set for applications 1,18 .The palette of established PhP materials does not cover the entire infrared and terahertz ranges, motivating the search of new PhPsupporting media.Another bottleneck is the difficulty to modulate the PhP properties in situ since they are dictated by the intrinsic crystal-lattice dynamics and the device geometry.
Although a certain degree of dynamic SPhP tuning via coupling to electrostatically controlled graphene plasmons 19 , atomic intercalation 20,21 and photoinjection 22 has been demonstrated, this direction of nanophotonics is still in its infancy.
Strontium titanate (SrTiO3) -a prototype perovskite commonly used as a substrate in oxide electronics -has been recently suggested as a promising PhPs medium 18,23,24,25,26,27 since it features two prominent Reststrahlen bands spanning from the mid-infrared to terahertz ranges 28 .In addition, SrTiO3 (STO) demonstrates a plethora of non-trivial physical phenomena, such as a soft-phonon mode behavior 28 , mid-infrared polaronic absorption 29 and superconductivity at low carrier concentrations 30 .Of particular interest are the surface properties of the STO, specifically a formation of a conducting two-dimensional electron gas at its interface with other insulating oxides 31,32,33,34,35 .Importantly, the lately demonstrated fabrication of ultrathin free-standing layers of STO and other oxide perovskites 36,37,38 places this family on a par with the two-dimensional van der Waals materials in the ongoing quest for subwavelength polariton confinement 27 .It is tempting to leverage the unique features of STO to achieve a better control of the phonon polaritons and build new functionalities in nanophotonic devices.
In this article, we explore the possibility of using temperature variation and electrostatic gating to tune the spectral and propagational properties of SPhPs in pristine STO and LaAlO3/SrTiO3 (LAO/STO) heterostructures.To this end, we use scattering-type scanning near-field optical microscopy (s-SNOM), which has already shown its potential for mid-infrared imaging of the two-dimensional electron gas (2DEG) in LAO/STO 23,39 above the phonon bands.Here we focus on the upper Reststrahlen band of STO (Fig. 1a) born out of the optical phonon mode υ4 as depicted in the inset 40 , and spanning from  = 546 cm -1 to  = 746 cm -1 (at 20 K) 29,41,42 .The frequency dependent dielectric function of STO,  (), is well known 18,28,29,41,42 and we use it as a reference for interpretation of our data.Figure 1a presents the real (red line) and imaginary (blue line) parts of  () at 20 K.Under the assumption that the dielectric function near the surface is the same as in the bulk, the complex-valued in-plane SPhP wavevector in pristine STO can be obtained from the standard relation , where  () = / = 2/ is the wavevector of light ( is the light velocity).As it is typical for surface modes in semi-infinite samples, Re( ) (green line) is only slightly larger than  (purple line) indicating a weak level of confinement 43 .

Results
Experimental technique.To image SPhPs in a broad range of temperatures and frequencies, we use a cryogenic (6 -300 K) s-SNOM platform equipped with a nanoscale Fourier transform infrared (nano-FTIR) module allowing us to obtain continuous spectra down to 600 cm -1 .
Developing upon the original method applied to SiC 44 , we use the experimental geometry depicted in Fig. 1b.A thick STO or LAO/STO sample is partly covered by a 30 nm thick layer of Au or Pt.The sample and a metal-coated s-SNOM tip are simultaneously illuminated with an infrared laser.The straight and sharp metal edge converts the incident field (Ein) into near fields providing the necessary in-plane momentum to launch the SPhPs.The field scattered by the tip (Esca) is captured with a remote infrared detector and demodulated at several harmonics of the tip tapping frequency.In this work we focus on the second and third harmonics of the s-SNOM amplitude,  and  .When the tip is sufficiently close to the edge, two paths (P1 and P2) of scattering are active.In P1, the edge launched SPhPs propagate along the surface towards the tip and scatter out to free space, while in P2 the incident beam is directly backscattered at the tip.The fields from P1 and P2 interfere constructively or destructively according to their phase difference, which depends on the tip-edge distance and the illumination direction as specified below.We note that an alternative path, where the tiplaunched SPhPs are reflected at the edge and scattered by the tip, can be neglected in the case of weakly confined surface polaritons 45,46 .
In our samples, two orthogonal metal edges (1 and 2) are used, allowing us to compare two illumination geometries (Fig. 1b and Supplementary Fig. S1).The projection of the incidentlight momentum to the sample surface is nearly parallel to edge 1 and, correspondingly, almost orthogonal to edge 2. In all cases, the tip is scanned along a line perpendicular to the edge (axis).In Fig. 1c, we present the hyperspectral x- map of the s-SNOM amplitude  obtained on edge 1 on top of pristine STO at 15 K and normalized to the signal on metal,  , (the map on edge 2 is shown in Supplementary Fig. S2a).Below approximately 725 cm -1 , a series of dispersive fringes are observed on the sample near the metal, which decay as the tip moves away from the edge.To extract the fringe spacing and the decay length, we fit the spatial profiles  ()/ , separately at each frequency with a damped-sine function  • sin 2 −  , where the amplitude , phase  , fringe spacing  and decay length  are adjustable parameters.In Fig. 1d and Supplementary Fig. S2b, the experimental profiles and the fits at 690 cm -1 are presented for edges 1 and 2 respectively.Figure 1e and 1f show  and  (symbols) as a function of ω for edge 1 (Supplementary Fig. S2c and S2d for edge 2).Notably, the fringe period is longer than the calculated phonon polariton wavelength  = 2/Re( ) (green line in Fig. 1e).Moreover, the fringe distance for edge 2 is larger than for edge 1.Both observations are in contrast to the case of edge-launched ultraconfined ( ≪  ) PhPs in van der Waals crystals, where  =  and is edgeindependent 47,48 .However, this is expected in our case since  ~ (purple line in Fig. 1e and Supplementary Fig. S2c) 43,44 .As shown in the Supplementary Note S3, the fringe spacing for an arbitrary illumination direction is given by the formula: where α and β are the incident and azimuthal illumination angles respectively and  is the angle between the SPhP momentum and the x-axis (Fig. 1b and Supplementary Fig. S3).The SPhP angle, in turn, is determined by the surface-polariton analogue of the optical Snell's law: The illumination angles are highly sensitive to the alignment of the laser-focusing mirror and hard to measure in the cryo-SNOM setup.To circumvent this problem, we model  simultaneously for both edges with equations ( 1) and ( 2), while treating α and β as adjustable parameters and keeping in mind that α on the two edges is the same, but the values of β differ by 90 °.Using this procedure, we obtain a good match between the experiment and theory (red solid lines in Fig. 1e and Supplementary Fig. S2c) for the values of α = 23.3° and β = 7.8 ° for edge 1 (97.8 ° for edge 2).The possibility to reproduce  () at many frequencies and for both edges with only two parameters corroborates that the fringe patterns are governed by the SPhP interference and not by optical artefacts unrelated to the sample properties.
The propagation length  = 1/Im( ) is a sensitive measure of the phonon scattering and is an essential figure of merit for applications 1 .As clearly seen in Fig. 1c and Supplementary Fig. S2a the interference persists at about hundred microns from the edge (80 μm for edge 1 and 140 μm for edge 2).This tells unequivocally that the SPhPs travel in our sample by at least the same distance before they become indistinguishable from noise (a similar observation at room temperature is made in Ref. [ 26 ]).We note that the maximum extent of the SPhP interference observed in SiC -a textbook PhP material -is about the same 44,49 .And yet, the theory based on the bulk dielectric function of STO (Fig. 1a) predicts a several times longer propagation length  than the experimental decay length  (Fig. 1f).A possible reason for this difference is the presence of additional electromagnetic losses on the surface due to oxygen vacancies or other defects.However, we believe that in the present case the decay length obtained by the damped-sine function fitting is artificially suppressed because of the limited size of the illumination spot 43 and can therefore serve only a lower bound for the true SPhP propagation length.
Thermal tuning of SPhPs in STO.To investigate the thermal variation of SPhP properties in bare STO, we analyze nano-FTIR profiles measured along the same path at various temperatures  from 15 to 300 K. Figure 2a presents the s-SNOM amplitude (symbols) and the corresponding damped-sine function fits (lines) at 690 cm -1 (the data at other frequencies show a similar trend).One can notice that the temperature variation of the amplitude profiles is rather small.Accordingly, the extracted values of  and  (purple and orange symbols in Fig. 2b) show no or a weak temperature dependence.For comparison, we present on the same graph the calculated fringe spacing and the propagation length based on the dielectric function of STO at different temperatures (Fig. 2c) 42 .The real part of  () is almost temperature independent in this frequency range, resulting in a rather weak variation of  , corroborating the observations.In contrast, Im[ ()] strongly decreases with temperature due to a reduction of the phonon damping 50 , which entails a strong increase of the theoretical value of  with cooling down not observed in our measurements.This difference can be naturally explained by the mentioned effect of the finite spot size which limits the decay length and masks its true temperature dependence.
Having established the existence of long-propagating SPhPs in STO in a broad range of temperatures, we focus on the frequency dependence of the s-SNOM signal.In Fig. 2d, we present the spectra of  ()/ , at 12, 100, 200 and 300 K collected 150 μm away from the metal edge, where the SPhP interference is absent.The signal increases abruptly below a certain onset frequency ~735 cm and forms a pronounced peak at ~700 cm .Since the onset frequency is close to  (indicated by arrow) and is just above the maximum frequency where SPhP fringes are observed, we assign the steep signal rise to the generation of SPhPs by the tip.Interestingly, the onset frequency barely changes with temperature while the peak intensity grows with cooling down.To understand this, we calculate the s-SNOM spectra (Fig. 2e) at various temperatures via the finite-dipole model of the sample-tip interaction 51,52 (See Methods and Supplementary Note S4) using the dielectric function of STO shown in Fig. 2c.
The remarkable agreement between the trends seen in the experimental and theoretical spectra indicates that the onset frequency is indeed linked to  , which is in turn determined by Re( ) and is therefore weakly T-dependent.On the other hand, the peak intensity is related to the optical losses, controlled by Im( ), and is higher at low temperatures.We note that the physical meaning of the peak frequency and the interpretation of its temperature dependence are less clear than the ones of the onset frequency.
Thermal tuning of SPhPs in LAO/STO.We move now to the discussion of the experiments performed on conducting LAO/STO heterostructures.The LAO thickness of our sample, 6 unit cells (about 2 nm), is above the threshold for the formation of the 2DEG at the interface with the STO substrate 32 .First, we notice that hyperspectral maps (Supplementary Fig. S5) clearly show interference fringes similar to the ones in pristine STO (Fig. 1c) telling us that longpropagating SPhPs persist in the presence of the 2DEG.Next, we focus on the analysis of the nano-FTIR spectra collected far away from the edge.In Fig. 3a, the spectra of  ()/ , at 12, 100, 200, and 300 K are shown.As in bare STO (Fig. 2d), the s-SNOM amplitude grows quickly below a certain frequency.However, in contrast with pristine STO, the onset frequency now shows a strong blueshift as the temperature is decreased.To quantitatively compare the thermal shifts of the phonon structure in the two systems, we adopt a common ad hoc definition of the onset frequency,  .We determine it as the intersection of the straight line fitting the high-frequency slope of the phonon peak with the horizontal axis as sketched in Fig. 2d and 3a.Since this fit involves a broad range of frequencies, the relative uncertainty of the onset frequency (for example, its shift as a function of temperature) is much better than the spectral resolution of the nano-FTIR spectroscopy (see Methods).One can see that  () for LAO/STO (red downward triangles in Fig. 3d) blueshifts by about 14 cm -1 between room temperature and 12 K, whereas it changes by less than 2 cm -1 in pristine STO (black circles).
This striking difference is likely caused by the 2DEG, since the thin insulating LAO layer cannot cause such a strong effect (Supplementary Note S6).To corroborate this hypothesis, we perform a finite dipole simulation of the s-SNOM spectra of the multilayer LAO/2DEG/STO system at different temperatures.We set the thickness  of the 2DEG layer to 2.7 nm in agreement with previous studies 53,54 .While the presence of the 2DEG can be detected using infrared ellipsometry 54,55 , no spectra of  () were reported, because the ultrathin conduction layer contributes weakly to the far-field optical properties.Instead, despite the subtle differences in the band structures of 2DEG and chemically doped STO (as detailed in Supplementary Note S7), we set the dielectric function of the 2DEG to be the same as in Nb doped strontium titanate with a comparable level of the charge carrier concentration.
Notably, the real part of  changes dramatically as a function of temperature, in a stark contrast with pristine STO.As it was shown in Ref.
[ 29 ], this effect is not caused by the temperature dependence of the carrier concentration (which is relatively weak) but by a polaronic screening of the charge carriers and a temperature dependent redistribution of the optical spectral weight between the low-frequency Drude peak to a mid-infrared polaron band above  .In Fig. 3c, we present the theoretical spectra 52 .The model perfectly reproduces the most salient features of nano-FTIR measurements: the increase of the phonon peak intensity and the blueshift of  (red triangles in Fig. 3d) with cooling down.This suggests that the electron-phonon interaction plays an important role in the infrared response of the 2DEG.
Electrostatic tuning of SPhPs in LAO/STO.The possibility of tuning the properties of the 2DEG is an important property of the LAO/STO heterostructures.In Fig. 4, we present the effect of back-gating of the 2DEG on the SPhP spectra.Due to the soft-mode instability -a hallmark of this material -the static permittivity of STO and the gating efficiency increase at low temperatures by several orders of magnitude, therefore we perform this experiment at 7.7 K. When the gate voltage  is swept from 0 to -200 V, the measured in situ sheet resistance (Fig. 4a) increases by more than a factor of 6, in agreement with previous measurements on similar devices 35 .This effect is due to a decrease of charge carrier density and a concomitant reduction of the charge mobility 23,35 .On the other hand, the resistance decreases very weakly when  is tuned from 0 to 200 V, likely due to a low gating performance at positive voltages in the SNOM setup, where the presence of visible light radiation with the photodoping effect influences the electrostatic doping process.The s-SNOM amplitude spectra (normalized to the peak value) collected on a same point of the sample far from the metal edge at VG = 0, -50, and -150 V are shown in Fig. 4b (the spectra at other gate voltages are presented in Supplementary Fig. S7).One can see that the SPhP onset slightly redshifts when the negative voltage is applied.
In Fig. 4c, we present the value of ω , obtained using the described ad hoc procedure, as a function of the gate voltage.The onset frequency drops from 743 to 738 cm -1 as VG is swept from 0 to -200 V, but it barely changes in the positive voltage side.The curves of ω ( ) and ( ) show correlated behavior indicating that the infrared SPhP frequency and DC transport properties are connected.
Below we check numerically if the gate-induced change of the dielectric function of the 2DEG close to the SPhP onset frequency (740 cm -1 ) may explain this connection.Lacking direct measurements of  () and especially of its variation with gating, we perform a simplified simulation.First, we assume, as before, that  () at zero gate voltage is equal to  () and calculate ω , using the same ad hoc method.Then we add a constant value  to the real part of  and compute the corresponding shift ω , ( ) = ω ( ,  = 0) − ω , (red symbols in Fig. 4d).Finally, we do the same calculation by adding  to the imaginary part of  , and obtain ω , ( ) = ω ( = 0,  ) − ω , (blue symbols).This modelling reveals that the onset frequency is affected much stronger by  than  .Specifically, the redshifts experimentally observed at  = -50 V and -150 V (Fig. 4c) correspond in the simulation to  =+12 and +24 respectively, as indicated by arrows in Fig. 4d.Keeping in mind that Re[ ( )] ≈-40 (Fig. 3b), these changes imply that Re[ ( ,  = −50V)] ≈-28 and Re[ ( ,  = −150V)] ≈-16.First, the lowering of the absolute value of  is qualitatively consistent with the expected decrease of the carrier density at negative voltages 35 .Second, the value remains negative, indicating the presence of carriers even at the highest absolute voltage applied.
Therefore, we conclude that the hypothesis of the connection between the density of the carriers and the shift of the onset frequency is reasonable.

Discussion
We have shown that the presence of the two-dimensional electron gas in the LAO/STO interface results in a stronger temperature dependence of the SPhP onset frequency, as compared to pristine STO, and it also enables a certain electrostatic control of this value: ω blueshifts by about 14 cm -1 from 300 K to 10 K and redshifts by 5 cm -1 when the maximum negative gate voltage is applied.The amount and the direction of both shifts are consistent with the expected variation of the real part of  () in these processes.While the physical reason of the gate-voltage related shift of ω is the reduction of the carrier concentration, the origin of the thermal effect is quite different, namely a redistribution of the spectral weight between the Drude peak and the polaronic mid-infrared band.
It is worth noting that applying gate voltage to the same samples does not significantly change the fringe spacing in hyperspectral ( − ) maps (See Supplementary Note S9).This indicates that the low-momentum SPhPs, which are responsible for the interference patterns in these maps, are much less affected by the gate voltage than the high-momentum SPhPs, which determine the onset frequency in the nano-FTIR spectra.Indeed, a simulation of the phononplasmon polariton dispersion in LAO/STO at zero gate voltage and  = −150 V (Supplementary Note S10) shows that the low-momentum SPhPs (~ , region A) are almost doping independent.The physical reason of this insensitivity is that the electromagnetic field of low- SPhPs penetrates into the STO substrate much deeper than the thickness of the 2DEG.
The same calculation clearly shows a redshift of the SPhPs with the higher momenta ( ≫  , region B), in agreement with our results.At even higher in-plane momenta , an extra surface polariton branch is seen above ω (region C), which is identified as plasmonpolariton in the 2DEG 23 .This branch demonstrates a much stronger dependence on the gate voltage than the SPhP branch.Even though the absolute s-SNOM amplitude above ω is small (see, e.g.Fig. 3a), the relative signal change as a function of the gate voltage in this spectral range has been shown to be very significant 23 .
Overall, our cryo-SNOM experiments supported by simulations suggest that strontium titanate is not only a new promising material supporting long-propagating phonon polaritons in the farinfrared range, but also a unique system where a two-dimensional electron gas, which can be easily created via heterostructuring, enables active tuning of surface PhPs, the idea initially realized in graphene-hBN system 19 .While we focused here on weakly confined SPhPs in bulk samples, the fast progress in fabrication of ultrathin crystalline layers of STO and other perovskite oxides 36,37,38 will allow studying and harnessing of highly confined volume waveguide PhP modes in this family of materials.We notice that the low-energy Reststrahlen band of STO, not covered by the present study, originates from a phonon mode, which softens dramatically at low temperature 28 potentially leading to a cavity-enhanced ferroelectric phase transition 56 thus realizing an important case of electrodynamic control of matter.The surface phonon polaritons from this band should be more sensitive to temperature than the ones studied here.On the experimental side, we believe that the method of two-edge interferometry developed in this work can be widely used to tackle uncertainties in the illumination geometry often present in the s-SNOM experiment.This approach is complementary to the use of circular disc launchers 57 , however, it does not require a 2D SNOM mapping and therefore is easier to realize in the case of nano-FTIR spectroscopy.Furthermore, with the future improvement of theoretical models of the near-field tip-sample interaction, we expect that this technique will allow direct extraction of the spectral optical properties of the 2DEG from the nano-FTIR spectra.

Methods
Sample preparation.The LAO/STO samples were grown by depositing approximately 6 unit cells of LaAlO3 on commercial TiO2-terminated (001)-oriented SrTiO3 substrates (Crystec GmbH) by pulsed laser deposition.The growth is performed in a pure oxygen atmosphere at a pressure of 8⋅10 -5 Torr while the substrate is heated to 820 °C during the deposition.A postannealing of 1 hour is performed at 575 °C in an atmosphere of 0.2 bar of oxygen.The samples are then cooled down in the same oxygen pressure.After the growth a back gate electrode was deposited over the whole sample by sputtering Au on the back of the substrate.The back gate electrode was electrically contacted using silver paste to bond a Cu wire to the Au electrode allowing to apply an electric field between the latter and the conductive interface.To measure the electrical transport properties, the LAO/STO interface was contacted in a van der Pauw geometry by ultrasonic welding using Al-wires.For both LAO/STO and bare STO devices, the metallic electrodes were deposited using a standard photolithographic process to pattern the