Transient photocapacitance spectroscopy on Au/TiO₂ Schottky diodes with rolled-up nanomembrane electrodes

Deep level transient spectroscopy (DLTS)¹ has proven to be a useful technique for determining parameters of traps in insulators, such as binding energies, cross sections, or number densities of the charge traps. Striking results have been obtained in the fields of doped or irradiated semiconductors, solar panels, or self-assembled quantum dots, to name just a few.^2–15 Since the first appearance of DLTS, almost 50 years ago, a large number of related methods have been developed extending the range of applications and taking into account the specific requirements of the material under study. The concept has been, for example, extended to conductivity experiments,^16,17 inverse Laplace transformations of the transients,¹⁸ or to capacitance transients under optical excitations,^19–21 which are particularly suited for levels deep inside the bandgap with small thermal activation rates. However, not all quantities of interest are accessible by each measurement concept, and the versatility of one method depends on the set of quantifiable parameters. Here, using the well-known oxygen vacancy in rutile titanium dioxide as a model trap, we show that the electron capture cross section can be determined by optical methods as well, namely, by deep level transient photocapacitance spectroscopy (DLTPCS) in combination with a rate equation model. The extracted parameter values are in agreement with those obtained by conventional lock-in DLTS.^22,23 This comparison is made possible by using rolled-up nanomembrane (rNM) electrodes,^24–26 which act as optically transparent gates²⁷ of the $Au / TiO 2$ Schottky diodes.

Rolled-up nanomembrane (rNM) vertical junctions were fabricated on $SiO 2$-coated (⁠ $2 μ m$ thick), $8×8 mm 2$ $Si$ (100) substrates. The device fabrication relies on photolithography, thin-film deposition, and etching methods, as recently reported elsewhere.^25,26 The preparation procedure is depicted in Fig. 1.

The vertical junctions are composed of a patterned $Ni$ bottom electrode, a $TiO 2$ thin film playing the role of active material, and an rNM $Au$ top electrode. The $Ni$ and $Au$ electrodes were deposited by electron-beam evaporation in an AJA International system in high vacuum (⁠ $≈ 10 − 7 Torr$⁠) and with substrates kept at room temperature. The $TiO 2$ film was grown by atomic layer deposition (ALD) on the patterned $Ni$ bottom electrode. The ALD process was carried out using a calibrated Oxford OpAL reaction chamber with titanium (IV) isopropoxide (TTIP) and $H 2 O$ as a precursor. The substrate was kept at $150 ° C$ during the $TiO 2$ deposition. This material was selected for our experiment since it contains a well-known deep level originating from a single oxygen vacancy, labeled as $V O$⁠. Such a level has a binding energy around $800 meV$ and acts predominantly as an electron emitter.^28–30 It is, thus, sufficiently deep for an unambiguous detection by optical spectroscopy but still accessible by conventional DLTS. To pattern the $TiO 2$ films (viz. on the bottom electrode and the strained nanomembrane), we employed reactive-ion etching (RIE), using a Plasma Pro NGP80 (Oxford Instruments). After the $TiO 2$ thin-film patterning, the roll-up process was carried out to form $Ni / Ti O 2 / Au$ vertical junctions. The interface at the bottom has an area of $80×10μ m 2$⁠, while the contact area of the interface at the top is $10μ m$ times the contact length of the rolled-up electrode, which has previously been estimated to $≈0.1μ m$⁠.³¹ To a good approximation, both the top and bottom electrodes form a parallel plate capacitor with the $TiO 2$ layer as the dielectric material, arranged in parallel to the capacitance from the junction.

The roll-up process employed here is a controlled release of a Cr/Ti strained nanomembrane (NM) from the $Si O 2$ substrate by means of the $Ge Ox$ sacrificial layer (SL) removal,²⁵ as illustrated by the sequential sketches in Fig. 1. The strained NM was patterned together with an Au thin film that is going to play the role of a top electrode after the roll-up process. On the top of the strained NM, a $TiO 2$ detaching structure was patterned to direct the NM release from the substrate. Within this framework, the SL selective removal triggers the strained NM release, which is directed by the detaching structure ripping out [Fig. 1(a)]. The strained NM proceeds curling itself up until forming a rolled-up Au-coated microtube. Still, the Au-coated microtube continues to roll until the rNM lands on the $TiO 2$ thin film, providing a reliable electrical top-contact for the vertical junction. Figure 1(b) shows a scanning electron microscope (SEM) image of the device architecture in which the rNM is contacting the $TiO 2$ film from the top, forming a cross-wire-like junction with the Ni layer as a bottom electrode. The reasons an rNM was chosen over a conventional flat wire contact lie in its optical properties. Incident photons are guided alongside the outer wall of the rNM all the way to the back of the microtube where they illuminate the oxide directly underneath the metal,^25,26 thus resembling a whispering gallery mode.³² After device manufacturing, the rNM vertical junctions were stored in a high vacuum (base pressure $≈ 10 − 5$ Torr) for at least $24 h$ to remove any water residue incorporated during the fabrication.

The sample morphology was characterized using optical and electron microscopy techniques. The optical microscopy images were acquired using a Nikon Eclipse ME600 microscope. The SEM images were acquired using an FEI Inspect F50 (viz. for imaging the entire device architecture). Transmission electron microscopy (TEM), annular dark field scanning transmission electron microscopy (ADF-STEM), and energy dispersive spectroscopy (EDS) were employed to elucidate the device morphology. The TEM and ADF-STEM images were acquired using a JEOL JEM 2100F. The EDS maps were obtained using an OXFORD EDS System with a $100 m m 2$ SDD coupled to the electron microscope. The lamella samples were prepared by a focused ion beam (FIB) using a HELIOS NanoLab 660, and Au thermally evaporated electrodes were employed to guarantee reproducibility in both FIB preparation and TEM imaging. These studies have revealed that our $TiO 2$ films crystallize in the rutile structure and have a thickness of $(53±2) nm$ (see the supplementary material). The layer sequence is expected to form an n-type Schottky diode since the $Ni / Ti O 2$ interface is known to form an Ohmic contact,³³ while the $Au / Ti O 2$ interface forms a Schottky barrier for sufficiently low defect densities.³⁴ 

The sample was mounted into a liquid nitrogen cryostat (Linkam HFS 600 with cooling unit LNP 95), which enables a temperature range from $150 K$ to $450 K$⁠. A dry nitrogen atmosphere is established by a constant flow of nitrogen gas via a gas flow controller (Bronkhorst EL-Flow series).

The capacitance of the sample was measured with the setup shown in Fig. 2(a). An AC excitation voltage with an amplitude of $100 mV$ and a frequency of approximately $28 MHz$ was generated by a HF2LI Zürich Instruments Lock-in amplifier and superimposed to the output of an arbitrary waveform generator (AWG, Keithley 3390). This relatively high frequency was used in order to obtain an acceptable signal-to-noise ratio for subsequent data analysis. The corresponding implications for the measured values of the Schottky barrier height and the doping density, both of which enter only qualitatively in our analysis, are discussed in the supplementary material. The bias voltage was applied to the top electrode with respect to the $Ni$ electrode, grounded virtually via the HF2TA transimpedance amplifier. The out-of-phase signal, measured with the lock-in amplifier, represents the capacitance, which can be measured with a time resolution of $10 μ s$⁠. A Picoscope 5444B digital oscilloscope was used to measure and average the transients.

To characterize the junction, current–voltage as well as capacitance–voltage measurements were carried out using the setup as exemplified in Fig. 2(b). The current–voltage characteristic reveals a rectifying relation. At applied voltages above $0.5 V$⁠, the trace is approximately linear, corresponding to the case that the current flow is limited by the bulk resistance rather than the rectifying junction. The resistivity is $1.1× 10 5 Ω m$⁠, orders of magnitude below the value for intrinsic rutile, indicating a large doping density, but still small enough to form a Schottky barrier. To a reasonable approximation, the rolled-up nanomembrane electrode and the $Ni$ layer form a parallel plate capacitor with the $Ti O 2$ layer as the dielectric material. The capacitance contribution of the sample can be estimated to $13 fF$⁠. This corresponds, for an estimated dielectric constant of 80, to an effective contact width of $≈100 nm$⁠. The built-in potential as determined from the Mott–Schottky analysis amounts to $V b i≈0.89V$ (see the supplementary material), using an effective electron mass of $m r u t i l e ⋆=20 m e$⁠,³⁵ where $m e$ denotes the free electron mass. The doping density is found to be $n D≈4.2× 10 19 c m − 3$⁠. The traces are consistent with the picture of an n-type Schottky junction, even at the selected measurement frequency.

For subsequent photocapacitance experiments, the sample was illuminated with a Thorlabs 910E LED (emission maximum at a wavelength of $910 nm$ at $300 K$⁠), controlled by a Keithley 3390 arbitrary waveform generator; see Fig. 2(a). Rectangular pulses with a frequency of $1 Hz$ were applied to the LED, switching it on and off. The LED was mounted outside the cryostat to keep it at room temperature, and the emitted light was guided by an optical fiber into the cryostat. In order to eliminate influences from the initial experimental conditions on the transients, the first four illumination cycles are excluded from the data analysis. In these experiments, the applied DC voltage was zero.

Typical photocapacitance transients are represented in Fig. 2(c) for three different temperatures. Apparently, the capacitance increases under illumination and decreases toward its initial value after the light has been switched off. The saturated capacitance change $ΔC(T)$ and the time constants $τ on / off(T)$ of the transients show non-monotonous evolution as the temperature $T$ is increased. Furthermore, the time constants in response to illumination are markedly shorter than those observed after the light has been switched off.

This behavior is tentatively interpreted in terms of photoionization of deep donor levels inside the depletion region of the Schottky contact. After the photoemission of a trapped electron into the conduction band, the charge density inside the depletion region and, thus, the capacitance increases. The time constant of this process is determined by the difference between the emission and capture rates, the latter of which dominates the duration of the transient after the light has been turned off. We cross-checked this interpretation by measuring the photocapacitance under illumination with light of a wavelength larger than $1550 nm$ and could not detect a response. This simple picture will be substantiated by the rate equation model described below.

The photocapacitance transients have been measured in a temperature range from $185$ to $340 K$ with a heating rate of $2 K min . − 1$⁠. For each temperature step, 30 individual transients were recorded and averaged. The data were processed using the lock-in weight function, Eq. (1), and are shown in Figs. 4(a) and 4(c) for the response after the light has switched on and off, respectively.

The capacitance change in response to the illumination, see Fig. 4(a), is positive, indicating electron emission from the traps. As the temperature is increased from $180$ to $≈280 K$⁠, $τ on$ decreases, while the amplitude $S$ of the lock-in signal increases. For $T⪆280 K$⁠, the photocapacitance vanishes, with a cut-off time constant that decreases as $T$ is increased, as visible in the temperature range between $≈260$ and $300 K$⁠.

The transients observed after the illumination has been turned off, see Fig. 4(c), are negative and, thus, indicative of trap repopulation. The observed time constants $τ off$ are significantly larger than $τ on$ for all temperatures and get smaller as the temperature is increased, with a larger temperature dependence. The capacitive response is strongly suppressed around $300 K$⁠, and a sharp cutoff can be observed toward longer time constants in the interval $270 K⪅T⪅320 K$⁠, which coincides with the thermal emission time obtained from the conventional DLTS experiment [dashed white lines in Figs. 4(a) and 4(c)]. Furthermore, the distribution of the time constants during capture is much broader as compared to the emission scenario.

For an explanation of this behavior, note that the measured time constants and amplitudes of the transients reflect the influences of up to three simultaneous contributions, namely, electron capture as well as thermal emission and, in the case of illumination, photoemission. In Sec. IV, we show how these influences can be disentangled and described by their characteristic quantities with the help of a rate equation model.

In this section, a system of coupled rate equations is used to determine the binding energy $E B$ of the deep level, its optical cross section $σ o$⁠, as well as the optical emission rate $e o$⁠. We do not use the information obtained from the conventional DLTS measurements in this sample characterization since it is not a priori known that the optically active trap is identical to that one dominating the thermionic emission. It could be, for example, a level with a significantly larger binding energy inaccessible to conventional DLTS.

The parameters determining the kinetics of the system can be obtained by a least-square fit of the analytic solution to the measured data in Fig. 4, with the conditions of a constant $e o$ under illumination and $e o=0$ in the dark. In each iteration step, the transients for all observed temperatures are calculated. We assume that prior to the first illumination cycle, all traps are occupied; i.e., $n 0 0=0$⁠, $n 1 0=1$⁠. The lock-in signals of the resulting transients, computed according to Eq. (1) on the same time grid as in the measured data, are represented in Figs. 4(b) and 4(d). The resulting fit parameters amount to $E b=(928±16) meV$ and $σ c=(4.4±2.6)× 10 − 16 m 2$⁠. Table I compares these fit parameters to those obtained from the conventional DLTS.

These values for the binding energy and the capture cross section are, within the error bars, the same as those obtained from the conventional DLTS. This finding provides strong evidence that the deep level traps we observe in the optical experiments are the same as the ones seen in the conventional DLTS measurement. Furthermore, the optical responses are well reproduced by the fit parameters for both photoemission and capture, see the full lines in Figs. 4(b) and 4(d), which represent the fit values.

We emphasize that hereby, a technique has been demonstrated that enables the determination of the electron capture cross section of deep traps from transient photocapacitance spectroscopy in combination with a suitable rate equation model. This may be particularly useful in samples where substantial recapture during emission is present. Experimentally, recapture could be suppressed by a high reverse bias voltage. This, on the other hand, is not possible in all samples and, furthermore, provides an additional source of electrons that may get captured, namely, from the leakage current.³⁷ Therefore, having the option to include capture during emission in the model is essential. We note, furthermore, that an extension to more complicated systems with several trap species along the lines of Ref. 38 is straightforward. Since each trap has its own characteristic time constant, the presence of multiple traps would be manifested in a corresponding number of peaks in the lock-in signals, which can be attributed to the traps by an adapted rate equation model.

We proceed with a discussion of the physics behind the observed phenomenology and begin with the capture process; see Figs. 4(c) and 4(d).

Below $≈280 K$⁠, the measured time constants show a broad distribution, with a mean value that decreases approximately exponentially with increasing $T$⁠. This dependence is in accordance with the ad hoc assumption of thermally activated capture, as expressed in Eq. (5). This suggests the following capture process. After emission (either thermally or by optical excitation), the electrons in the conduction bands drift toward the Ni back electrode in the built-in electric field. Recapture into a trap located at a certain position requires kinetic energy, which allows the electron to reach again this uphill location. The characteristic energy $ε c$ introduced in Eq. (5) is, therefore, a measure for the energy difference between the conduction band edge at the trap location and the flat-band region, which equals the quasi-Fermi level at the back contact. It represents possibly a non-trivial average of a form to be determined in future work. Within this picture, the broad distribution of capture times reflects the capture in a certain interval of the space charge region, starting from the onset of the depletion region and extending toward the rNM electrode, where $E b$ is referred to the local, varying conduction band bottom. Relating $ε c$ to the conduction band profile, we can estimate that the electrons are captured within an interval $Δz≈5 nm$ from the onset of the depletion region. Due to the stronger temperature dependence of the thermal emission rate in comparison with the capture rate, the two rates cross each other as $T$ is increased. For $τ c≫ τ e , t h$⁠, the traps are essentially empty and $ΔC$ vanishes. Thus, $τ e , t h(T)$ represents a transition line above which $ΔC$ is suppressed.

With this interpretation at hand, we move on to a discussion of the photocapacitance transients measured in response to illumination; see Figs. 4(a) and 4(b). Below $≈280 K$⁠, the measured time constant of the transients decreases approximately exponentially with increasing $T$⁠, while its amplitude increases. Since the optical emission rate should, to a first approximation, be temperature independent, one is tempted to attribute this decrease to a thermal contribution, according to Eq. (3). Note, however, that $τ e , t h≫ τ e , o$ in this temperature range. Heating of the sample by the light is an unlikely explanation, considering that $T$ should increase by as much as $≈100 K$⁠. Also, we exclude enhanced emission via tunneling after photon absorption since the photon energy (1.37 eV) is significantly larger than $E b$⁠. Whether the optical cross section or the intensity of the light coupled into the junction via the rNM gate, do have a temperature dependence, is an open issue to be addressed in future studies. Likewise, the technique can be readily extended to determine the temperature dependence of the optical cross section^19,39 as well as Franck-Condon shifts by frequency-dependent photocapacitance measurements.^39,40 As soon as the thermal emission rate becomes the dominant term at larger temperatures, however, the photocapacitance gets suppressed because the photons meet essentially empty traps. Furthermore, both emission processes are expected to show no pronounced dependence on the trap location. This explains why the emission time constants show a much sharper distribution than the capture time, which reflects the distribution of the traps in the accessible energy interval of the space charge region. Finally, we comment on the increase of the emission amplitude with increasing $T$⁠. It has its origin in the decreasing capture time as $T$ increases. Hence, photons find more traps occupied, and a larger transient amplitude emerges.

It has been demonstrated how the capture cross section of a deep trap can be determined by deep level optical spectroscopy, which works also for traps deep inside the bandgap and, thus, inaccessible by transport spectroscopy. This was exemplified on the well-known oxygen vacancy in rutile, which is deep enough for optical spectroscopy and at the same time still accessible by conventional DLTS. Using a Schottky contact formed by a rolled-up nanomembrane allowed us to couple the light with sufficiently high intensity into the space charge region of the corresponding Schottky barrier. The conventional DLTS and the DLTPCS methods in combination with a suitable rate equation model gave identical quantitative results within the error bars. Furthermore, the combination of these methods allows an unprecedented interpretation of the measured time constants and their temperature dependence. In particular, the measurements contain information regarding the capture process, for which an ad hoc model has been developed. Further work may comprise the application of DLTPCS to deep traps, which are inaccessible for pure transport techniques, the simultaneous determination of the optical cross section, and the Franck-Condon shift by looking at the photocapacitance as a function of the light energy,⁴⁰ depth-resolved measurements,⁴¹ or determination of the dependence of the capture cross section on the applied electric field.

The preparation of the rolled-up nanomembrane electrode is described in more detail in the supplementary material. Furthermore, a schematic illustration of the electron capture and emission processes is given, and the influence of the frequency dependence of the transient capacitance on the analysis is discussed.

The authors have no conflicts to disclose.

L. Berg: Data curation (equal); Investigation (lead); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (supporting). L. Schnorr: Data curation (equal); Software (lead); Writing – original draft (supporting). L. Merces: Investigation (equal); Methodology (equal). J. Bettini: Investigation (equal); Resources (supporting). C. C. Bof Bufon: Conceptualization (equal); Project administration (equal); Resources (equal); Writing – original draft (supporting). T. Heinzel: Conceptualization (equal); Funding acquisition (equal); Project administration (lead); Resources (equal); Supervision (equal); Validation (equal); Writing – original draft (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.