1. Introduction
Metal oxide (MOX) gas sensors and sensor arrays, i.e., E-Noses [
1], are widely employed in the detection of gases and gas concentrations in the ambient air to aid in a multitude of safety, security, medical, automotive and industrial control scenarios [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13]. All of these applications have been enabled by making progress along the three main directions of research, namely by striving for higher sensitivity, selectivity and stability [
14].
As is well known, MOX gas sensors respond to a multitude of gases with high sensitivity; however, these also suffer from poor selectivity and sensor drift. While selectivity problems could be dealt with by forming arrays of sensors with different cross-sensitivity profiles, much less progress has been made in the capability of detecting, compensating and mitigating the effects of sensor drift in MOX gas sensor arrays. With properly chosen sensors and sensor operation conditions, arrays of cross-sensitive sensors can be constructed to detect odor patterns that are characteristic of certain application scenarios and to distinguish these from interferent patterns that might turn up in these same scenarios. Problems with selectivity can arise in the case individual sensors within an array drift over the course of time at different speeds and in different directions. In such an event, odor recognition capabilities of E-Nose devices could deteriorate and eventually lead to a misfunctioning of such devices. Making progress in the direction of higher device stability is therefore essential for arriving at higher-performance E-Nose devices.
In view of this situation, we have decided to identify physical mechanisms which underly the processes of gas sensor drift and to eventually arrive at more stable and higher-performance gas sensors. Working in this direction, we have shown in a recent paper [
15] that experimentally observed long-term drifts in the baseline conductance of high-temperature deposited SnO
2 nanowire gas sensors can be understood in terms of an innovative equilibrating donor approach. The key idea in this approach is that the electronic properties of such high-temperature-deposited materials are not determined by metal impurities whose concentrations are permanently fixed during materials preparation but rather by oxygen-vacancy donors thermally generated during high-temperature deposition and quenched in as the temperature of the nanowires had been lowered into the range of normally employed sensor operation temperatures. With the quenched-in oxygen vacancy donors being key elements of the thermal lattice disorder, the concentration of such donor defects and their electrical charge states can adapt to the specific sensor operation conditions even after sensor preparation had taken place. The key enabling effect in such equilibration processes is that the formation energy of positively charged vacancy donor defects is Fermi-energy-dependent, which establishes a strong link between the structural and electronic degrees of freedom inside the MOX materials. With this link in place, the baseline electronic conduction of MOX sensor materials was shown to be continually driven towards lower levels as quenched-in non-equilibrium oxygen vacancy donors slowly anneal out and approach their sensor-operation-specific equilibrium concentrations.
Whereas in our previous paper [
15], our focus had been on the bulk property of long-term drift of the sensor baseline conductance in MOX materials, the focus in our present paper is on the surface and sub-surface effects of oxygen adsorption and sub-surface junction formation. Dealing with thin sheets of MOX materials with nanometric cross sections and infinitely extended surfaces, at which oxygen adsorption processes can take place, we study the effects of equilibration on the bulk donor and oxygen adsorbate concentrations as the cross-sectional dimensions
of the MOX sensor sheets are scaled down to increasingly lower values. Following this road towards lower
, we show that—under conditions of sufficient internal oxygen mobility—the coupled processes of bulk thermal generation of oxygen vacancies and the formation of oxygen ion adsorbates cooperate in such a way as to fit the ensuing sub-surface depletion zones into the increasingly narrower spaces of the MOX sensor sheets. In this way, a state of perfect volume depletion [
16,
17,
18] from mobile charge carriers is maintained down to the lowest levels of
. This process of downscaling comes to a natural end as cross sections of
are approached, where a cube of
side length would contain only one single oxygen-vacancy donor and two associated surface oxygen ions on average. At and around this level of miniaturization, the concept of homogeneously spread-out donor charges and ensuing electrical field and potential distributions arising from Poisson’s equation solutions becomes overstrained, possibly calling for new explanations of the MOX gas sensitivity at the single-
scale that go beyond the familiar double-Schottky-barrier model (DSBM) approaches, widely discussed in the literature [
19,
20,
21,
22,
23].
In presenting our arguments, the paper is organized as follows: in
Section 2, the close coupling of bulk thermal donor generation and the formation of surface oxygen ion adsorbates is explained.
Section 3 presents arguments that the rate of generation of positively charged oxygen vacancy donors in the bulk not only depends on the momentary values of the sensor operation temperature
but also on the value of the band bending potential
at the sites of vacancy formation. With
controlling the rate of generation of oxygen vacancy donors and the local densities of oxygen vacancy donors determining the degree of local band bending, an element of self-organization emerges that controls the formation of sub-surface depletion zones. This proposition of a self-organizing formation of sub-surface depletion zones inside MOX nanostructures is tested in
Section 4, where thin and infinitely extended sheets of MOX materials are analyzed by integrating Poisson equations with band-bending-dependent local donor charge densities. In this way, the internally arising electrical field and electron potential distributions are derived. The results show that at typical sensor operation temperatures, sub-surface space charge zones with approximately triangular potential profiles will develop, which optimally fit into the narrow spaces inside the MOX nanostructures. The limits to this kind of geometrical downscaling are discussed in
Section 5. There, a brief look at nanogranular materials [
24,
25,
26,
27,
28,
29,
30,
31,
32,
33] and the issue of a possibly minimum gas-sensitive grain size in MOX materials [
34,
35,
36] is undertaken.
Section 6 finally summarizes our results and points out opportunities for further research.
2. Coupling of Bulk and Surface Processes
In metal oxides, the key elements of lattice disorder are oxygen vacancies
which are formed by thermally exciting oxygen atoms out of oxygen lattice sites
[
37,
38,
39,
40,
41,
42,
43,
44,
45]. The emitted oxygen atoms, on the other hand, are injected into the nearby interstitial
lattice sites, thus turning them into mobile species:
As the initial
lattice sites had been occupied by
ions, the empty vacancy sites form double donors, which still leave two electrons to be emitted out of the vacancies into empty conduction band states. In this way, mobile electrons and two kinds of positively charged vacancies can be formed:
In
Figure 1a, a
donor vacancy is thermally generated inside a thin sheet of MOX material, leaving behind in the bulk an
interstitial and two mobile electrons. As in this configuration, the negative charge of the two electrons is compensated by the positive charge of the immobile
ion, a configuration with bulk electrical conductivity and zero gas sensitivity has emerged.
In
Figure 1b, the two electrons and the interstitial oxygen atom have migrated against the attractive force of the central
ion towards both nearest-neighbor surfaces, thus forming surface oxygen ion adsorbates
on both surfaces and creating sub-surface electrical fields. As shown on the left-hand side of
Figure 1b, an additional
-atom had to be captured from the air ambient to achieve full oxidation on both surfaces. While the internal electrical fields tend to re-establish the initial configuration shown in
Figure 1a, it has to be kept in mind that the state of oxidized surfaces shown in
Figure 1b is stabilized by the release of adsorption energy of about
per ion [
46,
47,
48] as the
adsorbates are being formed. Due to the transfer of the two electrons to surfaces and their immobilization at oxygen ion adsorbates, the configuration in
Figure 1b is electrically non-conducting but potentially gas sensitive as reducing gas species in the air ambient can interact with the surface oxygen ions, thus releasing an electron which is mobile inside the bulk [
40,
44,
45]:
As with the exception of Equation (4), all reactions are reversible; it becomes evident that—given close surfaces and sufficient oxygen mobility—MOX materials are able to alternate between internally conductive but non-gas sensitive and non-conducting but gas sensitive states. Further considering that upon cooling to
, all vacancies inside the bulk will anneal out, re-forming a fully coordinated MOX lattice with all metal atoms on
and all
-atoms on
sites [
15], it is clear that any gas sensitivity will vanish in this limit as well. In practice, however, such stable but insensitive states will never be formed as oxygen diffusion rates
are exponentially activated and thus also tend to zero in the limit
[
49,
50,
51,
52]:
In this latter equation
stands for the activation energy of oxygen diffusion,
for Boltzmann’s constant and
for the sensor operation temperature;
, finally, is the diffusion constant as extrapolated towards infinite temperature. Considering a thin sheet of MOX material of thickness
, surface diffusion across a distance of
at a temperature
will take characteristic times:
which will exponentially rise upon cooling to increasingly lower temperatures. We have already shown in our previous paper [
15] that such time constants are very short during high-temperature preparation of nanometric structures but very long in the order of months or even years in the range of normally employed sensor operation temperatures. For the purpose of illustration, some of those time constants
have been quantitatively evaluated and listed in
Table 1.
In the following, we focus on cases of complete equilibration of all electronic and structural degrees of freedom. In order to indicate this, we consistently denote equilibration temperatures by to distinguish these from directly accessible sensor operation temperatures and situations in which only partial equilibria had been attained where the available electronic degrees of freedom had equilibrated towards quenched-in and slowly relaxing structural non-equilibrium states.
3. Coupling between Electronic and Structural Degrees of Freedom
In the above section, we have shown that the internal processes of vacancy formation in the bulk and oxygen diffusion toward surfaces can create activated configurations with adsorbed surface oxygen ions that are gas sensitive. In this section, we provide evidence that the electrical fields generated by such oxygen ion adsorbates also react back onto the bulk, thus modifying the ease with which donor vacancies and their different charge states can be created there.
Upon starting our considerations, we graphically illustrate in
Figure 2a the ability of oxygen vacancies in the MOX bulk to function as double donors of electrons. With two electrons being initially bound to neutral oxygen vacancies, such vacancies are free to transfer their electrons either to unoccupied conduction- or empty valence-band states. Considering large ensembles of transferred electrons, such transfers statistically correspond to simple downward transitions towards fictive and unoccupied electron states positioned at the equilibrium Fermi energy
. As indicated in
Figure 2b, the extra energy added to the reservoir of MOX lattice vibrations enables electrons to be re-excited to the conduction band edge from where they can contribute to the electronic conduction inside the MOX material.
In the analysis of MOX sensor properties, it is often implicitly assumed that the total density of oxygen vacancy donors is permanently fixed during materials preparation and that thereafter only redistribution processes of electrons among all available electron states can take place. As this scenario fails to account for the post-deposition phenomena of baseline conductivity drift and ensuing changes in the reducing gas response, we have proposed in our previous publication [
15] that—with oxygen vacancy donors being key elements of the thermal lattice disorder in MOX materials—the densities of the donor vacancies and their different charge states can adapt to changes in the sensor operation temperature even after sensor preparation had taken place. The key idea in this proposal was that the energy releases in the downward electronic transitions in
Figure 2b can be directly re-invested in alleviating the formation of oxygen vacancies, thus turning them directly into singly and doubly ionized positive oxygen vacancy donors. Due to the electronic energy gains that are immediately realized during the vacancy formation process, the formation energies of positively charged vacancies are lowered, with the result that the densities of charged oxygen vacancy donors tend to significantly outnumber the densities of neutral vacancies:
In these latter equations
stands for the formation energy of neutral donor vacancies, which in the case of SnO
2 is about
[
53] and
and
for the electronic energy gains indicated in
Figure 2b. The factor of two in Equation (8), on the other hand, takes care of the spin degeneracy of singly ionized oxygen vacancy donors [
15].
With the charged donor vacancies taking part in the electronic redistribution processes inside the MOX materials, the condition for overall charge-neutrality at each temperature
reads [
15,
42,
43]:
There
and
are the electron and hole densities in the conduction and valence band states and
and
the volume densities of positively charged donor vacancies as defined in Equations (8) and (9). With the simplifications
and
the position of the common Fermi energy
relative to the conduction band edge
emerges as [
15]:
There,
is the effective conduction band density of states [
42,
43] and
the effective donor vacancy formation energy [
15]
With
being fixed, the concentrations of mobile electrons and of two-fold positively charged oxygen vacancy donors can be calculated that would turn up in a piece of MOX material that is operated at a constant temperature
under flat-band conditions and for a time long enough so that all redistribution processes of electrons and oxygen interstitials have come to an end. Up to this point, our arguments have been repeating all those ideas already introduced in our previous paper dealing with the bulk effect of sensor baseline conductivity drift [
15].
In order to get insight into situations in which flat-band conditions no longer apply, consider
Figure 3, where a n-p junction scenario with an abrupt interface has been envisaged. While on the left-hand side of
Figure 3, the MOX material is n-type doped by the naturally occurring oxygen vacancy donors, the right-hand side had additionally been doped by a large number of compensating deep acceptor impurities during preparation. As shown in our previous paper [
15], these latter impurities allow the Fermi energy to move somewhat deeper into the band gap but not to the extent that a true conversion to p-type conduction can be attained. Upon bringing both sides into contact, the higher electron densities on the left-hand side will tend to spread over onto the less electron-conducing right-hand side, thus producing a potential step several tenths of an electron volt high. Whereas in a fixed donor scenario, parabolic potential profiles with different slopes are expected to arise at the two sides of the interface, the situation changes once an equilibrating donor scenario is envisaged. In this latter case, vacancy donors forming on the right-hand side will benefit from larger electronic energy gains than those on the left-hand side, thus causing the junction to become more abrupt and the potential profile at the interface to become non-parabolic. Returning to
Figure 2 and Equations (9) and (10) and considering that the donor binding energies
and
are small compared with the energy gains
) and
) the energy gains in Equations (8) and (9) can be approximated by the magnitude of the local band bending potential
at each position
. In this way, Equations (8) and (9) simplify into [
15]
With the help of such band bending-dependent charge densities, the ensuing changes in potential profiles and internal electrical field distributions can be obtained by solving Poisson´s equation with space charge densities which by themselves depend on those band bending potentials which are induced by the negative ion adsorbate layers on both surfaces.
5. Limits to Geometrical Downscaling
In this section, we address possible physical reasons that limit the downscaling of solutions of Equation (15) towards increasingly smaller dimensions. What we should also like to do is making a digression towards nanogranular MOX materials with monolayer morphologies. In this way, we should like to contribute to the discussion of grain size effects and claims that there might be a lower limit of grain size beyond which the gas sensitivity of MOX materials does vanish [
34,
35,
36].
Starting point of our considerations is the inset in
Figure 7, where we have sketched a thin monocrystalline layer of MOX material with a thickness
. Subdividing the MOX sheet into squares with lateral side lengths
, a mental step can be taken in which the monocrystal layer is broken up into an array of weakly interlinked nanocrystals. Building on the results shown in
Figure 6a, the numbers
of oxygen vacancy donors can be calculated that would fit into small crystals of size
. Turning to the data in
Figure 7, it can be seen that crystals with side lengths in the order of
would contain roughly
oxygen vacancy donors on average, while ones with side lengths of
would contain only
donors on average. Considering the fact that oxygen vacancy donors and oxygen ion adsorbates can bind only multiples of the elementary charge
, it becomes evident that at the size level of
and below, the effects of granularity of electronic charge become dominant. In the following, we call this limit the granularity limit. As argued below, some interesting consequences may arise regarding the understanding of gas sensitivity phenomena when the granularity limit of MOX nanostructures is approached.
At length scales far above the granularity limit, donor numbers per grain are relatively large and associated statistical spreads are relatively small. In this limit, the concept of homogeneously spread-out donor charge densities does make sense, and Poisson equation approaches can be used to draw up images of electrical field and electron potential energy distributions throughout each grain and across grain boundaries. Electrons generated in a reducing gas-solid interaction and traveling across several of such grain boundaries need to absorb thermal energy from the MOX grains to surmount the individual potential barriers on their way, thus leading to a thermally activated electron mobility
where
represents the average barrier height along the transport path and
the mean thermal energy at the sensor operation temperature
. Such a kind of transport is depicted in
Figure 8, assuming that the grain-internal potential barriers have a triangular shape and surface barrier heights in the order of tenths of electron volts, as illustrated in
Figure 4d. Within this point of view, our equilibration approach largely reproduces the widely accepted DSBM pictures of the MOX gas response in nanocrystalline materials [
19,
20,
21,
22,
23]. The present equilibration approach, however, does represent a step beyond the present state of the art in that it can satisfactorily explain the occurrence of volume depletion of free carriers across wide ranges of MOX sheet thicknesses and/or grain diameters and also the surprising long-term stability of such volume-depleted structures.
Things change, however, when grain sizes approach the granularity limit defined above. In this case, the statistical spread in the average number of donors per grain increases towards the average numbers themselves and even beyond as grain sizes are reduced below the granularity limit. In such a situation, crystalline structures emerge where single crystals contain one or two donors while others will not contain any donors at all. In this case, it becomes evident that the concept of homogeneously spread-out donor charge densities will lose meaning, and Poisson equation approaches aimed at arriving at smoothly varying electrical field and electron potential energy distributions across grains and grain boundaries will become increasingly more questionable. Such concerns clearly apply to the conventional DSBM approaches in the published literature [
19,
20,
21,
22,
23] as well as to the equilibration approach presented in this paper.
Trying to arrive at an electron theory for the transport in nanocrystalline MOX materials, Zaretskiy et al. [
34,
35,
36] proposed that electronic transport processes inside increasingly fine-grained MOX materials would be more appropriately described in terms of discrete electron hops rather than in terms of classical band transport theory. Hop** transport has first been intensively investigated in the field of amorphous semiconductors [
59,
60,
61], where hop** rates between adjacent localized electron sites were found to be
in this equation
is a typical phonon frequency,
and
stand for energy differences and spatial separations between adjacent states and
for the localization length of the involved states. More recently, similar kinds of transport have been observed in various kinds of nanocrystalline materials, including MOX ones [
62,
63,
64]. It, therefore, appears plausible that in such materials, the grain-boundary modulated band transport proposed in the various DSBM approaches can be replaced by hop** transport approaches when grain sizes in the range of the granularity limit or below are approached.
Returning to Zaretskiy’s electron theory, a particularly interesting proposal was that gas sensitivity in MOX materials should vanish when hop** conduction becomes dominant and when average grain sizes are decreased below a critical value a
. Estimates of
were estimated to be in the range of a few nanometers. Following their arguments, it is suggested that the equilibration approach introduced in our previous paper [
15] and extended to the junction and sub-surface situations in this paper has pushed the limit of gas sensitivity loss from several nanometers down into the
range.
Experimental observations pointing in such a direction were recently published by Prof. Jianqiao’s group working on SnO
2-based quantum dot materials [
32,
33]. These researchers observed a reduction in the gas sensitivity at grain sizes in the range of several
and a complete loss at grain sizes of about
, i.e., at length scales consistent with the granularity limit defined in
Figure 7 above. Interestingly also, Density Functional Theory (DFT) calculations of this same group predicted the appearance of a considerable density of localized bandgap states when grain sizes were reduced into this same range. As their films had been prepared using a hydrothermal method and low-temperature annealing only, it is, however, not yet clear whether these films had progressed to a state of complete equilibration of all structural and electronic degrees of freedom as discussed in this paper.
Returning to Zaretskiy’s proposal of variable range hop** in tiny MOX crystals, we propose that the loss of continuous percolation paths in nanogranular MOX sensing materials may cause a loss of gas sensitivity at the single nanometer scale. Aiming at a resolution of the limiting grain-size issue, we show in
Figure 9 two examples of gas sensing interactions in a fine-grained MOX material with grain sizes in the range of the granularity limit. Whereas in the first case (
Figure 9a), all grains had been assumed to be equal, and each grain containing one single donor, the second case considers a situation in which one of those grains—due to the effects of statistical scatter—does not contain any donor at all. Whereas in the first case, hop** transport through adjacent donor sites successfully connects the gas-solid interaction and electrode collection sites, no continuous conduction path is formed in the second case as neighboring hop** sites are separated by distances
significantly longer than the electron localization lengths of the nearest-neighbor donor sites. In this kind of interpretation, a decrease in the MOX gas response at grain sizes
would arise from increasingly smaller numbers of electron percolation paths that successfully connect gas-solid interaction and electrode collection sites.