Electrical properties of strained off-stoichiometric Cu–Cr–O delafossite thin films

Off-stoichiometric Cu–Cr–O delafossite thin films with different thicknesses were grown by metal organic chemical vapor deposition on substrates with different coefficients of thermal expansion. Seebeck thermoelectric coefficient and resistivity measurements were performed on the range of 300–850 K. A qualitative change in the temperature-dependence of the resistivity is observed at the temperature corresponding to the deposition process, where the transition from tensile to compressive strain takes place. Arrhenius plots reveal different slopes in these two thermal ranges. The fact that the shift is more pronounced for the thinner films might indicate the induced strain plays a role in changing electrical behaviour. Furthermore, changes below 0.1% in electrical mobility were measured when the strain is induced by mechanical bending.


Introduction
The presence of stress (σ) in thin films is a significant research topic due to its impact on the performance of technological applications.Most films are investigated or used at temperatures that differ from those during the synthesis process.The potential significant mismatch between the coefficients of thermal expansion (CTE) of the deposited film and the substrate might lead to a substantial amount of thermal stress accumulated in the film.When stress reaches certain levels, it can cause detrimental effects such as cracking, peeling off, Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.buckling, or blistering [1].On the other hand, stress can have beneficial effects in tailoring or controlling some physical properties.Classical examples include magnetoelasticity [2] or piezoelectricity [3,4], where mechanical energy is converted into magnetic or electrical energy, respectively.The present article focuses on how the strain affects the carrier's mobility (µ).Strain (ε) changes the bond lengths (and the lattice parameter for crystals) at elementary cell level, consequently altering the material's energetic band structure through the addition of perturbation terms in the averaged potential V(r) [5]: r iJ (1) where U(r) represents the time-averaged potential (in which the electrons move independently), the summation represents the interaction of the electron with nuclei in the lattice at a distance r iJ (q denotes the elementary charge at position i and Z J the atomic number of nuclei at position J).These interactions lead to changes in the effective mass (m * ), which is related to the dispersion of the band edges.This is important as free carriers are located at the minimum or maximum of the conduction or valence bands (for electrons or holes, respectively).
In the simple quadratic approximation, m * is given by: where h represents the reduced Planck constant, and the second derivative d 2 E/dk 2 represents the dispersion of the band edges.µ is related to m * , according to [6,7]: where τ represents the relaxation time corresponding to scattering mechanisms.The engineering of µ through strain has been a subject of long-standing studies.In the case of certain materials, such as monolayers of phosphorus allotropes, the density functional theory estimates a 3000 % increase in µ for a strain of −6 % [8].However, in typical experimental cases, an enhancement up to one order of magnitude in µ is observed with increasing stress in semiconducting thin films [9].The dependence of µ on strain has been utilised to optimise active channels of silicon metal-oxide-semiconductor fieldeffect transistors.For instance, enhancement factors of 2.2 [10] and 1.5 [11] were reported for n-and p-channels, respectively, in the early 1990s.There are numerous factors that can induce strain in a thin film deposited on a substrate, and the majority of these are difficult to control.During the preparation of thin films, internal stresses develop to maintain the mechanical and thermodynamic equilibrium of film growth on a substrate.The distribution of total stress (σ tot ) in a film-substrate system can be described as follows: where the subscripts i, ext and th denote intrinsic, external and thermal stresses, respectively.Intrinsic stresses encompass those induced during film preparation and have several origins: strained regions within the films due to defects that appear during film growth (such as grain boundaries, dislocations, voids, impurities, etc); lattice mismatch between substrate and film; film/vacuum interface (surface stress, adsorption, etc); or due to dynamic processes (recrystallisation, interdiffusion, etc).The other two terms of the equation are associated with the adhesive interaction between film and substrate: σ ext represents any stress intentionally applied from an external load, while σ th is an extrinsic stress arising from the mismatch in CTE between the film and substrate, combined with temperature changes after the film growth process [12][13][14].In the absence of external loads, the stresses present in the film are referred to as residual stresses [15].The influence of film thickness (h f ) on its mechanical properties should also be taken into consideration.In bulk films, triaxial stresses are present.However, in the thin film approximation-where the substrate has a much greater thickness (h s ) than the film (h s ≫ h f ), and the total thicknesses of both are negligible compared to their lateral dimensions-the stresses develop biaxially [16,17].When the films grow epitaxially, mechanical anisotropy needs to be considered; for polycrystalline films with no preferred orientation, isotropic averaged material parameters can be assumed, and the stress has the same magnitude along all three directions [14].However, to effectively study or manipulate the strain, it must be induced in a controlled manner.The compliant techniques utilized in this work are based on the strain generated by the mismatch between the CTE of the film (α f ) and substrate (α s ), or external strain by orderly bending the substrate.When the first technique is used, considering the thin film approximation, a thermal strain (ε th ) arises from the difference between α f and α s , resulting in a tensile or compressive strain depending on whether α f is larger or smaller than α s , respectively (figure 1), at room temperature.For a non-epitaxially grown continuous film deposited at a temperature T d , ε th can be calculated using the following expression (only on the elastic regime) [1,14,18]: where T m represents the temperature at which the thermal strain is measured after the deposition, typically at room temperature.Moreover, the strain distribution along the z-axis should also be considered.The maximum strain occurs at the film-substrate interface, and then the films start to relax.The thickness-averaged stress (σ avg ) can be obtained by integrating the in-plane stress (σ in ) at height z from the substrate (z = 0) over the film thickness (z = h f ) [1]: where E f and ε avg are the Young modulus and the average strain of the film, respectively.For thick films, the effects of strain might be overshadowed by the overall elastic behaviour of the entire film.
The measurement methods of stresses can be categorised into two groups: those that determine the external stress applied onto a substrate from bending the substrate (such as beams with strain gauges [19]) and methods that measure residual stress, such as the sin 2 Ψ method [15] and the curvature method [20].In the latter group, the stress is calculated from Hooke's law, which requires knowledge of the elastic constants (E f and Poisson's ratio (ν f )) of the thin film material.When a film is strained, its geometrical shape is modifying (shrinking for example), that also affects the resistance of a film, overlapping over the piezoresistive effects.The gauge factor (k) represents the ratio of relative change in electrical resistance to the mechanical strain.This change in resistance has two components: one due to change on the size of the resistor (length and transversal area) and one due to semiconducting effects, which involve change in resistivity (∆ρ) relative to the unstrained resistivity (ρ).
Here R 0 represents the resistance of the unstrained sample, ∆R denotes the variation of R 0 after inducing strain on, and n is the charge carrier concentration.The terms 1 + 2ν f are directly associated with changes in dimensions of the semiconductor by its Poisson ratio, while ∆ρ/(ρ.ε) is attributed to the piezoresistive effect.In semiconductors, k is primarily influenced by ∆ρ, with the geometric factor playing a minor role.For example, in silicon and germanium, the piezoresistive term is 50-100 times larger than the geometric term.In this context, strain gauges are commonly used for experimental stress analysis [3,4].
The present work is dedicated to the study on how the strain is affecting µ in off-stoichiometric (Cu 2/3 Cr 4/3 O 2 ) copper chromium oxide (CCO) with a 3R polytype structure.This material is an interesting p-type transparent conductive oxide (TCO), with the one of the highest conductivities for a non-intentional doped semiconductor [21,22].Long chains of copper vacancies were observed and accounted for the high level of doping [23,24].The drawback of this material is the low values of µ, a legacy of conduction mechanism relying on small polaron hopping [25].This impedes the use of this material in fast electronic devices.In the literature, there are very limited reports on mechanical or thermal properties of stoichiometric CCO.The limited availability of previously reported values for elastic constants in the literature adds a further challenge to the present study.There are only two reports for the CTE of stoichiometric CCO (α CCO = 10.26 [26] and 7.5(1) ppm K −1 [27]).Both correspond to pellets of standard CCO stoichiometry.Moreover, a bulk modulus if 153 GPa is reported for the stoichiometric CCO [28] but, again, this value is based on studies involving pellets.These values might not be valid in the present case of the thin films, especially where a significant amount of Cu atoms is missing.
Thin films with different thicknesses were deposited on various substrates with different CTE.When films are mechanically bended, small variations, under 0.1% are estimated for µ.Seebeck coefficient (S) and ρ are measured in a temperature range from room temperature up to 600 • C.Then, the µ is estimated within a small polaron conduction mechanism assumption.A non-monotonous behaviour is observed at the temperature corresponding to the deposition temperature, where the strain is passing from the tensile to compressive.This effect is more pronounced for the thinner films, that might furthermore support the hypothesis that strain influence the µ behaviour.Different activation energies are measured in the films compressed or tensile.

Electrical characterisation
Electrical measurements at room temperature were performed using a Multiheight probe stand consisting of a linear fourpoint probe system with 1 mm distance between probes (Jandel Engineering Limited, UK), connected to a Keithley 2401 SourceMeter ® (Keithley Instruments, USA).Seebeck coefficient (S(T)) and electrical resistivity (ρ(T)) measurements were performed using a LSR-3 (Linseis Seebeck Resistivity from LINSEIS Messgeräte GmbH, Germany).The setup comprises a 4-point probe terminal, with a distance between inner probes of 8 mm, a gradient temperature of around 2 K, with the absolute S value being corrected to Platinum wire reference.

Strain bench setup
A home-built strain gauge bench setup was used to measure the variation of the electrical resistance of the semiconducting sample as function of the strain induced externally by controlled bending (figure S1).The bench is composed of a test structure made of an aluminium double clamped cantilever (140 mm length, 30 mm width, 0.5 mm thickness), where the sample to test is glued (Loctite 401, RS) in the middle of the beam.The deflection in the middle of the cantilever is controlled by a differential micrometer drive (DRV304, ThorLabs) with a coarse adjustment of 13 mm travel with a resolution of 5.0 µm, and a fine adjustment of 300 µm travel with a resolution of 0.5 µm.Reference strain gauges (RS PRO, 120 Ω, ref. : 865-6226, RS) made of metal Cu/Ni electrodes embedded within a polyimide foil are glued on the same cantilever location to calibrate the deflection-tostrain values.The controlled applied strain has a sensitivity of 3 × 10 −7 µm −1 with a maximum limit of strain applied of 4 × 10 −3 .A 200 nm film was deposited on the D 263 ® T eco glass (length = 55 mm, width = 7 mm and R 0 = 4.9 kΩ), that was then glued in the middle of the aluminium bar with a test cantilever attached.

Structural, morphological and chemical composition characterisation
The surface morphology and thickness of the films were both inspected by scanning electron microscopy (SEM) (FEI Helios 50 High Resolution).X-ray Photoemission Spectroscopy (XPS) was performed with a Kratos Axis Ultra DLD system using a monochromated (Al Kα: hν = 1486.7 eV) x-ray source coupled with a monoatomic argon gun for depth profiling.The following peaks were used for elemental quantification: Cu 2p 3/2 , Cr 2p 1/2 and O 1s and their corresponding relative sensitivity factors are depicted in table S1.The raw areas from the XPS spectra after sputtering for 400, 800 and 1200 s are depicted in table S2.A more detailed interpretation of XPS results is given in [29].The crystalline structure of Cu-Cr-O was studied using a Brucker D8 Discover diffractometer with Cu Kα radiation (λ = 1.541 84 Å) operating at 40 kV and 40 mA, in grazing incidence (0.5 • ) configuration (GIXRD).

Electrical characterization of mechanically bended films
Films glued on the aluminium bars were bent upwards and downwards, inducing tensile and compressive strain, respectively.The calibration gauge measured permanently the values of the strain while the Wheatstone bridge measured the change in resistivity ∆R/R 0 (R 0 corresponding to the unstrained state).The measurements are done at a fixed temperature and hence these changes are result of either size or piezoresistive effects.The carrier concentration remains constant in this case.Figure 5 depicts the normalised µ relatively to the unstrained one (µ/µ 0 ) in (a) compressive and (b) tensile strain regimes.Same trend is observed in both cases.µ increases with the increase of the distance between the atoms during the deformation.The insets within the two graphs show the ε dependence of k.The gauge factor value remains always negative, indicating that the piezoresistive effects prevail.Indeed, from equation ( 7), a negative k implies that the piezoresistive term is greater than the term corresponding to size effects.However, the change in µ is small, bellow 0.1 %; the flexible substrate breaks at around ε = 0.06 %, in absolute value, that makes impossible to evaluate the behaviour at higher ε values.

Thermal strain characterisation
As previously explained the thermal strain (ε th ) arises from the difference between the CTE of the film (α f ) and the CTE of substrate (α s ).Considering the CTE of delafossite   measurements during the heating of the sample).The resistivity data shows a typical semiconducting behaviour with values decreasing from around 0.1 Ω cm to 0.01 Ω cm at higher temperatures.Different values are measured for different substrates.Here might intervene to small errors in a precise measurement of the thicknesses as SEM micrographs reveal small differences over the area.Inhomogeneities might also appear during the chemical deposition.The ρ(T) evidence also a small 'kink' around 600 K (vertical lines), a temperature close to the deposition temperature, that is the cross-over from tensile to compressive stress.This feature is less pronounced in the case of thicker samples.The Seebeck coefficient is showing positive values around 100-200 µV K −1 .S(T) curve indicate an almost linear increase with temperature.Some inadvertences occur in the case of SrTiO 3 substrates at higher temperatures, but that might be related to artefacts of measuring setup.In order to elucidate the origin of the behaviour of the resistivity around the deposition temperature, we start from the classical equation of resistivity ρ = (neµ) −1 .The carrier concentration is related to the Seebeck coefficient that has a linear behaviour in that temperature range.That leaves mobility as the possible source of the kink on the ρ(T) curve.In order to estimate the values of the mobility we used the small polaron hoping mechanism model [30], where the conduction occurs via Cu + /Cu 2+ hole mechanism.Using the Heikes formalism [31] and accounting for the off-stoichiometry, the carrier concentration is calculated and then the mobility is estimated: where k B is the Boltzmann constant, g 1 and g 2 are the electron spin degeneracy of Cu + and Cu 2+ states, respectively, and c is the fraction of polaron sites occupied by polarons (c = n/N Cu , where N Cu (≈ 1.5 × 10 22 cm −3 ) is the total density of sites of Cu atoms in Cu 2/3 Cr 4/3 O 2 crystalline structure, considering the missing Cu atoms).A thorough discussion about the validity of small polaron model is beyond the scope of this article.It is generally accepted and used in the case of delafossites.However, we point out here the dependence on the temperature of the Seebeck coefficient.Generally, within the small polaron model the coefficient should be temperature independent, indicating the fact that the charge carrier concentration is not changing with temperature.The chained copper vacancies (reported as the source of conductivity on the case of Cu 2/3 Cr 4/3 O 2 ) are not healing in the temperature range investigated, a fact confirmed by the S(T) reversibility.The enlargement of Seebeck with temperature might be attributed to the hopping of charge carriers between states which differ both in their energy and in their coupling to the atomic The reference values of αs are their respective suppliers, and the α f value is from [26].
vibrations [32].This 'semi-metallic' behaviour was previously observed in the case of pure or doped delafossite [33].The mobility values for the films from the batches with two different thicknesses are presented in figure 8.At room temperature, the mobility has values of 0.1 up to 0.4 cm 2 V −1 s −1 .For the thinner sample it increases faster almost linearly with temperature, with a cross-over at a temperature corresponding to the deposition temperature.In the case of thicker films, this crossover between the two linear regimes is missing or much less  pronounced.In addition, the values of mobility at higher temperatures are lower than in case of thinner films.This particular behaviour for films with different thicknesses was observed previously reported by Rastogi et al in the case of spray deposited Mg doped p-CuCrO 2 semiconductor oxide thin film [34].This is a rare report where the electronic properties are studied in a temperature range covering the deposition temperature.In that work, properties of thin films with thicknesses of 155 and 305 nm are studied.A similar 'kink' is observed in σ(T) curve and again the change is less pronounced for thicker films.However, the 'kink' was not observed at the deposition temperature and the authors suggested the influence of grain boundaries on the conduction.Following the same idea, we calculated the activation energies for the film studied here.In the adiabatic regime of small polarons, the electrical conductivity follows an Arrhenius relationship [35]: where A is the pre-exponential factor and E H is activation energy for carrier hopping from site to site.The latter can be extracted from the slope of a logarithmic of equation ( 9), as shown in figure 9.A linear behaviour was obtained for thick films in the whole measured temperature range; the same does not hold true for the thinner 70 nm thick Cu 2/3 Cr 4/3 O 2 thin films, where two different slopes observed again, with a cross-over around the deposition temperature.70 nm films have E H = 126-147 meV, and 300 nm films 131-151 meV.This behaviour leads to a hypothesis that ε th might influence the transport properties of the films: according to equation ( 5), the T d corresponds to the cross-over between compressive and  tensile ε th regimes.The ε th induced by the substrate leads to a variation of the Cu-Cu interatomic distances (lattice parameter) [35].An increase (decrease) of ε th means a longer (smaller) distance for the polarons to 'jump', increasing/decreasing the µ.In figure 8(b) the µ(T) slope is almost constant.From equation ( 6), the ε th is inversely proportional to the film thickness, therefore it is expected the ε th will have less impact on the electrical properties of thicker films, which is in good agreement with our observed data.In figure 10, the µ(T) data was normalised to the µ at deposition temperature (µ 0 ), as at this temperature there is no effect from strain (ε th = 0).Another point to mention is the electrical properties of Cu 2/3 Cr 4/3 O 2 grown on MgO: this was the only studied substrate with α s > α f .If the ε th has a role on the electrical properties of Cu 2/3 Cr 4/3 O 2 thin film, the trend on MgO is expected to be the opposite compared with the other studied substrates.For the same temperature T ̸ = T d , in MgO the Cu 2/3 Cr 4/3 O 2 is in a tensile (compressive) regime when for the other substrates it is in compressive (tensile) regime, and for thin films the real value might be in fact higher than the CTE of MgO (12.8).Future investigations using substrates with even higher CTE values shall be put into perspective.Furthermore the variation of mobility for different strain values at fixed temperatures was inspected (figure 11): at fixed T < T d the µ/µ 0 (T) values are similar regardless the strain; the only pronounced discrepancies at fixed T are visible at high T for thinner substrates: if strain has a major effect on mobility, it would more likely to be at high T > T d .

Conclusions
Off-stoichiometric Cu 2/3 Cr 4/3 O 2 films with different thicknesses were grown by DLI-MOCVD.The strain was induced thereafter by using the mechanical bending for films deposited on flexible substrates or by using rigid substrates with different thermal expansion coefficients.In the first case the negative values of the gauge factor indicated that piezoresistive effects predominate over geometric ones.Small changes in mobility, around 0.1% were estimated for the investigated range strain.In the case of thermally induced strain, the ρ(T) dependence indicates a semiconductive behaviour whilst the S(T) dependence indicates the presence of an energy dispersion within the polaronic states in these polycrystalline films.A peculiar feature is observed in the ρ(T) curve at a temperature corresponding to the deposition temperature.This corresponds to the transition between the tensile and compressive regimes in the thin films.This feature is more prominent for thinner films and almost disappears in thicker films, where the averaged strain over is smaller, that might imply the strain affect the mobility.However, no quantitative assessments can be drawn at this level and experiments with much larger thermal expansion coefficients are envisaged.

Figure 1 .
Figure 1.Schematic of the development of thermal strain of a thin film grown on a substrate with different CTEs.The films are relaxed at the temperature corresponding to the deposition temperature T d .Depending on the CTE film/substrate ratio, films are compressed or tensiled at temperatures different from T d .

Figure 6 .
Figure 6.Expected ε th (T) of Cu 2/3 Cr 4/3 O 2 on different substrates.The reference values of αs are their respective suppliers, and the α f value is from[26].

Figure 7 .
Figure 7. Measured and S(T) for (a) 70 and (b) 300 nm thick Cu-Cr-O films on different substrates.

Table 1 .
CTE of the substrates used throughout this work.