# Dimensional Roadmap for Maximizing the Piezoelectrical Response of ZnO Nanowire-Based Transducers: Impact of Growth Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{17}cm

^{−3}to 10

^{20}cm

^{−3}[28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]. The main reason is related to a large incorporation of residual impurities (i.e., Al, Ga, In, etc.) which act as shallow donors in the vapour phase deposition technique [31], as well as to the specific role of hydrogen which can form a wide variety of defects acting as shallow donors in the wet chemistry techniques [37]. Although ZnO NWs exhibit a top polar c-face and six non-polar m-plane sidewalls regardless of the growth methods used, the main characteristics of these surfaces (e.g., surface roughness) are not equivalent. Surface trap density is also, to a significant extent, affected by the growth method used [28,33,49,51,52]. It is thus very important to take the growth method into consideration in the design and optimisation of piezoelectric mechanical transducers in view of its effect on doping level and surface trap density. However, this line of research has not been explored so far.

## 2. Simulation Framework

^{®}environment, which provides fully flexible description of geometry, differential equations to be solved, and boundary conditions.

#### 2.1. Device under Study and Simulated Structure

_{d}) and surface trap density at the interface between ZnO and PMMA (N

_{it}), were also varied as a function of growth method.

#### 2.2. System of Equations

_{d}is the concentration of ionized donor atoms. In previous studies [20], the free carrier concentration n was computed using Boltzmann statistics. However, depending on growth method, doping level can reach quite large values with respect to degeneracy level, which is around 10

^{18}cm

^{−3}in ZnO. Here, n was thus computed by considering Fermi-Dirac statistics:

_{1/2}(x) is the Fermi-Dirac integral function of order ½, kB is the Boltzmann constant, q is the electric charge of one electron, and T is the temperature, considered here equal to 300 K.

_{Final state}− V

_{Initial state}.

#### 2.3. Boundary Conditions

_{Top}was obtained by averaging V on the top surface. At the interface of ZnO and PMMA (i.e., the diagonal line pattern on the ZnO surface from Figure 2b), we introduced a surface charge Q

_{s}under the assumption of a uniform trap density (N

_{it}) at thermal equilibrium. In this paper, the potential used to calculate Q

_{s}was taken from the initial state, which simulates ideally slow traps, with a charge that remains frozen during the transition from initial to final state. Q

_{s}was expressed as a function of the local potential V

_{init}as:

_{Fi}is the difference between Fermi level and intrinsic level.

## 3. Simulation Results and Discussions

#### 3.1. Input Experimental Data for the Simulation

_{d}), and a typical surface trap density (N

_{it}) value were selected in that purpose. As regards the dimensions of ZnO NWs, their typical radius was varied over a similar range of 4 to 150 nm while their length was firstly kept fixed to 5 µm in all devices. These dimensional properties of ZnO NWs are very typical and similar for each growth method. In contrast, the range of doping level in ZnO NWs strongly depends on the growth method used, as represented in Figure 3. From the large number of experimental data reported in the literature using field-effect transistor (FET) measurements [30,35,40,44,45,46,47], I–V measurements on four-terminal contacted ZnO NWs [28,29,31,36,37,50], terahertz spectroscopy [34], conductive AFM (i.e., SSRM and SCM measurements) [32,43], and electrochemical impedance spectroscopy [39,41], a range of charge carrier density values was inferred for each growth method when ZnO NWs are grown using standard conditions (i.e., typical chemical precursors, typical growth temperature and pressure). Overall, the charge carrier density of ZnO NWs typically lies in the range of 10

^{17}to 10

^{20}cm

^{−}

^{3}. The vapour phase deposition techniques including TE, CVD, and MOCVD methods result in the formation of ZnO NWs with a lower mean charge carrier density ranging from 10

^{17}to 5 × 10

^{18}cm

^{−3}at maximum [28,29,30,31,32,33,40,44,45,46,47,48,49,50]. In present deposition techniques, the incorporation of residual impurities (i.e., Al, Ga, In) acting as shallow donors is mainly responsible for this range of charge carrier density values [31]. Residual impurities usually occur as contaminants in the materials sources (i.e., TE) or in the growth chamber (i.e., CVD, MOCVD). The high growth temperature used in MOCVD is also favourable to the diffusion of residual impurities from the substrate, like Al from sapphire, into ZnO NWs. In contrast, wet chemistry deposition techniques, including CBD and electrodeposition, lead to the formation of ZnO NWs with a higher mean charge carrier density, ranging from 5 × 10

^{17}cm

^{−3}at minimum to 10

^{20}cm

^{−3}[34,35,36,37,38,39,41,42,43]. In the electrodeposition process, the use of zinc choride as the typical chemical precursor to enhance the morphology of ZnO NWs is favourable to the massive incorporation of chlorine acting as a shallow donor [42]. In the CBD process, the massive incorporation of hydrogen-related defects acting as shallow donors is mainly responsible for this range of charge carrier density [36]. The growth medium in water is full of hydrogen and the crystallization process resulting in the elongation of ZnO NWs through the development of their c-plane top facet basically involves a dehydration process [64]. A large number of hydrogen-related defects (e.g., interstitial hydrogen, substitutional hydrogen on the oxygen lattice site, zinc vacancy–hydrogen complexes) acting as shallow donors are thus formed systematically [37].

^{12}cm

^{−2}(i.e., about 10

^{12}eV

^{–1}cm

^{−2}) [28,33,49,51]. In contrast, the surface trap density of ZnO NWs grown by wet chemistry is about one decade larger (1–4 × 10

^{13}cm

^{−2}i.e., about 10

^{13}eV

^{–1}cm

^{−2}) [52]. The higher value when using wet chemistry is expected because their surfaces are typically rougher and thus present a larger density of defects.

#### 3.2. Piezoelectric Performance as a Function of the Nanowire Growth Method

_{d}and N

_{it}corresponding to each NW growth method, as described in the previous section. A numerical simulation was also made to evaluate the effect of varying the length (L) for one particular growth method that is of high interest, namely the CBD growth of O-polar ZnO NWs, and with a typical radius set to 50 nm.

#### 3.2.1. TE Method

^{17}cm

^{−3}and a value of N

_{it}= 1 × 10

^{12}eV

^{–1}cm

^{−2}was taken. These parameters correspond to ZnO NWs grown by the TE method as presented in Figure 3. Figure 4a shows a strong radius dependence of the piezoresponse, further influenced by the doping level. At the minimum value of N

_{d}, a step in the piezoresponse was observed, with a strong increase (by about 13 times) in absolute value, as NW radius was reduced from 140 nm to 120 nm. A further reduction of the radius had little effect on the piezoresponse. A similar dependence was also found for the maximum value of N

_{d}, but the step in piezoresponse was observed at lower radius (i.e., for a reduction from 50 nm to 40 nm). The region between the two curves in Figure 4a thus represents the range of optimization of the VING devices. VING devices integrating NWs with a radius larger than 120 nm for low doping levels (blue curve), and larger than 40 nm for high doping levels (red curve), result in poor piezoresponse. This is due to the screening effect originating from free carriers in the ZnO NW [20,54,56,68,69,70]. An example of this effect is shown in Figure 4b. It depicts a qualitative map of the free carrier distribution in a VING device under compression. The VING transducer integrates NWs with a radius of 140 nm and a low doping level (1 × 10

^{17}cm

^{−3}). A depletion region is created from the PMMA/ZnO interface and a neutral core starts from the bottom and extends towards the top of the NW. In this neutral region, the free carriers screen the piezoelectric response and the contribution of polarization electric charges is largely reduced, thus the overall voltage is reduced as well. Figure 4c shows the effect of the reduction of the NW radius down to 80 nm at the same doping level. In this case, the depletion region is large enough to fully deplete the NWs from its sides and from its top, drastically increasing the performance of the device. We can thus identify a critical value for the ZnO NW radius, below which the performance of VING transducers can be largely improved, for given doping level and trap density. This particular NW radius will be called “NW critical radius” all along the article and summarised for every growth method in Table 1. The critical radius can be evaluated analytically from charge neutrality between surface traps and surface depletion under the additional condition that depletion region reaches the center of the NW by solving Poisson equation in cylindrical coordinates (Figure S1 and Equation S5 in the Supplementary Materials).

#### 3.2.2. CVD Method

_{it}. The doping level lies in the range from 1 × 10

^{17}up to 1 × 10

^{18}cm

^{−3}, corresponding to the blue and red curves in Figure 5a, respectively. Figure 5a shows the weak piezoresponse obtained with NWs exhibiting radii larger than about 140 nm for low doping levels (blue curve) and 25 nm for high doping levels (red curve). A higher performance was obtained for NW radii below 120 and 20 nm for low and high doping levels, respectively.

#### 3.2.3. MOCVD method

_{d}goes from 1 (blue curve) to 5 (red curve) × 10

^{18}cm

^{−3}. This method shares the same Nit as TE and CVD methods as a first approximation. According to the results, the piezoresponse is improved when the ZnO NWs radius is smaller than 22 nm for minimum N

_{d}(blue curve) and smaller than 5 nm for maximum N

_{d}(red curve). A poor performance is expected for NWs with radius larger than about 30 and 7 nm for low and high doping levels, respectively. It should be noted that the simulation of a device integrating NWs with 5 nm radius could reach the limit where continuum medium equations do not apply anymore. In this sense, it should be considered as a rough approximation. Indeed, according to studies based on first-principles calculations, ZnO nanoscale materials with radius lower than 3 nm could present significantly higher piezoelectric coefficients compared to bulk material, namely there would be a radius-dependent size effect with its piezoelectric properties [71]. However, the formation of ZnO NWs with a radius smaller than 5 nm has not been experimentally shown yet and there has been thus no experimental evidence yet of the improvement of the piezoelectric coefficients in that range of radii. A comparison of the performance obtained when using these last three growth methods involving vapour phase deposition techniques (TE, CVD, MOCVD) show that smaller radii are gradually required to obtain the optimal performances: 120 nm, 120 nm, and 22 nm respectively, for low doping levels, as well as 40 nm, 20 nm, and 5 nm, respectively, for high doping levels. This can be explained as the doping level of NWs grown by MOCVD is higher than with the other growth methods, and the doping level of the NWs grown by TE is the lowest one, while keeping the same surface trap density. Because of the higher doping levels, NWs with smaller radii are required in order to obtain the full depletion of free charge carriers and thus to reduce the screening effect.

#### 3.2.4. CBD Method

_{it}is equal to 1 × 10

^{13}eV

^{–1}cm

^{−2}. Figure 6a shows the radius dependence of the VING performance for the CBD (O) method, namely for O-polar ZnO NWs. A positive piezo response is obtained in this case because the c-axis in the NW is oriented along the [000–1] direction (i.e., O polarity). This also changes slightly the interaction between the piezoelectric and semiconducting effects, producing a less sharp transition from the fully screened piezo response state to the fully depleted NW state where a maximum performance is obtained. The doping level range goes from N

_{d}= 5 × 10

^{17}cm

^{−3}(blue curve) to N

_{d}= 5 × 10

^{18}cm

^{−3}(red curve). Owing to the higher surface trap density of the CBD method, a better performance can be obtained for larger radius compared to that of the MOCVD method. For instance, half the optimal piezo response (~0.6V in absolute value) can be obtained for a NW grown by the CBD (O) method with a radius of ~55 nm. In contrast, a radius of ~23 nm is needed for a NW grown by MOCVD, which is larger by a factor of more than 2. Interestingly the critical radii are very similar for both methods. The best performance can be obtained for NWs with radius below 20 nm and 5 nm for low and high doping levels, respectively. For NW radius larger than 70 nm (low doping level) and 22 nm (high doping level), we expect that a poor performance is obtained.

#### 3.2.5. Electrodeposition: Analytical Evaluation of the Critical NW Radius

_{d}(~10

^{20}cm

^{−3}) which would lead to NWs with a wide neutral core over a wide range of NW radius. Instead, an analytical approach was considered to address the case of ZnO NWs grown by electrodeposition. To assess the agreement with the numerical simulations reported in the last sub-sections, the optimal NW radius to obtain fully depleted ZnO NWs was also calculated using an analytical model proposed in [72,73] for Si and GaAs NWs. The model considered only semiconductor equations and was developed to assess the critical radius a

_{crit}(Supplementary Materials), which marks the boundary between a fully depleted NW (r < a

_{crit}) and a NW that is only depleted at its surface (r > a

_{crit}). Figure 7 shows the values calculated within the analytical model correspond very well to the values reported in the last sub-sections for the different growth methods. In the case of the CBD (O) method, the analytical value is larger compared to the values extracted from our simulations. A theoretical value of ~70 nm and ~20 nm is calculated for low and high doping concentrations. From our simulations, we extract the values of ~20 nm and ~5 nm, respectively. This difference can be explained because the analytical models do not take into account the piezoelectric effect and the orientation of the NWs. The analytical model allows us as well to estimate the possible optimal radius of NWs grown by the electro-deposition method. The values lay well below the limit of our numerical model (estimated to be 5 nm with the piezoelectric coefficients used). This confirms that our numerical model would provide inacurate piezopotential results on NWs grown by this last method.

#### 3.2.6. Effect of the Variation of NW Length on the VING Performance

_{d}= 5 × 10

^{17}cm

^{−3}. Figure 8 shows a linear increase in the piezoresponse as the NW length increases up to 3 μm. For larger values, the piezoresponse saturates to about 0.75 V in absolute value. This means that longer NWs are not required to obtain optimal devices. This theoretical result is in accordance with the trends of experiments in energy generation [17], and output potential [18], although the output potential values are not the same, which can be due to different electrical and structural parameters. Furthermore, the inferred optimal length of O-polar NWs around 4 µm does not represent any technical challenge: it is typically achieved by the CBD method when the synthesis conditions using zinc nitrate, hexamethylenetetramine (HMTA), and chemical additives including polyethylenimine are optimized [74].

#### 3.3. Summary and Discussion about the Mechanisms at Work

_{d}allowed by each method based on reported experimental data. The range of values between the critical radii obtained for the low and high values of N

_{d}defines an optimization process window. This Table shows that the different growth methods do not offer the same potential for VING device optimization when using standard conditions. The TE and CVD methods exhibit the widest optimization process window by controlling both the radius and the doping of the NWs. In particular, they allow the use of wider NWs with radius not exceeding 120 nm for the low doping level. It should be noted here that ZnO NWs grown by the TE and CVD methods present a typical radius in that range, such that the targeted critical radius is not a technological challenge. However, for the high doping level, the critical radius drops to 40 and 20 nm for the respective TE and CVD methods, respectively. In that case, the growth of ZnO NWs with this small radius is still feasible, but deserves a particular effort to be reached. On the other hand, the MOCVD method has much smaller critical radii for the low and high doping level down to 22 and 5 nm. Although the MOCVD method is a well-known technique to get high aspect ratio ZnO NWs [66], the present small radii require working in dedicated conditions such as low VI/II ratio and relatively high growth temperature [75,76]. This can be quite challenging to be reached. In contrast, ZnO NWs grown by MOCVD have significantly been developped for optoelectronic devices [77]. The CBD method has the great advantage of emphasizing the importance of considering polarity as a critical quantity for VING devices. While ZnO NWs grown by CBD with the Zn-polarity have the narrowest optimisation window and small critical radii of 18 and 12 nm for the low and high doping levels, respectively, O-polar ZnO NWs grown with the same technique exhibit one of the largest optimization window with a significant increase in the radii to obtain increased performance, half the optimal performance is reached for NWs with radii of 55 and 15 nm for the low and high doping levels. In this technique, the typical radius of ZnO NWs lies in the range of 50–100 nm when using a zinc salt as a source of zinc ions and HMTA [78]. The further decrease in the radius of ZnO NWs is more required for Zn-polar ZnO NWs than for O-polar ZnO NWs. This is typically achieved by using chemical additives such as PEI [74] and ethylenediamine [79] to inhibit the radial growth of ZnO NWs and hence to limit their radius. However, it is evident here that the addition of chemical additives is likely not sufficient for Zn-polar ZnO NWs, which points out the strong interest in developing O-polar ZnO NWs for VING devices in the capacitive configuration. It is worth noticing that these O-polar ZnO NWs also outperforms the Zn-polar ZnO NWs to get high quality Schottky contacts with Au on their top [80], which is critical in the Schottky configuration consisting in making Schottky contacts at the bottom and at the top of the VING structure [6].

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Structure of the VING device; (

**b**) unit cell of the VING and (

**c**) cross section of the VING unit cell around the axis of symmetry.

**Figure 3.**Schematic diagram summarizing experimental data for charge carrier density in ZnO NWs grown by TE [29,40], CVD [28,44,45,46,47,48,49,50], MOCVD [31,32,33], CBD (O) [36], CBD (unknown polarity) [34,37,43], CBD (Zn) [36], and electrodeposition [38,39,41]. A logarithmic scale is used for doping level. The coloured solid stars represent the experimental data point reported in the literature for each growth method, as an average value [28,29,33,34,38,39,40,41,44,49,50] or an interval of minimum and maximum values [31,32,36,37,43,45,46,47,48]. The coloured solid lines represent the deduced range of charge carrier density used in the numerical simulation for each growth method.

**Figure 4.**(

**a**) Variation of the piezoresponse of a VING as a function of the ZnO NW radius for a range of doping level (N

_{d}) from 1 (△) to 5 (▲) × 10

^{17}cm

^{−3}typical in the TE method. The shadowed zones represent the regions of interest for a given doping level. Free carrier distribution of a NW with N

_{d}= 1 × 10

^{17}cm

^{−3}and a radius of (

**b**) 140 nm and (

**c**) 80 nm. A trap density N

_{it}= 1 × 10

^{12}eV

^{–1}cm

^{−2}was considered in the simulation.

**Figure 5.**Variation of the piezoresponse of a VING as a function of ZnO NW radius taking into account two vapor deposition techniques for the growth: CVD and MOCVD. (

**a**) A range of doping levels (N

_{d}) from 1 × 10

^{17}( ) to 1 ( ) × 10

^{18}cm

^{−3}is considered, which is typical in the CVD method, (

**b**) a range of doping level (N

_{d}) from 1 ( ) to 5 ( ) × 10

^{18}cm

^{−3}is considered, which is typical in MOCVD method. A trap density N

_{it}= 1 × 10

^{12}eV

^{−1}cm

^{−2}was considered in the simulations.

**Figure 6.**Variation of the piezoresponse of a VING as a function of ZnO NW radius taking into account the CBD deposition technique. (

**a**) A range of doping level (N

_{d}) from 5 × 10

^{17}( ) to 5 ( ) × 10

^{18}cm

^{−3}is considered, which is typical in CBD (O), (

**b**) a range of doping level N

_{d}from 5 × 10

^{18}(○) to 1 (●) × 10

^{19}cm

^{−3}is considered, which is typical in CBD (Zn) method. A trap density N

_{it}= 1 × 10

^{13}eV

^{−1}cm

^{−2}was considered in the simulations.

**Figure 7.**Critical radius a

_{crit}as function of N

_{d}for two ZnO NWs with values of N

_{it}from 10

^{12}(green curve) and 10

^{13}eV

^{−1}cm

^{−2}(red curve). The black marks on the curves indicate the radius values for which full depletion was obtained as calculated using FlexPDE. The mark ( ) corresponds to the TE method, ( ) to the CBD (Zn) method, ( ) to the CBD (O) method, ( ) to the CVD method and ( ) to the MOCVD method. The blue and magenta arrows indicate the limiting value of N

_{d}for N

_{it}= 10

^{12}eV

^{−1}cm

^{−2}and 10

^{13}eV

^{−1}cm

^{−2}. Beyond this value of doping, the critical radius goes below 5nm and we consider that surface effects could modify piezoelectric coefficients. The mark ( ) corresponds to the conditions of NWs grown by the electron-deposition method, which was not simulated with FlexPDE.

**Figure 8.**Variation of the piezoresponse of a VING device as a function of the NW length. The parameters of the NW used in the model correspond to ZnO NWs grown by the CBD (O) method.

**Table 1.**Summary of NW radius values to achieve full depletion or optimal performance for the different growth methods (NW critical radius). In the case of the CBD (O) technique, the values correspond to the radius to obtain half the optimal performance.

Growth Method | ZnO NW Radius for Full Depletion | |
---|---|---|

$\mathbf{Min}.{\mathit{N}}_{\mathit{d}}\left(\mathbf{nm}\right)$ | $\mathbf{Max}.{\mathit{N}}_{\mathit{d}}\left(\mathbf{nm}\right)$ | |

TE | <120 | <40 |

CVD | <120 | <20 |

MOCVD | <22 | <5 |

CBD (O) | <55 | <15 |

CBD (Zn) | <18 | <12 |

Electrodeposition | Not simulated (estimated < 4) | Not simulated (estimated < 4) |

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**MDPI and ACS Style**

Lopez Garcia, A.J.; Mouis, M.; Consonni, V.; Ardila, G.
Dimensional Roadmap for Maximizing the Piezoelectrical Response of ZnO Nanowire-Based Transducers: Impact of Growth Method. *Nanomaterials* **2021**, *11*, 941.
https://doi.org/10.3390/nano11040941

**AMA Style**

Lopez Garcia AJ, Mouis M, Consonni V, Ardila G.
Dimensional Roadmap for Maximizing the Piezoelectrical Response of ZnO Nanowire-Based Transducers: Impact of Growth Method. *Nanomaterials*. 2021; 11(4):941.
https://doi.org/10.3390/nano11040941

**Chicago/Turabian Style**

Lopez Garcia, Andrés Jenaro, Mireille Mouis, Vincent Consonni, and Gustavo Ardila.
2021. "Dimensional Roadmap for Maximizing the Piezoelectrical Response of ZnO Nanowire-Based Transducers: Impact of Growth Method" *Nanomaterials* 11, no. 4: 941.
https://doi.org/10.3390/nano11040941