# Fundamental Definitions for Axially-Strained Piezo-Semiconductive Nanostructures

^{*}

## Abstract

**:**

^{16}cm

^{−3}) and under compression. Here we give the definitions for the enhancement, depletion, base and tip piezopotentials, their characteristic lengths and both the tip-to-base and the depletion-to-enhancement piezopotential-ratios. As an example, we use these definitions for analyzing the local piezopotential and free charges in n-type ZnO truncated conical nanostructures with different doping levels (intrinsic, 10

^{16}cm

^{−3}, 10

^{17}cm

^{−3}) for both axial compression and traction. The definitions and concepts presented here may offer insight for designing high performance piezosemiconductive nanotransducers.

## 1. Introduction

^{16}cm

^{−3}) and only under compression [22]. In fact, though, at least for the case of compression, the potential difference between the tip and the base has also been computed for different doping levels [23], both the local electrical potential and free charges have not been reported (for both compression and traction). We compute the local piezopotentials as well as the carrier concentrations along the nanostructure both in case of traction and compression for 2 µm long ZnO truncated conical nanowires with different doping levels (intrinsic, 10

^{16}cm

^{−3}and 10

^{17}cm

^{−3}), base radius equal to 150 nm and radius of the tip ranging from 25 to 125 nm.

## 2. Materials and Methods

^{16}cm

^{−3}, 10

^{17}cm

^{−3}).

## 3. Results

#### 3.1. Depletion Piezopotential (ΔV_{PZ-DEPL}) and Enhancement Piezopotential (ΔV_{PZ-ENHANC})

_{PZ-DEPL}, and the enhancement piezopotential, ΔV

_{PZ-ENHANC}, as the potential drops in the depletion and in the enhancement regions, respectively. For consistency with literature, both these potentials must be conventionally taken as the difference between the extremity of the (depletion or enhancement, respectively) region closer to the tip and the extremity closer to the base. With these definitions, the total voltage drop between the tip and the base is the sum of the depletion piezopotential and of the enhancement piezopotential. Since we ignore other effects (e.g., the band bending due to the metal-semiconductor interface) and only consider the piezopotentials due to strain, the depletion piezopotential and the enhancement piezopotential will always be concordant. Though the total potential drop already provides important information, in many cases and, for instance, when designing piezotronic devices, both the depletion piezopotential and the enhancement piezopotential are crucial as the charge transport properties at a certain junction will depend on the correspondent local piezopotential.

#### 3.2. Tip Piezopotential (ΔV_{PZ-TIP}) and Base Piezopotential (ΔV_{PZ-BASE})

_{PZ-TIP}, and the base piezopotential, ΔV

_{PZ-BASE}, as the potential drops developed at the tip and base, respectively. Similar to the depletion piezopotential and the enhancement piezopotential, as schematically shown in Figure 1, the tip piezopotential and the base piezopotential will also be conventionally taken as the difference between the tip and the center or between the center and the base, respectively. Clearly, depending on the sign of the axial force and on the type of doping, the base and the tip will be depleted and enhanced, respectively, or vice versa. In cylindrical nanowires, the base and tip have the same radius, but in tapered nanowires the tip area may be much smaller than the base area, which may translate in higher strains which will tend to result in higher piezopotentials at the tip.

#### 3.3. Characteristic Lengths of the Piezopotentials (Tip, Base, Depletion or Enhancement Piezopotential)

#### 3.4. Depletion-to-Enhancement Piezopotential Ratio (r_{PZ,DEPL-TO-ENHANC})

_{PZ,DEPL-TO-ENHANC}, defined as the ratio between the depletion piezopotential and the enhancement piezopotential. Since both the depletion piezopotential and the enhancement piezopotential are always concord, r

_{PZ,DEPL-TO-ENHANC}is always positive.

#### 3.5. Tip-to-Base Piezopotential Ratio (r_{PZ,TIP-TO-BASE})

_{PZ,TIP-TO-BASE}, defined as the ratio between the tip piezopotential and the base piezopotential. Since the tip piezopotential and the base piezopotential are always concord, r

_{PZ,TIP-TO-BASE}is always positive.

#### 3.6. Piezopotential in Truncated Conical Dielectric Nanowires under Vertical Compression or Traction

#### 3.7. Piezopotentials, Free Carrier Concentrations, Piezopotential Ratios and Characteristic Lengths of the Piezopotentials in Truncated Conical Nanowires with 10^{16} cm^{−3} Doping under Vertical Compression or Traction

^{16}cm

^{−3}doping.

#### 3.8. Piezopotentials, Free Carrier Concentrations, Piezopotential Ratios and Characteristic Lengths of the Piezopotentials in Truncated Conical Nanowires with 10^{17} cm^{−3} Doping under Vertical Compression or Traction

^{17}cm

^{−3}doping.

## 4. Discussion and Conclusions

_{PZ-DEPL}) and the enhancement piezopotential (ΔV

_{PZ-ENHANC}), the tip piezopotential (ΔV

_{PZ-TIP}) and the base piezopotential (ΔV

_{PZ-BASE}). Depending on the type (p or n) of doping and on the sign of the force (traction or compression) [22,34] the tip (base) is enhanced (depleted) or depleted (enhanced) and vice versa. All these piezopotentials can be conventionally taken with their positive terminal closer to the tip (i.e., coincident with the tip if the tip is within the region where the piezopotential is or, otherwise, with the center) and the negative terminal closer to the base (i.e., coincident with the base if the base is within the region where the piezopotential is or, otherwise, with the center). These distinct piezopotentials can only be defined if the nanostructure is sufficiently long so that there is region in the center where the piezopotential is almost perfectly constant (i.e., there is almost zero electric field) and the depletion and enhancement regions are clearly distinguishable at the extremities of the nanowires. If the length of the nanowire is insufficient, the depletion and the enhancement regions will overlap and these piezopotentials may not be defined. Clearly, these definitions only apply to semiconductive nanostructures as in dielectric nanowires there is no depletion or enhancement so that it is not possible to define these piezopotentials.

_{PZ,DEPL-TO-ENHANC}) and the tip-to-base piezopotential ratio (r

_{PZ,TIP-TO-BASE}) which quantitatively express the asymmetrical ability of the nanostructure to develop the piezopotential in different regions. With our definitions of the fundamental piezopotentials, both these ratios are always positive. As to the expected values for these ratios, the piezopotential is more easily created in depleted regions (rather than in enhanced regions, because of the reduced number of free charges) and at the tip (rather than at the base, because of the reduced area and, therefore, of the higher strains). As a result, when the tip is depleted, these ratios will be identical and will tend to be very high (much higher than 1), whereas if the tip is enhanced these ratios will be reciprocal (multiplicative inverse) and may be larger or smaller than 1.

^{16}cm

^{−3}and 10

^{17}cm

^{−3}), base radius equal to 150 nm and radius of the tip ranging from 25 nm to 125 nm. In these simulations, as expected [22,23,34], the piezopotential tend to be higher in the depletion region and at the tip so that when the depletion region is at the tip (i.e., the enhancement region is at the base), as it is the case for n-type ZnO nanowires under compression, the ratios r

_{PZ,DEPL-TO-ENHANC}and r

_{PZ,TIP-TO-BASE}are identical and tend to be very high (up to 170), with the highest values found when the tip radius is minimum (i.e., higher pressure and strains at the tip). By contrast, if the depletion region is at the base and the enhancement region is at the tip, as it is the case for n-type ZnO nanowires under traction, the higher strains at the tip do not necessarily result in significantly higher piezopotentials because of the higher concentrations of free charges at the tip. The characteristic lengths of the piezopotentials are always of the same order of magnitude, though, of course, when the depletion region is at the tip and, therefore, under higher strains, the characteristic length of the tip (or depletion) piezopotential may extend significantly more (e.g., up to about 2 times) more than the characteristic length of the base (or enhancement) piezopotential.

^{16}cm

^{−3}to 10

^{17}cm

^{−3}), for low tip radius the depletion piezopotential is very well preserved, thus resulting in an even increased depletion-to-enhancement piezopotential ratio. This work can great facilitate systematic analyses of axially strained piezosemiconductive nanostructures and can provide insight for the design of better piezoelectric nanotransducers.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic representations of the truncated conical nanowires illustrating (

**left**) the boundary conditions and the coordinate system, with definitions of L

_{NW}(length), R

_{NW}(base radius), R

_{TIP}(tip radius) and (

**right**) the tip piezopotential, the base piezopotential and the central region with an almost zero potential.

**Figure 2.**Piezoelectric potential within conical truncated dielectric nanowires with length 2 µm, base radius 150 nm and compressive force 442 nN. (

**a**–

**c**) Color maps of the piezopotential for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Base piezopotential (

**d**) and tip piezopotential (

**e**) for different values of the tip radius.

**Figure 3.**Piezoelectric potential within conical truncated dielectric nanowires with length 2 µm, base radius 150 nm and traction force 442 nN. (

**a**–

**c**) Color maps of the piezopotential for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Base piezopotential (

**d**) and tip piezopotential (

**e**) for different values of the tip radius.

**Figure 4.**Piezoelectric potential within conical truncated nanowires with length 2 µm, base radius 150 nm, compressive force 442 nN and n-type 10

^{16}cm

^{−3}doping. (

**a**–

**c**) Color maps of the piezopotential for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Base (enhancement) piezopotential (

**d**) and tip (depletion) piezopotential (

**e**) for different values of the tip radius.

**Figure 5.**Free charge concentrations within conical truncated nanowires with length 2 µm, base radius 150 nm, compressive force 442 nN and n-type 10

^{16}cm

^{−3}doping. (

**a**–

**c**) Color maps of the free charge concentrations for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Charge concentration at the base (enhancement) (

**d**) and tip (depletion) (

**e**) for different values of the tip radius.

**Figure 6.**Piezoelectric potential within conical truncated nanowires with length 2 µm, base radius 150 nm, traction force 442 nN and n-type 10

^{16}cm

^{−3}doping. (

**a**–

**c**) Color maps of the piezopotential for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Base (enhancement) piezopotential (

**d**) and tip (depletion) piezopotential (

**e**) for different values of the tip radius.

**Figure 7.**Free charge concentrations within conical truncated nanowires with length 2 µm, base radius 150 nm, traction force 442 nN and n-type 10

^{16}cm

^{−3}doping. (

**a**–

**c**) Color maps of the free charge concentrations for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Charge concentration at the base (enhancement) (

**d**) and tip (depletion) (

**e**) for different values of the tip radius.

**Figure 8.**Piezoelectric potential within conical truncated nanowires with length 2 µm, base radius 150 nm, compressive force 442 nN and n-type 10

^{17}cm

^{−3}doping. (

**a**–

**c**) Color maps of the piezopotential for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Base (enhancement) piezopotential (

**d**) and tip (depletion) piezopotential (

**e**) for different values of the tip radius.

**Figure 9.**Free charge concentrations within conical truncated nanowires with length 2 µm, base radius 150 nm, compressive force 442 nN and n-type 10

^{17}cm

^{−3}doping. (

**a**–

**c**) Color maps of the free charge concentrations for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Charge concentration at the base (enhancement) (

**d**) and tip (depletion) (

**e**) for different values of the tip radius.

**Figure 10.**Piezoelectric potential within conical truncated nanowires with length 2 µm, base radius 150 nm, traction force 442 nN and n-type 10

^{17}cm

^{−3}doping. (

**a**–

**c**) Color maps of the piezopotential for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Base (enhancement) piezopotential (

**d**) and tip (depletion) piezopotential (

**e**) for different values of the tip radius.

**Figure 11.**Free charge concentrations within conical truncated nanowires with length 2 µm, base radius 150 nm, traction force 442 nN and n-type 10

^{17}cm

^{−3}doping. (

**a**–

**c**) Color maps of the free charge concentrations for tip radius equal to 125 nm (

**a**), 75 nm (

**b**) and 25 nm (

**c**). (

**d**,

**e**) Charge concentration at the base (enhancement) (

**d**) and tip (depletion) (

**e**) for different values of the tip radius.

Young’s Modulus | Poisson’s Ratio | Piezoelectric Coefficients (e33) |
---|---|---|

129 GPa | v = 0.349 | 1.22 C m^{−2} |

**Table 2.**Piezopotentials, piezopotentials-ratios and characteristic lengths of the piezopotentials for 2 µm long n-type ZnO truncated conical nanowires with 150 nm base radius and 10

^{16}cm

^{−3}doping under axial compression.

N_{D} = 10^{16} cm^{−3}, Compression (442 nN) | |||
---|---|---|---|

R_{TIP} | 25 nm | 75 nm | 125 nm |

ΔV_{PZ-DEPL} =
ΔV_{PZ-TIP} | −1540 mV | −180 mV | −52 mV |

ΔV_{PZ-ENHANC} = ΔV_{PZ-BASE} | −21 mV | −20 mV | −19 mV |

r_{PZ,DEPL-TO-ENHANC} = r_{PZ,TIP-TO-BASE} | 73 | 9 | 2.7 |

Characteristic length (ΔV_{PZ-DEPL} = ΔV_{PZ-TIP}) | 165 nm | 115 nm | 94 nm |

Characteristic length (ΔV_{PZ-ENHANC} = ΔV_{PZ-BASE}) | 83 nm | 80 nm | 80 nm |

**Table 3.**Piezopotentials, piezopotentials-ratios and characteristic lengths of the piezopotentials for 2 µm long n-type ZnO truncated conical nanowires with 150 nm base radius and 10

^{16}cm

^{−3}doping under axial traction.

N_{D} = 10^{16} cm^{−3}, Traction (442 nN) | |||
---|---|---|---|

R_{TIP} | 25 nm | 75 nm | 125 nm |

ΔV_{PZ-DEPL} = ΔV_{PZ-BASE} | 131 mV | 58 mV | 30 mV |

ΔV_{PZ-ENHANC} = ΔV_{PZ-TIP} | 30 mV | 28 mV | 27 mV |

r_{PZ,DEPL-TO-ENHANC} = 1/r_{PZ,TIP-TO-BASE} | 4.4 | 2.1 | 1.1 |

r_{PZ,TIP-TO-BASE} = 1/r_{PZ,DEPL-TO-ENHANC} | 0.23 | 0.48 | 0.9 |

Characteristic length (ΔV_{PZ-ENHANC} = ΔV_{PZ-TIP}) | 60 nm | 74 nm | 79 nm |

Characteristic length (ΔV_{PZ-DEPL} = ΔV_{PZ-BASE}) | 90 nm | 89 nm | 89 nm |

**Table 4.**Piezopotentials and piezopotentials-ratios for 2 µm long n-type ZnO truncated conical nanowires with 150 nm base radius with 10

^{17}cm

^{−3}doping under axial compression.

N_{D} = 10^{17} cm^{−3}, Compression (442 nN) | |||
---|---|---|---|

R_{TIP} | 25 nm | 75 nm | 125 nm |

ΔV_{PZ-DEPL} = ΔV_{PZ-TIP} | −1028 mV | −38 mV | −12 mV |

ΔV_{PZ-ENHANC} = ΔV_{PZ-BASE} | −6 mV | −6 mV | −6 mV |

r_{PZ,DEPL-TO-ENHANC} = r_{PZ,TIP-TO-BASE} | 170 | 6.3 | 2 |

Characteristic length (ΔV_{PZ-DEPL} = ΔV_{PZ-TIP}) | 66 nm | 31 nm | 30 nm |

Characteristic length (ΔV_{PZ-ENHANC} = ΔV_{PZ-BASE}) | 29 nm | 29 nm | 29 nm |

**Table 5.**Piezopotentials, piezopotentials-ratios and characteristic lengths of the piezopotentials for 2 µm long n-type ZnO truncated conical nanowires with 150 nm base radius and 10

^{17}cm

^{−3}doping under axial traction.

N_{D} = 10^{17} cm^{−3}, Traction (442 nN) | |||
---|---|---|---|

R_{TIP} | 25 nm | 75 nm | 125 nm |

ΔV_{PZ-DEPL} = ΔV_{PZ-BASE} | 75 mV | 22 mV | 10 mV |

ΔV_{PZ-ENHANC} = ΔV_{PZ-TIP} | 7 mV | 7 mV | 7 mV |

r_{PZ,DEPL-TO-ENHANC} = 1/r_{PZ,TIP-TO-BASE} | 10.7 | 3.2 | 1.4 |

r_{PZ,TIP-TO-BASE} = 1/r_{PZ,DEPL-TO-ENHANC} | 0.093 | 0.32 | 0.7 |

Characteristic length (ΔV_{PZ-ENHANC} = ΔV_{PZ-TIP}) | 25 nm | 26 nm | 29 nm |

Characteristic length (ΔV_{PZ-DEPL} = ΔV_{PZ-BASE}) | 30 nm | 30 nm | 30 nm |

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Amiri, P.; Falconi, C.
Fundamental Definitions for Axially-Strained Piezo-Semiconductive Nanostructures. *Micromachines* **2021**, *12*, 20.
https://doi.org/10.3390/mi12010020

**AMA Style**

Amiri P, Falconi C.
Fundamental Definitions for Axially-Strained Piezo-Semiconductive Nanostructures. *Micromachines*. 2021; 12(1):20.
https://doi.org/10.3390/mi12010020

**Chicago/Turabian Style**

Amiri, Peyman, and Christian Falconi.
2021. "Fundamental Definitions for Axially-Strained Piezo-Semiconductive Nanostructures" *Micromachines* 12, no. 1: 20.
https://doi.org/10.3390/mi12010020