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Thermal-electric model for piezoelectric ZnO nanowires

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Nanotechnology Nanotechnology 26 (2015) 265402 (11pp)

doi:10.1088/0957-4484/26/26/265402

Thermal-electric model for piezoelectric ZnO nanowires Rodolfo Araneo1, Fabiano Bini2, Antonio Rinaldi4, Andrea Notargiacomo3, Marialilia Pea3 and Salvatore Celozzi1 Electrical Engineering Division of DIAEE, University of Rome ‘Sapienza’, Rome, via Eudossiana, 18–00184, Italy 2 Mechanical and Aerospace Engineering Department—DIMA, University of Rome ‘Sapienza’, Rome, via Eudossiana, 18–00184, Italy 3 Institute for Photonics and Nanotechnology—CNR, Via Cineto Romano 42, 00156, Rome, Italy 4 International Research Center for Mathematics & Mechanics of Complex System, University of L’Aquila, 04012 Cisterna di Latina (LT), Italy, and with ENEA, Research Center Casaccia, 00123 Rome, Italy 1

Received 20 March 2015, revised 30 April 2015 Accepted for publication 8 May 2015 Published 10 June 2015 Abstract

The behavior of ZnO nanowires under uniaxial loading is characterized by means of a numerical model that accounts for all coupled mechanical, electrical, and thermal effects. The paper shows that thermal effects in the nanowires may greatly impact the predicted performance of piezoelectric and piezotronic nanodevices. The pyroelectric effect introduces new equivalent volumic charge in the body of the nanowire and surface charges at the boundaries, where Kapitza resistances are located, that act together with the piezoelectric charges to improve the predicted performance. It is shown that the proposed model is able to reproduce several effects experimentally observed by other research groups, and is a promising tool for the design of ultrahigh efficient nanodevices. Keywords: ZnO nanowires , Schottky contacts, Kapitza resistance, pyroelectric effect, piezotronic effect (Some figures may appear in colour only in the online journal) 1. Introduction

(e.g. wires [7], tubes [8], belts [9], rods [10], rings, pagodas, etc.) by allowing the exploitation of diverse functional properties of ZnO for nanotechnologies. When a stress is imposed on the ZnO wurtzite crystal, the polarization of inner ions generates an electric charge (the piezoelectric effect) and thus produces a piezoelectric potential in the material (piezopotential hereafter) that can ultimately be used to control an external current that can flow through the ZnO material (the piezotronic effect). On one hand, the field of piezoelectrics [11–17] seeks to use this behavior for direct power generation, for example by harvesting waste mechanical energy from the environment to self-power electrical devices [18–20]. On the other hand, the field of piezotronics deals with applications for the production of control signals from mechanical stimuli [21, 22]. Piezotronics has indeed been broadly identified as a new electronics which uses the piezopotential as a gate voltage to modulate through external strain the charge transport across a metal/ semiconductor Schottky interface or a p–n junction [23–27].

Over the past decade, ZnO has enjoyed substantial interest within the wurtzite semiconductive materials, for example as compared to GaN, InN, and CdS. The underlying reason is its attractive low cost coupled with many outstanding functional properties; primarily semiconductivity, piezoelectricity, pyroelectricity, direct bandgap, and biocompatibility [1–3]. Its applications in electronics and photonics were initially limited by its extreme brittleness; ZnO in bulk form exhibits a brittle behavior with maximum deformations well below 1%, thus hindering any highly relevant application. Eventually, giant size effects were discovered in ZnO nanostructures, which can differ dramatically from conventional bulk samples in many ways; in particular, ZnO nanowires (NWs) can withstand large elastic deformations up to 15% without breaking and acquire a ductile behavior at failure. These superior mechanical properties have enabled breakthroughs in the development of novel one-dimensional nanostructures [4–6] 0957-4484/15/265402+11$33.00

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© 2015 IOP Publishing Ltd Printed in the UK

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Figure 1. Possible architectures of synced NWs in actual piezoelectric/piezotronic devices to harvest electrical power or control an electrical response from tensile/compressive loads based on ultra strength semiconductor ZnO nanowires: (a) device under no strain; (b) device under tensile strain (colored NWs). (c) Triangle generally used to describe the pathways among mechanical, electrical, and thermal energies in a class of noncentrosymmetric materials exhibiting direct and converse effects.

properties of ZnO [45–47] (Young modulus and failure strength) at the nanoscale relative to the bulk is fundamentally associated to surface effects in samples with high surface-tovolume ratios, which also cause piezoelectric coefficients [48, 49] and thermal properties [50, 51] to be sample-size dependent; the latter issues have been much less researched and acknowledged so far. In this work, we report on the fully coupled thermomechanical-electric problem, with the goal being to estimate the current-voltage (I-V) characteristic [26, 31, 42, 43, 52–54] of piezoelectric ZnO NWs under a purely uniaxial compressive or tensile strain (figures 1(a), (b)). Although some devices rely on different types of mechanical loadings (e.g. lateral deflection), the scope is here limited to NWs under purely uniaxial compression/traction, which continues to be one of the most relevant reference arrangements for piezoelectric and piezotronic nanodevices [30, 52]. Tensile/compressive uniaxial strain can be achieved in a vertically loaded freestanding NW [24, 43, 45, 55, 56], and also in a NW lying flat on a flexible, much thicker substrate and fixed to it at the two ends, such that uniaxial stress arises when the substrate is bent (assuming a radius of curvature much larger than the NW length, as is usually the case). ZnO NWs seem ideal for such devices as they can be synthesized by relatively easy, lowtemperature, cheap, CMOS/MEMS-compatible hydrothermal growth methods. Regardless, other wurtzite piezoelectric materials could also be used and can benefit from the methodology and results discussed here. Our investigation demonstrates that thermal effects in the NW may greatly impact the predicted performance of piezoelectric and piezotronic nanodevices. The proposed

Although an actual device includes a large number of electrically synced nanostructures [28, 29] to generate a significant and reliable effect, the single nanostructure can be conceived as the building block for novel advanced devices. Various novel devices based on ZnO NWs have been fabricated so far by utilizing the piezotronic properties, such as piezoelectric field effect transistors, piezoelectric diodes [30], triggers, actuators, and flexible nanosensors [31] including strain sensors [32, 33], photodetectors [34, 35], biosensors [36], and gas sensors [37]. Notably, in the biomedical field arrays of ZnO NWs packaged on a flexible thin polymer could also be used to harvest energy [38], as well as to stimulate cellular response [39, 40]. Despite the impressive research and technological trends from the past few years, right after the early seminal work by Wang and co-workers [13, 41], the existing piezoelectric and piezotronic nanodevices still present major aspects [17, 25, 26, 30, 31, 42] that deserve better theoretical insights and explanations. The electro-mechanical characterization of piezoelectric structures at nanoscale, for instance, remains a challenging and involved task: the physical framework is inherently complex and multidisciplinary, calling for a multiphysics approach [43] to account for the mutual coupling among mechanical, electrical, and thermal effects. In addition, the numerical resolution schemes are complicated both by the presence of nonlinearities in fundamental equations and boundary conditions (e.g. Schottky contacts, thermal boundary resistances), and by the need to directly account for the size effects that have been experimentally observed [8] or theoretically predicted [44] by first-principles calculations in ZnO. The aforementioned enhancement in mechanical 2

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physically based numerical approach leads to unique insights about the behavior of such devices by providing a means to address aspects that are crucial but hardly measurable in real devices, thus yielding major advantages for design.

of the carrier concentrations through the diffusion coefficient Dn, p . The constitutive relations for a linear piezoelectric material are [57–60]

2. Numerical model and methods

(1a)

i Di = q ( ND+ + p − NA+ − n)

(1b)

i Jn, i = −qUn

(1c)

i Jp, i = +qUn

(1d )

̇ i + k ij (T ) j = 0, QS,

(1e)

(2a)

Di = eijk ε jk + κ 0 κij E j + γi θ,

(2b)

where Cijkl is (a component of) the elastic stiffness tensor, eijk is the piezoelectric strain tensor, βij is the tensor of stresstemperature moduli, γi is the (primary) pyroelectric coefficient vector, and κij is the relative dielectric tensor (κ0 is the vacuum permittivity). Furthermore, εkl is (a component of) the strain tensor, Ek is the electric field vector, and θ = T − T0 is the temperature change with respect to the initial temperature, T0 . Remarkably, while derived in the framework of a classical continuum matter theory, the set of fundamental equations (1) and the set of constitutive relations (2) are sufficient to account for all the relevant physical phenomena that concern the NW: the elastic behavior, the dielectric properties, the thermal conduction, and—more importantly— the piezoelectric and the pyroelectric effects, as shown in figure 1(c). In the hexagonal ZnO wurtzite crystal, electric dipoles are mainly generated along the polar c-axis in response to mechanical strain components that are parallel to the polar axis, through the e33 coefficient (Voigt notation is here used for conciseness) and placed on the basal plane, through the e31 = e32 coefficient. A polarization could also be induced on the basal plane by a shear strain through the e15 = e24 coefficient but it does not play any role in the present case, due to the application of a uniform loading and to the symmetry of the problem. For simplicity, the generation of the piezoelectric field along the polar axis is here treated as a linear effect of the sole strain, even if the influence of non-linear components can be seamlessly accounted for, as highlighted for several semiconductors [61–63]. A signature aspect of our model is the excess generation of bond charges in response to a temperature variation along the c-axis, the so-called pyroelectric charges, which are here modeled through the primary and secondary pyroelectric effects [64, 65]. The sum of the latter two effects is the experimental quantity usually measured in the laboratory under constant stress conditions. The primary pyroelectric effect is related to spontaneous polarization charges and is modeled through the pyroelectric coefficient vector γi (the primary pyroelectric polarization vector can be readily expressed as Pipyro,I = γi θ ). The secondary pyroelectric effect, which is the result of the piezoelectric contribution originated by the elastic strains (crystal deformation) caused by restrained thermal expansion, is accounted for through the thermal stress tensor βij (computed from the thermal expansion tensor αlm as βij = Clmij αlm ) and the direct piezoelectric tensor eijk (the secondary pyroelectric polarization vector can be readily expressed as Pipyro,II = eijk α jk θ ). As a word of caution, the ternary pyroelectric effect that originates from an inhomogeneous temperature distribution that leads to

Modeling the thermo-electro-mechanical behavior of the piezoelectric semiconductor NW entails the description of the transport of charge carriers under the simultaneous influence of a time-independent potential distribution, mechanical (elastic) strain, and the direct/inverse thermal effects in the NW. Although a detailed theoretical investigation of the matter at nanoscale is usually best pursued by means of semiclassical approaches based on the numerical solutions of the Boltzmann transport equation (BTE) or—even better—by means of atomic-level/molecular-dynamics simulations, which are capable of accounting for subtleties and complexities in volume-confined samples, here we propose a classical engineering approach based on continuum modeling that has proved satisfactory in modeling many nanodevices [4, 43]. This approach leads to a system of fully coupled nonlinear partial differential equations that can be conveniently crafted to account for the specific transport properties and constitutive relations of nanomaterials, which are modeled as effective continuum media. Given this general framework, for a piezoelectric NW the elastic properties, the electric voltage, the charge current transport, and the thermal behavior are governed by the following system of transport equations, comprising Newton’s law, Poisson’s equation, the current continuity equations, and the Fourier equation,

i σij = 0

σij = Cijkl εkl − e kji E k − βij θ

where σij is the stress tensor, Di is the electric displacement vector, ‘ND+ − NA+’ denotes the net ionized impurity concentration (consisting of ionized positive donors ND+ and negative acceptors NA+ ), n and p are the free electron and hole concentrations in the conduction and valence band, Jn, i and Jp, i are the electron and hole current densities, Un = Gn − Rn and Up = Gp − R p are the net differences between the generation (G) and the recombination (R) rates for the charge ̇ i is the source volumetric carriers, T is the temperature, QS, heat flux, and k ij is the thermal conductivity tensor (q is the positive electric charge). The two current densities Jn, i and Jp, i can be expressed as a sum of two terms, i.e. Jn, i = q ⎡⎣ nμn Ei + Dn (n )i ⎤⎦ and Jp, i = q ⎡⎣ pμp Ei − Dp (p)i ⎤⎦ under the so-called drift diffusion approximation, where the first term is the drift term that is proportional to the driving electric field through the electron motilities μn, p and the second term is the diffusion term that depends on the gradient 3

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polarization variation through the flexoelectric effect tensor fijkl is discarded here and is not present in the constitutive equations (2). The main reason for this simplification is that the ternary effect is usually negligibly small in pyroelectrics and ferroelectrics in comparison with the primary and secondary effects, especially when the temperature variations ∂T /∂xi are small. In support of this assumption, our results will indeed show that the temperature gradients inside ZnO NW are small. Since the hexagonal crystal of ZnO wurtzite can be assumed to have a transverse isotropic material symmetry from a mechanical standpoint [66], the elastic behavior is isotropic on the basal plane and the elastic stiffness tensor Cijkl is expressed according to the Voigt notation as a 6 × 6 matrix where only five independent elastic constants are present, i.e. C11 = C22, C12, C13 = C23, C33, and C44 = C55, being C66 = ( C11 − C12 ) /2. The dielectric relative permittivity tensor is considered uniaxial with only two independent dielectric constants, i.e. κ11 = κ22 and κ33, because refractive index analyses have shown only two different refractive indices for polarization parallel (ordinary) and perpendicular (extraordinary) to the caxis [67]. Finally, the thermal properties of the NW were simulated using the thermal conductivity tensor k ij and the thermal stress tensor βij . Only a few papers [51, 68, 69] are available in literature about the thermal properties of individual NWs, which is a field that is largely unexplored nowadays due to both theoretical and measurement challenges. On one hand, the understanding of the transport dynamics at the nanoscale needs to address fundamental conceptual issues, such as for example the appropriate definition of temperature and thermodynamic equilibrium in a nanostructure [68] Consequently, there is an incredible number of computational approaches to heat-transfer problems that span the range from classical approaches based on the numerical solutions of Fourier’s law (here adopted) to simulations at the atomic level based on the BTE. On the other hand, direct thermal conductance measurements are made difficult [50] by the lack of robust tools to eliminate or account for extrinsic effects, such as that of the unknown thermal contact resistance between sample and substrate, or the possible presence of a surface coating on the NW that introduces relevant surface interactions with the NW core. Hence, the thermal expansion tensor is here assumed to be uniaxial (as the relative dielectric tensor) and completely described by two coefficients, one along the c-axis and the other on the basal plane. More importantly, the thermal conductivity is taken as a scalar quantity, and for the determination of its correct value we referred to available studies conducted directly on ZnO NWs, since the measured thermal conductivity was observed to be reduced by more than one order of magnitude compared to that of bulk ZnO [50]. This thermal size effect of ZnO NWs is due to the strong suppression effect that is exerted by the enhanced phononboundary scattering on phonon transport in quasi onedimensional nanostructures [70].

For the successful application of our numerical model, we point out the importance of choosing proper boundary conditions across the junction interfaces between the ZnO and the metal contacts, at the top and bottom of the NW. Under the thermionic emission theory of the current conduction mechanism, in accordance to Wang’s results [42], two Schottky barriers are assumed at the metal–semiconductor junctions, whose heights are modulated by the polarization (piezoelectric and pyroelectric) charges that appear at the end surfaces of the NW, as discussed later (figure 2). For the thermal problem, we assume the presence of two Kapitza thermal contact resistances [71] across the interfaces. Several significant studies have been conducted on the temperature distribution in wires carrying electrical current [72] by assuming that the temperature profiles and its first derivative are continuous across the wire–contact junction. Nevertheless, this common assumption would result in a marked departure from reality, as previously shown for nanotubes [73] where Kapitza thermal resistances at the ends were found to heavily influence the heat transport mechanisms of the whole nanotube. In that case, both numerical simulations and experimental investigations highlighted a sharp drop in the average kinetic energy at the boundaries that is consistent with the concept of a thermal contact resistance. In fact, the Kapitza resistance assumes the normal component of the heat flux to be continuous across the interfaces, while the temperature undergoes a discontinuity that is proportional to the normal component of the heat flux through an interface ̇ n = ζ ( T − Text ), where ζ is the inverse of parameter, i.e. QS, the interface resistance. Finally, all the fundamental differential equations (1) are strictly coupled through the constitutive relations (2), which are strongly nonlinear according to the quasi-Fermi formulation [43]. We recall that in the quasi-Fermi approach the independent variables are the electric potential Ψ , the displacement vector ui , the temperature T , and the two Fermi potentials Φn and Φp, which allow expression of the electron and hole concentrations through the Fermi–Dirac statistics or the Maxwell–Boltzmann statistics when the semiconductor is not degenerate. The resulting differential equations (1) have been solved through a Finite Element Method (FEM) with appropriate iterative nonlinear solvers [43]. The relevant parameters used in the simulations are reported in table 1. In addition, in our calculations the effective electron and hole masses [74] are respectively m e = 0.28m 0 and m h = 0.68m 0, m 0 being the free electron mass; the levels of the donorsacceptor concentrations [67] are ΔℰD = ΔℰA = 35 meV, and the bandgap is always assumed to be larger than the possible band shift with the gap energy equal to [6] ℰg = 3.4 eV.

3. Discussion of the results 3.1. Physical mechanisms

Figure 2(a) shows the model of the NW under consideration, which was assumed to be negatively doped with a uniform donor concentration ND and interconnected by two gold metal 4

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Figure 2. Schematic pictures and diagrams showing the piezotronic effect inside a NW subject to tensile/compressive strains: (a) sketch of the NW under an applied voltage Va and force F with two Schottky contacts at both the ends (color qualitatively represents the electric potential inside the NW); (b) band diagram inside the NW under no strain; (c) band diagram inside the NW under an applied positive voltage Va; (d) band diagram inside the NW under an applied positive voltage Va and a compressive negative strain. Computed profiles of the conduction band edge Ec (z ) inside of the NW under zero strain (e), negative strain (f) and positive strain (g). The numerical results confirm the theoretical model which predict that the non-polar piezotronic effect is able to modulate the band structure inside the NW and consequently the current flow through it. (h) Difference ΔΦ between the two Schottky barrier heights ΦS and Φ D plotted with respect to the applied voltage under different strains. (i) I–V characteristics of the NW under different tensile/compressive strains.

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Table 1. Material properties of ZnO (Voigt notation is used).

Tensor components

11 = 22

12

13 = 23

33

44 = 55

Elastic moduli Cij [GPa] Relative electric permittivity κij Thermal expansion tensor αij [10−6 K−1] Pyroelectric tensor γij [10−6 C m−2 K)]

209.7 7.77 4.75

121.1

105.1

210.9 8.91 2.9 −12

42.47

Thermal conductivity tensor k ij [W/(m K)] Tensor components Piezoelectric constants eij [C/m2]

18 31 = 32 −0.51

contacts at both ends. When the structure is under thermal equilibrium ( Va = 0), two Schottky barriers are formed at the contacts with the same barrier height (SBH) Φ B, numerically equal to the difference between the working function of the metal Φ M and the affinity χ of the ZnO, i.e. Φ B = Φ M − χ (ref. figure 2(b)). Near the gold contacts, free electron charges migrate from the semiconductor into the metal, leaving behind two depletion regions in the NW. Consequently, the conduction band Ec ‘bends’ away from the Fermi level of the semiconductor E fs (taken as reference) and two voltage drops, ΦS and Φ D, equal to the built-in diffusion potential Φ bi (i.e. ΦS = Φ D = Φ bi ), arise across the two space-charge regions. Such a structure can also be regarded as two back-to-back Schottky diodes, as represented in figure 2(a). When a positive external voltage Va is applied (without applying an external force F) with the polarity shown in figure 2(a), electrons move from the left contact (the source) to the right contact (the drain) and a positive current flows through the NW along the positive c-axis, producing two voltage drops, VS and VD, across the source and drain contacts that are respectively reverse (VS is negative) and forward biased (VD is positive). The negative voltage drop VS at the source makes carriers move from the semiconductor to the metal, thus widening the depletion region at the metal contact. Conversely, the positive voltage drop VD at the drain makes carriers flow from the metal to the semiconductor, thus narrowing the depletion region. Thus, the Fermi levels in the metal contacts, which were initially equal to those in the semiconductor, move readily from the Fermi level in the ZnO: the Fermi level E fm,S in the source contact increases by − qVS (figure 2(c)) and the Fermi level E fm,D in the drain contact decreases by qVD, giving rise to a difference that depends on the externally applied voltage as E fm,S − E fm,D = q ( VD − VS) = qVa. We stress that in this theoretical discussion we discard the finite dimensions of the NW (interaction between the two interfaces) and other secondary effects in this phenomenon, such as the Schottky barrier lowering. The flow of electrons alters the equilibrium-band diagram inside the NW by changing the curvature of the bands (figure 2(c)) and modifying the potential drops ΦS and Φ D, such that the potential drop is ΦS = Φ bi − VS at the source and Φ D = Φ bi + VD at the drain. According to the thermionic emission-diffusion theory, the current transport mechanism is dominated by the reverse-biased source Schottky barrier via

18 33 1.22

15 = 24 −0.45

its potential drop ΦS, which drives the current IDS as ⎡ ⎛ ΦB ⎞ ⎤ IDS = ⎢ SA R T 2 exp ⎜ − ⎟⎥ ⎝ Vth ⎠ ⎦ ⎣ ⎛ ⎞ 2qND 1 q × exp ⎜⎜ ΦS − Vth ) ⎟⎟ , ( κs ⎝ Vth 4πκ s ⎠

where AR is the Richardson constant, S is the surface of the contact, Vth is the thermal voltage, and κ s is the dielectric constant of the semiconductor. To gain physical insights about the piezoelectric and pyroelectric behavior of the NW, let us first consider only the piezoelectric effect and neglect thermal effects (i.e. the temperature in the NW is assumed fixed at room-temperature T0 ). The application of a compressive force F = − F u z (figure 2(a)) produces a uniform strain inside the NW that causes a negative piezoelectric surface charge to appear at the source contact σpiezo = Ppiezo ⋅ u n ≃ − e33 ε33 and an opposite surface charge at the drain. The deformability of the metal contacts is negligible for a long NW with high aspect ratio L /R. The volumetric charge inside the NW can also be discarded, being ρpiezo = −  ⋅ P = 0 in the absence of a strain gradient. The piezoelectric charges are distributed inside the NW within a depth δp, and produce two piezopotentials at the junctions (negative at the source and positive at the drain) that are partially or fully shielded in the body of the NW by free carriers, depending on the doping concentration and on the dimensions of the NW. Consequently, the SBH at the source increases ( Φs′ = Φ B + ΔΦS) and the SBH at the drain decreases ( Φ D′ = Φ B − ΔΦ D ), as shown in figure 2(d), causing the space-charge region to widen at the source and narrow at the drain. Since the reverse-biased source junction predominates, as previously explained, and its barrier increases under a compressive negative strain, a decrease is observed in the current IDS through the NW. Under a positive tensile strain, a reverse situation occurs, such that the piezoelectric charge at the source is positive, the piezopotential is positive, the barrier height decreases, and the current IDS increases.

(

)

3.2. Model results without the pyroelectric effect

Figure 2(e), f, and g present the levels of the conduction band in the NW obtained through our numerical scheme without 6

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considering the pyroelectric effect, respectively under zero applied strains (figure 2(e)), negative strain (figure 2(f)), and positive strain (figure 2(g)), for an external voltage Va varying from 0 to 10 V (geometrical parameters of NW: L nw = 4 μm, Rnw = 150 nm ). Throughout, the model consistently captures the physical phenomenology. In addition, all the secondary effects overlooked in the above explanation are also accounted for, i.e. the finite size of the NW, the effect of the contacts assumed to be non-deformable, the Schottky barrier leveling, and the screening effects of the free carriers, all of which reduce the actual potential difference ΔΦSD = ΦS − Φ D. For instance, under zero applied strain, figure 2(e) renders that ΔΦSD is not equal to but lower than the external voltage Va. On this aspect, the plot in figure 2(h) highlights the dependence of ΔΦSD on the voltage Va for different values of the strain, which expectedly plays a modulating effect on the potential difference ΔΦSD (the so-called piezotronic effect) and on the difference of the barrier heights at the contacts; the thermal influence appears to be negligible for low Va. Finally, the I-V characteristics of the NW in figure 2(i) indicate that the effect of a positive (tensile) strain can be also obtained by reversing the applied voltage (clearly, the direction of the current IDS is also reversed). Under a positive applied voltage, a negative (compressive) strain reduces the current flow, while a positive (tensile) strain increases it. The reverse occurs under a negative voltage. It should be noted that the I-V characteristics exhibit a characteristic saturation behavior that was previously identified and discussed [43].

σpyro = Ppyro ⋅ u n ≃ γ33 θ appears at the source (noting that the pyroelectric coefficient γ33 is negative) and a positive surface charge appears at the drain. It is significant of the pyroelectric effect that these charges are of the same sign as the piezoelectric charges under negative compressive strain; thus, based on the foregoing, their net effect is the increase of the SBH at the source and the reduction of the current amount at a prescribed applied positive voltage Va. It should also be appreciated that this effect is not polar in our model, since the NW always increases its temperature with respect to T0 under an electrical current. When looking in greater detail at the parabolic-shaped temperature profile in figures 3(c)–(e) for different applied voltages and strains, the associated volumetric pyroelectric ∂ charge ρpyro = −  ⋅ P pyro = ∂z γ33 θ inside the NW has, on the first order, a linear behavior that is positive in the first half of the NW length and negative in the second half (figure 3(a)). Such a charge distribution resembles the linearly graded junction that, under the depletion approximation, leads to a band structure that varies as the third power of the position z, pushing the conduction band edge Ec near the Fermi level on the positively charged half and far from the Fermi level in the second half [75]. Although the depletion approximation may appear as a radical approximation here, it could readily be demonstrated that the solution of the Poisson equation, accounting for the free electron charges with a firstorder approximation, leads to the same conclusions [75]. Hence, the volumetric charge should lead to a small asymmetry on the conduction band in the semiconductor body, as shown in figure 3(b), which is favorable for improving the current rate. Nevertheless, looking at the conduction band edge shown in figure 3(f) at zero strain under different positive voltages Va, these conclusions may not be self-evident because they are mainly concealed by the fact that the increase of temperature in the NW, with a parabolic shape, leads to a bending of the conduction band edge with a similar shape (see for comparison the case without thermal model in figure 2(e)). This effect is very complex because all the parameters in the model are dependent on the temperature, but it can be explained as follows. At medium temperature (T ≃ 300 K), the system is in the so-called extrinsic range [75], where the electron concentration is approximately equal to that of the donors ( n ≃ ND ). Therefore, since n (z ) is exponentially related to the energy difference between the conduction band edge Ec and the Fermi level E f through the density of states NC in the conduction band (which is a 3 function of temperature as T 2 ), the difference between the conduction band and the reference Fermi level fixed to zero is Ec − E f = k B T ln ( NC /n ) ≅ k B T ln ( NC /ND ). We neglect here the narrowing of the bandgap since the temperature difference distribution θ (z ) in the NW is, in general, quite small. We can conclude as a first approximation that an increase in temperature moves the conduction band edge far from the Fermi level. It is interesting to observe that, consistent with the described effects, the temperature profiles in figures 3(c)–(e) change from the compressive to the tensile strain state. In fact,

(

3.3. Adding pyroelectric effect

The inclusion of the pyroelectric effect and thermal properties makes the multi-physics nature of our NW model substantially more complex, because (i) the pyroelectric polarization vector produces new contributions to the equivalent surface and volumetric charges in the NW, and (ii) all the quantities of the model (e.g. free charges densities, intrinsic densities, ionized dopants, motilities, diffusivities, etc) need to be regarded as temperature dependent. Nevertheless, the device behavior rendered by such an enhanced model can be grasped from some simple observations. The inclusion of thermal behavior affects the piezoelectric response of NWs mainly in two fundamental ways. First, while setting the room temperature, a temperature gradient arises inside the NW and that increase makes the equivalent surface and volumetric charges increase, together with a corresponding temperature drop across the Kapitza resistances at the metal contacts. Second, changing the room temperature affects all the physical properties of the NW in a number of ways. As shown in figure 3(a), the current flow IDS under the electric field E in the NW produces a volumetric heat source that leads to a parabolic-like temperature distribution over the length of the NW (there are no temperature gradients in radial direction due to the high aspect ratio of the NW). The Kapitza resistances at the two metal junctions are responsible for a lumped temperature drop across the junctions, with the metal thermostatted at room temperature T0. As a result, in figure 3(b) a negative surface pyroelectric charge 7

)

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Figure 3. Schematic pictures and diagrams showing the pyroelectric effect inside a NW under the of the thermal model. (a) Temperature

distribution inside the NW, with two temperature drops at the contacts due to the Kapitza resistances, which produce equivalent surface pyroelectric charges at the contacts and volumic pyroelectric charge inside the NW body. (b) Band diagrams inside the NW deformed by the pyroelectric charges. Computed temperature profiles inside the NW under negative strain (c), no strain (d), and positive strain (e). The strain modules the current flow inside the NW through the piezotronic effect and, consequently, the power dissipated by Joule effect, changing the temperature profiles inside the NW. (f) Computed band diagram of the NW under different applied voltages Va and no strain. (g) I–V characteristics of the NW under different tensile/compressive strains computed with and without thermal model. (h) Difference between the current that flows through the NW under strain and the current that flows when the strain is zero, with and without thermal model.

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Figure 4. Computed temperature dependence behavior of I–V characteristics the NWs. The transport characteristics are shown at 77 K (a), 200 K (b) and 300 K (c). The temperature also affects the SBH at the source contact (d) and drain contact (e). The temperature dependence of the difference between the SBH under tensile +1% strain and compressive −1% strain (i.e. Δ = SBH(+ 1%) − SBH(− 1%)) is shown in (f).

the strain modulates the current flowing in the NW by the piezotronic effect and, consequently, the electric power density is actually converted to thermal power by the Joule effect. In turn, this changes the temperature distribution in conjunction with the applied voltage and, quite remarkably, causes the temperature drops at the two interfaces where two Kapitza resistances are located. In addition, figure 3(g) shows the I–V characteristics of the NW under different strains and confirms that the introduction of the thermal behavior of the NW reduces the current that flows through the NW at a given voltage Va. Yet, figure 3(h) suggests that the net piezotronic effect due to strain, i.e. the difference between the I–V characteristic under a prescribed strain and the I–V characteristic with no applied strain (taken as reference), is quite insensitive to the pyroelectric contribution. Finally, we can use the model to investigate the impact of the room temperature on the piezotronic effect, which has been shown experimentally to have a significant temperature dependence [52]. Figures 4(a)–(c) show the I–V characteristics computed with our model at room temperatures

respectively equal to 77 K and 200 K. Two main observations are possible: as the temperature decreases, the characteristics under different strains are more and more distinct, revealing an enhanced role of the piezotronic effect at lower temperatures. Furthermore, as the temperature decreases, the saturation-knee shifts upwards to higher voltages and become sharper, while the characteristics present a larger linear-like region. These results can be explained by bearing in mind the fundamental ways (all accounted for in our model) through which the temperature affects the NW system. As the temperature reduces, we move from the extrinsic region to the ionization region [75], where the free carrier density drastically decreases, since there is only enough latent energy in the material to push a few dopant carriers into the conduction band, and because the bandgap expectedly increases as predicted by the Varshni equation [75]. Moreover, the temperature affects in complex ways the carrier current density predicted by the drift-diffusion equations via the temperature dependence of the carrier concentrations, mobilities, and diffusion coefficients. Decreasing the temperature decreases 9

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References

the diffusion component of the total current, since in the ionization region the reduction in the carrier density prevails over the greater charge mobility due to reduced lattice vibrations. The freezing-out effect on the free carriers, which are reduced both in number and thermal energy, causes the Schottky barriers to thicken, such that fewer carriers are able to migrate through the contacts [52]. Accordingly, we observe in figures 4(a)–(c) that the current flow decreases as the temperature decreases. In addition, when a NW is elastically strained, the screening effect by the free charges on the straininduced piezoelectric charges at the contacts is also reduced, since the population of free electrons that can migrate through the NW is reduced at lower temperatures. Hence, without screening by free charges of opposite sign, the piezoelectric charges can play a more pronounced role on the higher Schottky barriers, thus affecting the current flow in a more evident way, as shown by our results. This finding also accounts for the greater linearity of the I–V characteristics at reduced temperatures, since at lower temperatures the NW depletes its population of free charges and resembles the linear behavior of a dielectric material, in which the screening effect by free carriers is completely absent [11]. The SBH at the two contacts also depends on temperature. Figures 4(d) and (e) show the estimated change in the SHB at the source and drain contacts, respectively, confirming that a temperature decrease leads to an increase in the barrier heights. Clearly the strain affects the two barrier heights in the opposite way, since piezoelectric charges of opposite sign are produced at the two contacts. In addition, figure 4(f) shows the temperature dependence of the difference Δ between the SHB(+ 1%) under a tensile strain of +1% and the SHB(− 1%) under a compressive strain of −1%, which renders a meaningful evaluation of the piezotronic effect [52]. The results, besides reproducing the experimentally observed physical behaviors [52], clearly show how the piezotronic effect would decrease for increased temperature. In conclusion, the above results confirm that the model proposed in this paper delivers a friendly theoretical tool to approach the subtleties of mechanical, electrical, and thermal effects in ZnO NWs and to evaluate their impact on reliability and electrical functions. While the calculations were implemented on a specific set of experimental data, the conclusions agree with experimental trends from piezoelectric force microscopy (PFM) tests on individual ZnO NWs across the size range of interest and bear general significance for closely related wurtzite semiconductor materials of interest, such as GaN. As such, this modeling strategy represents one step further towards a comprehensive multi-physics design framework for micro- and nanodevices based on ZnO and wurtzite materials.

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Acknowledgments Funding for this research is provided through the Italian Ministry of Research and Education, grant ‘FIRB RBFR10VB42’. 10

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