Because electrical mobility is much larger than hole mobility, we ignore the contribution of holes to the photocurrent [4]. According to Ohm’s Law, we can write the photocurrent of a ZnO NW as expressions (1):

$$Delta I = frac{US}{L}Delta sigma ,Delta sigma = Delta nmu e$$


where (U) is the voltage, (L) is the length of an NW, (S) is the cross section of an NW, (e) is the electron charge, (mu) is electrical mobility, and (Delta n) denotes the difference of electron density between dark and illumination conditions. Ignoring the effect of OAC, (Delta n) is (n_{{{text{pe}}}}) which is the concentration of photogenerated electrons. Assuming reasonably that electrical mobility is constant in the NW length range we study, the change in the photocurrent of the NW can be reflected reasonably by that of (Delta n) as its length varies. At the same time, the Schottky barrier height (SBH) of the NW is inversely dependent on its length, which can be observed clearly in their experiment. The SBH is closely related to the electrical transport of the NW. Generally speaking, the SBH can be modified by three factors: (1) the difference in work function between the semiconductor and contact metal, which is influenced by the level of doping concentration in the semiconductor when keeping the contact metal constant, (2) the surface state of the semiconductor, and (3) the oxide film on the semiconductor surface. However, in this paper, the SBH is only affected by the first factor because the latter two factors are unchanged for the same growing conditions. The high level of average electron density in ZnO NWs will cause low SBH [6]. As a result, the above-mentioned questions are equivalent to explaining why (Delta n) and the average electron density in a ZnO NW depend on its length.

It is widely known that the process of oxygen adsorbing and desorbing on a ZnO NW surface: ({text{O}}_{{text{2(g)}}} + {text{e}}^{ – } rightleftharpoons {text{O}}_{{text{2(ad)}}}^{ – }) and ({text{O}}_{{text{2(g)}}} + 2{text{e}}^{ – } rightleftharpoons 2{text{O}}_{{text{(ad)}}}^{ – }) [7,8,9]. The more detailed process has already been described [10]. In a stable environment, the quantity of adsorbed oxygen molecules dominates the number of electrons captured inside the NW, significantly impacting the NW’s conductivity under dark and illumination. It is necessary to know what dominates the quantity of adsorbed oxygen molecules.

As growing ZnO NWs, varied defects usually appear in NWs, resulting in unintentional doping [11, 12]. Defects such as OVs, as we all know, usually dominate the electronic and chemical properties and adsorption behaviors [13], because OVs could provide many sites for oxygen molecules to adsorb on the surface and seize electrons [14,15,16]. We can find that the higher density of OVs can lead to more oxygen molecules adsorbing on the surface of a ZnO NW [16,17,18]. Hence, the OAC can be characterized by the density of OVs. Additionally, it is also reasonable to take the density of adsorbed oxygen to describe the OAC [15, 19] because the adsorbed oxygen molecules occupy the position of OVs in the lattice for ZnO materials [20]. During growing a ZnO NW, with its length increasing, the density of its OVs increases and finally saturates [21]. Kayaci et al. observed that the density of OVs began to increase at a ZnO thickness of approximately 40 nm [20]. In addition, the literature [4] shows that the top shape of ZnO NWs grown by Song et al. is a micro-pyramid whose OAC is weaker than that of flake and column [19]. The proportion of the micro-pyramid-shaped surface in the total surface area of the NW increases as its length decreases, which will result in the weaker OAC of the NW. Besides the effects caused by the growing process, OAC can also be affected by temperature and other external factors [22]. Sanghwa observed the oxygen re-adsorption process on a ZnO NW surface under UV illumination in the air because of the Joule heating effect [23]. It shows that a suitable high temperature will enhance the OAC of a ZnO NW. To analyze conveniently, we first do not consider the effects of Joule heating and other factors due to the low Joule power and constant environmental conditions in Ref. [4]. So, given the length-dependent OAC, it is easy to recognize that the adsorbed oxygen density (N_{s}) on a ZnO NW surface increases when its length increases, leading to an increase in the density of captured electrons which can be proved by the scanning results of surface potential in the report [5].

Hence, we have the length-dependent density of electrons adsorbed (n_{c}) on the surface of a ZnO NW as Eq. (2) [24]

$$n_{c} = alpha N_{s}.$$


Here, (alpha) is the charge transfer coefficient denoting the number of electrons captured by one chemisorbed oxygen molecule. The expression (n_{c}) means the number of electrostatic carriers confined on a ZnO NW surface, which can be characterized by the surface potential (V_{sp}). Then, we can derivate the depleted width (r) by the following expressions (3)–(5) [25].

$$V_{sp} = frac{{2pi (ealpha N_{s} )^{2} }}{{varepsilon N_{d} }},$$


$$lambda_{D} = left( {frac{varepsilon kT}{{2pi e^{2} N_{d} }}} right)^{{{raise0.7exhbox{$1$} !mathord{left/ {vphantom {1 2}}right.kern-nulldelimiterspace} !lower0.7exhbox{$2$}}}},$$


$$r = lambda_{D} left( {frac{{eV_{sp} }}{kT}} right)^{{{raise0.7exhbox{$1$} !mathord{left/ {vphantom {1 2}}right.kern-nulldelimiterspace} !lower0.7exhbox{$2$}}}}.$$


Here, (varepsilon) is the dielectric constant, (N_{d}) is the concentration of donor impurity, (k) is the Boltzmann constant, (T) is the absolute temperature. To calculate the average dark electron density (n_{{{text{dark}}}}) in a ZnO NW, we should know the average adsorbed electron density of a ZnO NW exposed to air in the dark which can be described by

$$n_{ac} = frac{{2int_{0}^{L} {alpha N_{s} dL} }}{RL}.$$


Therefore, (n_{{{text{dark}}}}) becomes

$$n_{{{text{dark}}}} = n_{0} – n_{{{text{ac}}}}$$


where (n_{0}) is the electron density of the NW before it is exposed to air, and (R) is the radius of the NW. When a ZnO NW length is close to 40 nm, (N_{s}) will become zero and (n_{{{text{dark}}}}) will approximately equal a constant of (n_{0}). In contrast, as its length is long enough, (N_{s}) becomes a constant value of (N_{s0}) and (n_{{{text{dark}}}}) will be approximately reduced to Eq. (8) [26]

$$n_{{{text{dark}}}} = n_{0} – frac{{2alpha N_{s0} }}{R}.$$


As shown in Fig. 1, the distribution of electrons inside a ZnO NW exposed in the air depends strongly on its length, which is different from that in vacuum, and the depleted region width (r) increases with its length increases and saturates at a long length. When the light which can generate electron-hole pairs illuminates ZnO NWs, the chemisorbed oxygen will be photon-desorbed. The additional photoinduced electron density is determined by the concentration of the electrons confined inside the NW and the light intensity. As the light intensity is large enough, all electrons captured by oxygen molecules on the surface will be desorbed. Thus, (Delta n) can be modified as the following expression:

$$Delta n = n_{{{text{pe}}}} + n_{{{text{ac}}}}.$$


Fig. 1
figure 1

The distribution of electrons of a ZnO NW in vacuum a and in the air b

On the contrary, the number of photon-desorbed electrons is approximately equal to that of the photogenerated holes under low light intensity. The dark electron density in ZnO NWs is about (10^{17} {text{cm}}^{ – 3})[4] which is much larger than the electron density ((10^{14} {text{cm}}^{ – 3})) that light induced based on our calculation results, as shown in Fig. 2. It reveals that there are enough adsorbed electrons to be photon-desorbed for the NW in Ref. [4]. Therefore, we reasonably believe that (Delta n) in Ref. [4] can be instead approximately by (2n_{{{text{pe}}}}).

Fig. 2
figure 2

The difference in electron density (Delta n) between illumination and dark conditions from the experimental data in Ref. [4]

Not only can the length of a ZnO NW affect OAC, but in any case, it likewise influences LPC. For ZnO NWs, the higher aspect ratio will result in fewer recombination centers with a small number of interparticle junctions and higher electron delocalization, allowing the electron-hole pairs to separate effectively and lower their recombination rate [15, 17]. The smaller the size, the higher the recombination rate of electron-hole pairs in ZnO NWs [1]. In addition, the decreasing length will cause an increase in electron concentration under dark [6], leading to a decrease in LPC [27]. It should also be noted that OVs can render defect energy levels in a ZnO NW to prevent electron-hole pairs from recombining and lead to stronger light absorption [17, 28,29,30,31]. Briefly, the high density of OVs contributes to reducing the recombination rate of electron-hole pairs and increasing the density of photoinduced electrons. More important is that Hong et al. found that the radiative recombination rate in ZnO NWs decreased as the length increased and became saturated at about 600 nm [32]. Amazingly, the length of 600 nm approaches the peak photocurrent length in Ref. [4], which points to the strong relevance between the interesting phenomenon mentioned above and the recombination rate of electron-hole pairs. It is known that LPC (tau) is determined by the radiative and nonradiative lifetime: (frac{1}{tau } = frac{1}{{tau_{r} }} + frac{1}{{tau_{nr} }}) [32]. Here, (tau_{r}) and (tau_{nr}) are the radiative lifetime and nonradiative lifetime constants, respectively. Considering the effects of the different diameters on the (tau^{ – 1}), we obtain the recombination rate per unit area (tau_{p}^{ – 1}) from the experimental data of Ref. [32]. The inset in Fig. 3a reveals that the (tau_{p}^{ – 1}) is more dependent on the length than the diameter from 29 to 40 nm. It can also be demonstrated in other literature [33]. We believe that (tau^{ – 1}) of diameter from 29 to 50 nm in the unit area is similar at the same length. Herein, (tau^{ – 1}) of a ZnO NW with a diameter of 50 nm can be calculated and described simply by

$$uptau ^{ – 1} = 2.5 times L^{ – 1.68} + 6.16,$$


Fig. 3
figure 3

a The length-dependent recombination rate of ZnO NWs with a diameter of 50 ({text{nm}}) we obtain from the model. Inset: the recombination rate (tau^{ – 1}) and the recombination rate per unit area (tau_{p}^{ – 1}) are calculated from the experimental data in Ref. [32]. b Comparison of the results calculated in this work with the experimental and calculation ones in Ref. [4]. Under a voltage of 10 V, the photocurrent as the function of the height of ZnO NWs with a diameter of 50 nm

as shown in Fig. 3a. Because test time is far more than (tau) which is often less than 1 (ns) for ZnO materials [34], the density of photogenerated carriers at steady-state follows as

$$n_{{{text{pe}}}} = betaupgamma frac{I}{{{raise0.5exhbox{$scriptstyle {{text{hc}}}$} kern-0.1em/kern-0.15em lower0.25exhbox{$scriptstyle lambda $}}}}uptau.$$


Here, (beta) is quantum efficiency, (I) is the light illumination intensity, (h) is Planck’s constant, (c) is the light speed in vacuum, and (upgamma) is the material adsorption factor at the light wavelength (lambda) [35], In conclusion, the photocurrent of a ZnO NW in Ref. [4] will be given by

$$Delta I = frac{USemu }{L}(2beta gamma frac{I}{{{raise0.5exhbox{$scriptstyle {hc}$} kern-0.1em/kern-0.15em lower0.25exhbox{$scriptstyle lambda $}}}}{uptau )}.$$


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