Electrorotation and Electroorientation of Semiconductor Nanowires

Article pubs.acs.org/Langmuir

Electrorotation and Electroorientation of Semiconductor Nanowires Pablo García-Sánchez* and Antonio Ramos Departimento Electrónica y Electromagnetismo, Facultad de Fı ́sica, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012, Sevilla Spain ABSTRACT: A number of experimental studies have shown electric-field manipulation of nanowires dispersed in liquids. We demonstrate from first-principles that the electrical response of semiconductor nanowires in liquids must be described by considering, at least, two mechanisms for interfacial polarization: the classical Maxwell−Wagner interfacial polarization and the formation of an electrical double layer. We compare the theoretical predictions with our experimental data and with data published elsewhere and show that both mechanisms play an important role.

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INTRODUCTION Small particles in suspension can be manipulated and characterized with the use of ac electric fields.1,2 Recent examples include the precise control of metal and semiconductor nanowires. These are small rods with typical length ranging, approximately, between 5 and 10 μm and diameter below 1 μm. Nanowires and nanotubes (specially semiconductor nanowires3,4) are employed in a wide range of applications such as biosensors,5−7 array elements for light absorption in solar cells,8−13 laser and light-emitting diodes,14,15 energy storage,16−19 and as building blocks for novel nanocircuits.20−23 Several experimental works show the ability of ac electric fields to rotate and displace metal nanowires suspended in aqueous electrolytes.24−26 The orientation of conducting nanowires with ac fields has been extensively studied both theoretically and experimentally.27−33 Silver nanowire assembly patterns depend on the frequency of applied ac fields.34 Semiconducting and conducting nanowires can be separated by using dielectrophoresis.35 Also, dielectrophoresis has been used for the alignment of GaN nanowires36 and assembly of semiconductor nanowires on prepatterned electrodes.37,38 In this work we focus on the electrokinetic behavior of semiconductor nanowires dispersed in electrolytes. The motivation for this study arises from recent experimental work that shows measurements of the rotation and orientation rate of semiconductor nanowires subjected to ac electric fields.39,40 We reproduce experimentally some of these measurements and study, theoretically and numerically, the mechanisms that give rise to the motion of the nanowires. In particular, we study the semiconductor nanowires from firstprinciples by solving the transport equations for charge carriers in both media (particle and electrolyte). We find that the behavior of the semiconductor nanowires is given by the combination of, at least, two distinct interfacial polarization mechanisms, one due to the electrical double layer (EDL) charging41 and the other to the classical Maxwell−Wagner interfacial polarization.1 The former (EDL charging) arises © 2017 American Chemical Society

from the induction of a bipolar layer due to accumulation of charge carriers of different signs at both sides of the particle− electrolyte interface. It is well-known that this mechanism accounts for the polarization of metal microparticles in electrolytes.42−45 For particles larger than the EDL thickness, the characteristic time for EDL charging can be described as the RC time for charging the EDL capacitor through resistive media. In the case of an electrolyte−semiconductor interface, the EDL capacitance (CDL) is estimated by the series association of the capacitances in each media. Using the Debye−Hückel theory,46 the diffuse layer surface capacitance is given by Cdiff = εi/λDi, where εi and λDi are, respectively, the dielectric permittivity and Debye length of medium i. Thus, for each medium, we can define a typical time as RCdiff = S εi/σiλDi, where S is a typical dimension of the particle and σi is the conductivity. The RC time for charging the EDL is a combination of these two typical times that depends on particle shape. For the case of Maxwell−Wagner polarization, the relaxation mechanism arises from the contrast in conductivity and permittivity between the particle and the electrolyte. The characteristic time is a combination of the charge relaxation times of the two media, τi = εi/σi.1 This time marks the transition from a dielectric to an ohmic behavior. For thin EDL (λDi ≪ S ), Maxwell−Wagner relaxation time (∼ε/σ) is much smaller than the EDL charging time (∼S ε/σλD), and therefore, Maxwell−Wagner relaxation mechanism appears at higher frequencies than EDL charging. The paper is organized as follows: experimental results with semiconductor nanowires are briefly reviewed and new experimental data are shown. Later, we present the theoretical analysis of the problem and obtain numerical predictions for the electrorotation and electroorientation rate in the limit of Received: June 12, 2017 Revised: July 26, 2017 Published: August 3, 2017 8553

DOI: 10.1021/acs.langmuir.7b01916 Langmuir 2017, 33, 8553−8561

Article

Langmuir thin EDL. In general, the predictions qualitatively agree with experimental data, but relevant discrepancies are found in the electrorotation of ZnO nanowires.

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REVIEW OF EXPERIMENTAL RESULTS

Among all ac electrokinetic phenomena, electroorientation (EOr) and electrorotation (ROT) have recently received attention in the field of semiconductor nanowires. EOr generally refers to the orientation of a nonspherical particle by the action of an electric field with a fixed direction.1 ROT refers to the asynchronous rotation of a particle subjected to a rotating electric field. In experiments, these electric fields are usually generated with microelectrodes embedded in a microfluidic system that contains the small particles dispersed in a liquid. In this section we describe the main experimental findings with semiconductor nanowires. Experimental Data on Electroorientation. Akin et al.47 have used ac electric fields to obtain the electrical conductivity of silicon nanowires dispersed in oil. Their method relies on the EOr phenomenon: the silicon nanowires are aligned with the direction of an homogeneous ac electric field, and upon a sudden change in the direction of the field, the orientation rate of the nanowires is measured. They report measurements of the orientation rate as a function of the frequency of the ac field, i.e., the EOr spectra, and they determine the nanowire conductivities from these spectra. More recently, they implemented this method in a microfludic system for high-throughput measurement and sorting of semiconductor nanowires.39 Figure 1

Figure 2. Electrorotation data in ref 40. ZnO nanowires change the direction of rotation for frequencies larger than ≈300 kHz. Note that the sign convention for the rotation velocity is opposite to the convention used in the rest of the present paper. Reproduced with permission from ref 40 (copyright National Academy of Sciences).

Interestingly, ZnO nanowires change the direction of rotation for frequencies larger than ≈300 kHz. It is well-known, experimentally and theoretically,31 that ROT of metal nanowires happens in opposite direction to the rotation of the applied electric field; this is known as counterfield rotation. Thus, data for ZnO nanowires in Figure 2 show a transition from counter- to cofield rotation as the frequency of the rotating field increases. In order to model this behavior, Fan et al.40 employed an expression for leaky dielectric ellipsoids in leaky dielectric media. This model only provides one relaxation, the Maxwell−Wagner polarization. Thus, it cannot explain the two relaxations as observed in experiments and, therefore, cannot predict the transition from counterto cofield rotation with frequency. We have reproduced the experimental observations of Fan et al.40 for the case of ZnO nanowires. To this end, we synthesized ZnO nanowires by following the hydrothermal growth method.48 Figure 3A shows SEM micrographs of the nanowires. They are polydispersed in length and thickness. In this work, we redispersed the nanowires in a KCl aqueous solution with conductivity 1.5 mS/m. The nanowire dispersion was sonicated before experiments for at least 10 min. For ROT measurements, we generate a rotating electric field by using an array of four coplanar microelectrodes as shown schematically in Figure 3A. The electrodes are made of platinum on a glass substrate and they have a circular shape. The distance between opposite electrodes is 500 μm and they are subjected to an ac signal with peakto-peak amplitude of 10 V and frequency in the range 10 kHz to 10 MHz. The maximum frequency is limited by the generator specifications. In order to achieve a rotating field, we impose a phase-lag of 90° between the ac signals of neighboring electrodes. A drop of the nanowire dispersion is placed on the electrode array and videos of the nanowire rotation are recorded with the aid of microscope. From the videos, we measured the rotation speed of nanowires with length between 5 and 6 μm. The thickness for these nanowires range between 250 and 400 nm and, thus, their aspect ratio is β ≈ 0.06. Figure 3B shows data for the angular velocity in units of revolutions per second (rps) as a function of signal frequency. Data points correspond to the average value of three measurements for a given frequency; error bars are the standard deviation of those measurements. The high dispersion is, presumeably, due to the dispersity in particle aspect ratio and particle position within the electrode array. Negative values for the angular velocity indicate counterfield rotation, confirming that ZnO nanowires exhibit a transition from counter- to cofield rotation for increasing signal frequency. Importantly, we confirmed that the magnitude of the cofield maximum is clearly smaller than the magnitude of the counterfield peak.

Figure 1. Electroorientation data of silicon nanowires by Akin et al.39 The alignment rate is determined as a function of the frequency of the ac electric field. Data are shown for three different liquids. Reproduced with permission from ref 39. Copyright Royal Society of Chemistry. shows their EOr results for silicon nanowires dispersed in different liquids: mineral oil, dipropylene glycol (DPG), and DPG with salt. For all liquids, they observed a strong decrease of the electroorientation rate at angular frequencies in the megahertz range. As shown below, the torque that drives the rotation arises from the action of the electric field on the induced particle dipole, and therefore, the decrease in orientation rate is due to a relaxation of the particle polarizability. The nanowire conductivity is obtained from the fitting of these data. Interestingly, when the liquid is not very insulating but shows some conductivity, they report another relaxation for lower frequencies. They attributed this relaxation to the charging of the EDL at the liquid−particle interface. We show below that this is in accordance with the predictions of our theoretical modeling. Electrorotation Experiments in the Literature. Electrorotation of nanowires and carbon nanotubes dispersed in DI water was reported by Fan et al.40 They measured the rotation speed of these small particles as a function of the applied signal frequency, i.e., the ROT spectra. Figure 2 reproduces the data in ref 40. For frequencies below 300 kHz, semiconductor ZnO nanowires rotate in the same direction than metal nanowires, although with a smaller rotation speed. 8554


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Langmuir E = Re[Ẽ eiωt ]

(1)

and the time-averaged of the torque is given by

τe = (1/2)Re[p̃ × Ẽ *]

(2)

where p̃ is the phasor of the particle dipole moment and * indicates complex conjugate. The induced dipole on the cylinder can be decomposed into one component along the cylinder axis and another component perpendicular to the axis: p̃ = 4πεa3[A(ω)Ẽ x + B(ω)Ẽ y]

(3)

where ε is the electrolyte permittivity, A(ω) is the nondimensional complex polarizability along the cylinder axis and B(ω) indicates the nondimensional complex polarizability perpendicular to the axis. In EOr experiments, the direction of the applied electric field is fixed and the phasor can be written as Ẽ = E0(cos θux + sin θuy), with θ the angle between the direction of the electric field and the particle axis. In this case, the timeaveraged torque results in τe(EOr ) = 2πεa3E02 Re[A(ω) − B(ω)]cos θ sin θ u z ≈ 2πεa3E02 Re[A(ω)]cos θ sin θ u z

(4)

In ROT experiments, the applied electric field phasor can be written as Ẽ = E0(ux − iuy) and the time-averaged torque results in τe(ROT) = − 2πεa3E02 Im[A(ω) + B(ω)]u z ≈ −2πεa3E02 Im[A(ω)]u z

Figure 3. (A) SEM micrograph of ZnO nanowires synthesized by the hydrothermal growth method. The nanowires are placed in the center of an electrode array and subjected to a rotating electric field. (B) Electrorotation data for ZnO nanowires.

(5)

For slender particles (a ≫ b), the polarizability along the axis dominates over the polarizability perpendicular to the axis.44 For this reason, we have neglected |B(ω)| in front of |A(ω)| in eqs 4 and 5. Inertia of a nanowire is minute and the rotation velocities correspond to the balance between electrical and viscous forces. The viscous torque on a cylinder that rotates with angular velocity θ̇ around an axis perpendicular to the cylinder axis can be written as τv = γθ̇uz, with γ the rotational friction coefficient. For a cylinder in the bulk of a fluid, γ is given by the following expression:49

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THEORETICAL ANALYSIS Let us consider a cylindrical particle of a semiconducting material with radius b and length 2a, thus the aspect ratio is β = b/a. The particle is immersed in an electrolyte and subjected to an homogeneous ac electric field (E) with angular frequency ω; see Figure 4. As a consequence, electric charge is induced at the particle−electrolyte interface. The interaction between the applied electric field and the induced charges gives rise to an electrical torque that drives the particle rotation. If the applied electric field is homogeneous, this torque is τe = p × E, where p is the dipole moment induced on the particle. Using phasors, the applied ac electric field can be written as

γ=

8πηa3 3(ln(1/β) + δ⊥)

(6)

with δ⊥ = −0.662 + 0.917β − 0.05β and η the liquid viscosity. Using the expressions for the EOr and ROT torques, the balance with viscous torque yields 2

θ(̇ EOr ) =

2πε 3 2 a E0 Re[A(ω)]cos θ sin θ = Γ(ω)cos θ sin θ γ (7)

̇ θ(ROT) =−

2πε 3 2 a E0 Im[A(ω)] γ

(8)

where the electroorientation rate has been defined as Γ(ω) = (2πε/γ) a3E20Re[A(ω)]. Then, it is necessary to calculate the polarizability of a semiconductor nanowire for comparing experiments with theoretical predictions. It is also important to notice that, in experiments, our nanowires are close to the device substrate and the presence of this wall increases viscous friction while it reduces the magnitude of the electrical

Figure 4. A semiconductor nanowire is modeled as a cylinder of diameter 2b and length 2a. The nanowire is subjected to an ac electric field of arbitrary direction. The electric currents induce an electrical double layer at the liquid-particle interface. 8555


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Langmuir torque.50 Thus, we expect that expressions 7 and 8 provide the maximum theoretical values for the electroorientation and electrorotation rate. Polarizability of a Semiconductor Nanowire. The applied electric field drives ions in the electrolyte and electrons/holes on the semiconductor side. We assume that no transfer reactions occur for charge carriers arriving at the semiconductor−electrolyte interface, although these reactions could happen for sufficiently high electric field that induces electrochemical reactions. In other words, we assume that the interface blocks the passage of charge carriers. In this situation, an electrical double layer (EDL) is induced at the interface. We can describe the EDL as comprising two different diffuse-charge layers with opposite signs: one in the electrolyte and another in the semiconducting material. The extension of the diffuse layer is given, in each medium, by the balance between electromigration and diffusion of the corresponding charge carriers. The Debye length (λD) is the parameter that determines the thickness of the diffuse layer. For the semiconductor, this length is indicated as λD1, while for the electrolyte is λD2 (see Figure 4). In both cases, the Debye length is calculated as follows:46 λDj =

(σ2 + iωε2)E 2 ·n = (σ1 + iωε1)E1·n

where ϕj, σj, and εj are, respectively, the electric potential, the bulk conductivity, and the dielectric constant of each medium. Equation 11 can be interpreted as the conservation of the total current and it will be valid as far as tangential currents at the semicondutor−electrolyte interface are negligible. n is a unit vector perpendicular to the semiconductor−electrolyte interface and Zj stands for an effective surface impedance due to the thin diffuse layer of medium j: Z͠ j =

(9)

where j = 1, 2; εj indicates the permittivity of the corresponding medium; kB, T, and e are, respectively, the Boltzmann constant, the absolute temperature, and the elementary charge; and c0j is the bulk concentration of charge carriers in the corresponding medium. Expression 9 assumes that charge carriers are monovalent and that there is, in each medium, only one type of positive charge carrier and one type of negative. Despite that confinement effects can play an important role in properties of semiconductor nanocrystals, we expect that expression 9 will be valid as far as the concentration of charge carriers is governed by the competition between electromigration and thermal diffusion. Our goal in this section is to obtain the polarizability of the nanowire along its axis, A(ω). First, we calculate the electric potential in the system since the particle polarizability can be derived from it. To this end, it is important to realize that the Debye length in the solid and in the liquid is typically much smaller than the nanowire diameter. For example, λD ≈ 30 nm for the ZnO nanowires23 and for the KCl water solution with conductivity 1.5 mS/m. It is then customary to apply the thin EDL limit and, thus, the liquid and solid bulk are electroneutral. In this case the electric potential satisfies Laplace equation with specific boundary conditions that account for the charge at the EDL that appears at interfaces.51 Zhao et al.52 studied the thin EDL limit for a semiconductor−electrolyte interface subjected to an ac field. They considered the transport equations (PNP equations) for charge carriers in the liquid and the solid, and assumed that no transfer reactions occur for charge carriers arriving at the semiconductor−electrolyte interface, i.e., the interface is impervious to charge carriers. For sufficiently small applied voltages, the equations are linearized and it is found that the electric potential satisfies the following boundary conditions at the interface: ϕ2 − ϕ1 = σ2

∂ϕ2 ∂n

Z̃2 + σ1

∂ϕ1 ∂n

Z1̃

λDj iωεj 1 + iωεj /σj

(12)

which corresponds with the expression found by Zhao et al.52 These equations assume that the semiconductor−electrolyte interface has no intrinsic surface charge; i.e., there is no EDL in the absence of an applied electric field. It is also possible to make the weak-field approximation in the case of an interface with intrinsic surface charge (see, for instance, ref 53). In addition, if the surface is highly charged, we can expect surface conduction that could lead to high Dukhin number54 and the associated phenomena of concentration polarization. These effects are beyond the scope of this work. Equation 10 accounts for the potential drop due to charge accumulation at the EDL and eq 11 represents the conservation of the total electrical current (ohmic plus displacement currents). In deriving these boundary conditions, it is assumed that the mobilities of positive and negative ions are the same. Likewise, positive and negative charge carriers in the semiconductor have the same mobility. While this approximation is correct for a KCl aqueous solution, charge carriers in semiconductors may have very different mobilities. Notice that for a sufficiently low frequency, ωε/σ ≪ 1, eq 12 simplifies to a purely capacitive impedance, as corresponds to the impedance of a quasiequilibrium EDL. We use the equations above to find the electric potential in the system for an applied field along the nanowire axis. We make use of the axial symmetry and reduce the problem to a 2D axisymmetric domain as shown in Figure 5. Equations and

εjkBT 2e 2c j0

(11)

Figure 5. 2D axisymmetric domain for computing the value of the nondimensional polarizability (not to scale). Dimensions are scaled with the nanowire length, so a = 1 and b = β. The applied electric field is along the nanowire axis and its magnitude is E0 = 1.

boundary conditions are also shown in the figure. We used a commercial software that implements the finite element method (COMSOL Multiphysics). From the solution of the phasor potential ϕ, we compute the nondimensional polarizability A as follows.31 Using spherical coordinates (r, θ), the solution for the electric potential can be written as the applied potential (−E0 r

(10) 8556


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Langmuir cos θ) plus the perturbation due to the polarized cylinder (ϕ′). The potential outside a sphere with radius larger than a can be expressed using the Legendre polynomials: ∞

ϕ = −E0r cos θ + ϕ′ = −E0r cos θ +

∑ l=1

Al r l+1

Pl(cos θ) (13)

where the polar angle θ is measured from the symmetry axis. We are interested in the term with l = 1 of the expansion, thus A1 = A. Taking into account the orthogonality of Legendre polynomials 3 ϕ′P1(cos θ ) dS A= (14) 4π S

Figure 7. Electric field lines around a semiconductor nanowire in an electrolyte: (A) frequencies much lower than τEDL, (B) intermediate frequencies, (C) frequencies much higher than the Maxwell−Wagner relaxation time for the two media τMW.

∫

where P1 (cos θ) = cos θ and S is a spherical integration surface which encloses the cylinder and is far from the particleelectrolyte interface. Figure 6 shows numerical results of the real and imaginary parts of A(ω) as a function of nondimensional frequency Ω of

small, leading to a correspondingly small polarizability. At intermediate frequencies the EDL is not charged and the electric field lines are almost perpendicular to the particle surface, corresponding to a particle much more conducting than the surrounding medium. In this case, the electric field lines are strongly perturbed and the particle polarizability is high. Finally, the behavior at high signal frequencies is determined by the dielectric character of the two media. Water has a much larger dielectric constant than the semiconductor particle, and therefore, it is much more polarizable. In this case, the electric field lines surround the particle and, again, it is only slightly perturbed. For this reason, the particle polarizability in Figure 6 is relatively small at high frequencies. The extrema of Im[A] occur at the characteristic frequencies of the transitions of the real part. The lowfrequency peak predicts counterfield ROT, while cofield ROT is expected from the high-frequency peak (see eq 8).

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COMPARISON BETWEEN THEORY AND EXPERIMENTS In this section we compare the experimental data with the predictions of the theoretical analysis. Figure 8A shows numerical results for the real part of the polarizability of a nanowire with aspect ratio 0.01. Other values of the parameters are given in Table 2 and correspond to the experimental conditions in Akin et al.39 for the case of silicon nanowires dispersed in dipropylene glycol, when a double layer is present. The Debye length for the liquid is chosen in order to fit data. We show Re[A] for the two conductivity values of Figure 1, 10−6 and 10−7 S/m. The ratio between Debye lengths for each conductivity is in agreement with eq 9, i.e., it decreases with the square root of conductivity. Note that there is no flat segment for the real polarizability at intermediate frequencies between the two relaxations. This means that the polarization spectrum is not just the superposition of two Debye relaxations. This would happen if eq 12 were simplified and written as a purely capacitive impedance. However, we do not make the low frequency approximation and maintain the square root term in the denominator of eq 12. Numerical results are plotted together with experimental data in Figure 8B). Note that units for the y-axis of the numerical results are arbitrary in this plot. Experimental data for the electroorientation rate are not absolute; they are normalized to the maximum rate obtained with each liquid. Thus, we cannot compare with our prediction for the magnitude of the electroorientation rate. The frequency dependence is in agreement with experimental data. It is found that the high frequency relaxation is not dependent on liquid conductivity

Figure 6. Numerical results for the real and imaginary parts of the nanowire polarizability A(ω). Aspect ratio of the nanowire is β = 0.06. Results are obtained under the assumption of a thin EDL.

the applied field. The nondimensional frequency is defined as Ω = (ε2/σ2)ω, where ε2/σ2 is the charge relaxation time in the liquid. We consider a nanowire with aspect ratio β = 0.06, which corresponds to the average aspect ratio of ZnO nanowires in ROT experiments. Values for the other parameters are shown in Table 1. As expected, the frequency Table 1. Values for the Physical Properties of the Liquid and the Semiconductor Nanowirea semiconductor (ZnO) electrolyte (KCl in Water)

λD/a

σ

ε/ε0

0.01 0.01

480 mS/m 1.5 mS/m

3.9 80

a

These values are used in the calculations of ZnO nanowires under the assumption of thin EDL. Data for ZnO are within the range reported in ref 23 for ZnO nanowires.

dependence of the polarizability shows two distinct relaxations: The polarizability is very small for frequencies much lower than the reciprocal of the typical charging time of the electrical double layer (τEDL). This can be understood according to the electric field lines shown in Figure 7A. At low frequencies there is sufficient time to charge the EDL and the electric field lines surround the nanowire. The perturbation of the electric field is 8557


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Figure 8. (A) Numerical values for the real part of the polarizability for a semiconductor nanowire with aspect ratio 0.01. The lowest conductivity (10−7 S/m) is taken for scaling the electric field frequency (Ω = ωε2/σ2). Properties of the solid and liquid are in Table 2. (B) Numerical results plotted together with the data of Figure 1. Note that units for the y-axis of the numerical results are arbitrary in this plot.

Table 2. Values for the Physical Properties of the Liquid and the Semiconductor Nanowirea semiconductor (Si) electrolyte (DPG+salt,DPG)

λD/a

σ

ε/ε0

0.01 0.03, 0.03·√10

0.06 S/m 10−6,10−7 S/m

11.7 21

a

These values are used in the calculations of silicon nanowires. Silicon permitivitty is taken from the literature. Debye length for DPG is changed to fit data. Other data correspond to experiments reported in ref 39.

and it appears for a dimensional angular frequency around ω = 105 rad/s. On the other hand, the low frequency relaxation increases with the square root of liquid conductivity. Interestingly, experimental data also show that the electroorientation at very low frequencies does not vanish. However, as shown in ref 30, this orientation is not due to the electrical torque on the induced dipole, but it is generated by the induced-charge electroosmotic flow around the nanowire.41,55,56 The imaginary part of the polarizability in Figure 6 shows a positive peak for low frequencies (Ω ≈ 0.5) and a negative peak for frequencies around Ω = 10. This means that, according to eq 8, our model predicts counterfield rotation at low frequencies and cofield rotation for higher frequencies, in accordance with experimental observations for the ROT of ZnO nanowires in KCl aqueous solutions. For a quantitative comparison between experimental data and numerical results, we take into account that the applied electric field is not homogeneous within the observation region. Thus, we have to use the average value of the electrical torque within the field of view. In ref 31, we used finite element methods to calculate the average torque for the same electrode geometry and applied voltages. The theoretical values for the ROT velocity can be calculated from eq 8 by using E20 = 112 × 106 V2/m2. For cylinders with aspect ratio β ≈ 0.06, eq 6 yields γ/a3 = 263.35 Pa·s. Thus, theoretical ROT velocities in units of revolutions per second are calculated as θ′(ROT)/2π = 20.9 Im[A]. Figure 9 shows the experimental data for the electrorotation of ZnO nanowires and the theoretical prediction for two different values of ZnO conductivity. For σ1 = 0.48 S/m, results for Im[A] are taken from Figure 6, and for σ1 = 4.8 S/mm, we

Figure 9. Numerical results for thin double layers are compared with experimental data for the electrorotation of ZnO Nanowires. The theory predicts the same amplitude for the two rotation peaks, although with different signs. Experimental data show that the cofield rotation is weaker.

calculated the nanowire polarizability for a Debye length that was rescaled according to eq 9, i.e., λD1 = 30/√10 nm, while the other parameter values were not modified (Table 1). The comparison shows that experimental data for counterfield rotation are close to the numerical predictions for nanowires within this conductivity range. As mentioned by Janotti et al.,57 the control of ZnO conductivity (and, consequently, its Debye length) is a major issue. For example, electrical properties of ZnO are very sensitive to surface modifications. This variability in conductivity might also be responsible for the large distribution in experimental data, together with the dispersity in nanowire aspect ratios and positions within the electrode array. When comparing experiments and theory it is important to keep in mind that the nanowires were close to the array substrate and the presence of this wall reduces the ROT velocity. For this reason, it seems reasonable to consider that the numerical prediction for the ROT velocity in the liquid bulk must overestimate measurements and, thus, σ1 = 4.8 S/m represents a more realistic value for the average conductivity of the nanowires. 8558


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dielectric behavior (Maxwell−Wagner relaxation). This relaxation must appear in experiments for increasing signal frequency, although beyond the capabilities of our current equipment. Therefore, from an experimental point of view, it is desirable to explore the electrorotation of ZnO nanowires for frequencies higher than 10 MHz.

Interestingly, numerical results for cofield rotation of ZnO nanowires clearly overestimate the experimental data. We have explored the numerical predictions for a wide set of parameter values and, in all cases, the theoretical cofield and counterfield peaks have the same magnitude. Experiments clearly show that cofield rotation has a smaller magnitude and we conclude that our theoretical modeling in the limit of thin double layers can not account for this observation. In the Appendix we show how to study the more general scenario in which the Debye length is not necessarily much smaller than the nanowire dimensions. Thus, we cannot apply the thin double layer approximation and the numerical computations become more involved. In order to show the effect of nanowire conductivity and its Debye length, we have included in the Appendix a number of numerical results for the imaginary part of the polarizability. For a relatively thick Debye layer in the particle, the two electrorotation peaks change their relative height, in contrast to the predictions of the thin double layer theory. However, the magnitude of the imaginary part of the polarizability is greatly reduced and the prediction for the ROT velocity decreases by orders of magnitude. For this reason we rule out the posibility that a finite Debye length in the nanowire can account for a lower cofield rotation as observed in experiments with ZnO.

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APPENDIX: NANOWIRE POLARIZABILITY ALONG THE AXIS FOR ARBITRARY DOUBLE LAYER THICKNESS In this section we do not make the thin double layer approximation. The electrolyte and the semiconductor bulks are not considered to be electroneutral and we have to solve the more general problem in which the electric potential satisfies the Poisson equation with a net volume charge given by the difference between the concentrations of positive and negative charge carriers. Following refs 45 and 61 in the limit of small applied voltages and for symmetric mobilities of charge carriers in each medium, the Poisson-Nernst−Planck (PNP) equations yield the following governing equations for the electric potential and charge density: ρj ∇2 ϕj = − εj (15)

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CONCLUSIONS Electrorotation and electroorientation spectra of semiconductor nanowires dispersed in electrolytes are modeled from firstprinciples by solving the transport equations for charge carriers. From the numerical results, we identify two distinct polarization mechanisms: the classical Maxwell−Wagner polarization and the charging of the electrical double layer. Numerical results predict two relaxations for the electroorientation spectra, in agreement with experimental data for silicon nanowires. The dependence of electroorientation data with liquid conductivity also agrees with the numerical solutions of the model. With respect to electrorotation, the model predicts counterfield and cofield rotations at low and high frequencies, respectively. A quantitative comparison with experimental data for the electrorotation of ZnO nanowires shows that the counterfield rotation is correctly described by the model, while the cofield rotation is overestimated. We have considered that this discrepancy could be explained by a thicker Debye length in our ZnO nanowires. To this end, we included in the Appendix a general analysis for the case of arbitrary Debye layer in the liquid and the particle. We found that the two electrorotation peaks show different heights for a relatively thick Debye layer in the particle. However, we rule out this explanation because the nanowire polarizability is greatly reduced for thick double layers and, consequently, the theoretical prediction for the angular velocity is orders of magnitude smaller than experimental data. Thus, it remains to find a satisfactory explanation for the cofield electrorotation of ZnO nanowires dispersed in aqueous solutions. It is well-known that metal oxides in water acquire a surface charge that depends on the solution pH and the type of ionic species in solution.58,59 In particular, the isoelectric point for ZnO is around pH = 9.60 This surface charge has important consequences such as the modification of the surface impedance and the appearance of a surface conductance at the particle−electrolyte interface.54 Future theoretical work for the electrokinetics of metal oxide nanowires in water should explore the influence of surface charge. With respect to the cofield rotation that we found in the numerical computations, this occurs as a consequence of the relaxation from ohmic to

∇2 ρj = (1 + iωτj)

ρj 2 λDj

(16)

where ρj = e(c+j −c−j ) is the electrical charge density and τj = εj/ σj is the charge relaxation time in medium j. c+j and c−j indicate, respectively, the concentration of positive and negative charge carriers in medium j. Equations 15 and 16 are supplemented with boundary conditions at the semiconductor−electrolyte interface. They are the continuity of the electric potential and the continuity of displacement current: ϕ1 = ϕ2

(17)

n ·(ε1∇ϕ1) = n ·(ε2∇ϕ2)

(18)

where n is a unit vector perpendicular to the semiconductor− electrolyte interface. Also, we impose that the semiconductor−electrolyte interface is impermeable to charge carriers (i.e., ions do not migrate into the solid and, likewise, electron−holes do not migrate into the liquid). This requirement yields to the following two boundary conditions (j = 1,2) n ·(σj∇ϕj + Dj∇ρj ) = 0

(19)

where σj and Dj are, respectively, the electrical conductivity and ionic diffusivity in medium j. We solve the equations in the problem domain of Figure 5B for an applied electric field along the nanowire axis. From the solution of the electric potential, we find the nanowire polarizability along its axis by integrating the perturbation of the electric potential on a sphere as in eq 14. Now, it is important that the radius of the sphere is large enough such as to contain all charges in the system, i.e., the radius of the sphere must be much larger than the size of the particle plus the liquid Debye length. Figure 10 shows the imaginary part of the polarizability for a thick EDL in the semiconductor λD1 = 0.5a and different values 8559


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REFERENCES

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Figure 10. Imaginary part of the nanowire polarizability (ZnO in KCl) for decreasing conductivity of the semiconductor. The electrical double layer is not thin and the two electrorotation peaks have different height. As conductivity decreases, the cofield peak disappears.

of the nanowire conductivity. Other values for the parameters are taken from Table 1 (ZnO in KCl aqueous solution). As the conductivity decreases, the peak for cofield electrorotation is weaker and appears at lower frequencies. Figure 11 shows the

Figure 11. Imaginary part of polarizability (ZnO in KCl) for increasing Debye length of the semiconductor. The two electrorotation peaks have different height when the electrical double layer is not thin.

imaginary part of the polarizability for a given conductivity of the nanowire σ1 = 10−3 S/m and different values of λD1. For a relatively thin EDL, the two electrorotation peaks are of the same magnitude. As λD1 increases, Im[A] becomes much smaller and the two peaks are not of the same magnitude (counterfield rotation is larger than cofield).

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Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Pablo García-Sánchez: 0000-0003-3538-2590 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge financial support from Spanish Government Ministry MECD under contract FIS2014-54539-P. 8560


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Langmuir

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Electrorotation and Electroorientation of Semiconductor Nanowires

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