Inhomogeneous Charge Distribution in Semiconductor Nanoparticles

Jun 3, 2015 - A set of equations to determine the total number of conduction electrons and the radial charge distributions inside a spherical nanopart...
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Inhomogeneous Charge Distribution in Semiconductor Nanoparticles M. A. Kozhushner,† B. V. Lidskii,† I. I. Oleynik,*,‡ V. S. Posvyanskii,† and L. I. Trakhtenberg†,§ †

Semenov Institute of Chemical Physics of RAS, 4 Kosygin Street, Moscow 119991, Russia University of South Florida, 4202 East Fowler Avenue, Tampa, Florida 33620-5700, United States § Moscow Institute of Physics and Technology (State University), 9 Institutskii per, Dolgoprudny, Moscow Region 141700, Russia ‡

S Supporting Information *

ABSTRACT: The inhomogeneous spatial distribution of charge carriers within semiconductor oxide nanoparticles is investigated by taking into account processes involving the interaction of conduction electrons with oxygen donor vacancies in the bulk and with oxygen adsorbates at the surface. The main characteristics of the semiconductor nanoparticles, such as the surface charge, the distributions of positive and negative charges in the bulk, and the temperature dependence of the concentration of conduction electrons, are determined selfconsistently by taking into account the interaction of all charges in the system under conditions of complete thermodynamic equilibrium. The developed statistical mechanics model allows us to determine the net surface negative charge on oxygen atoms, the spatial distributions of the conduction electrons, the positively charged ionized donors, and the electrical potential inside the nanoparticle as a function of nanoparticle radius and temperature.

■

INTRODUCTION Due to scientific and technological importance, nanostructured materials have been studied intensively over the past two decades.1−7 Specifically, electronic characteristics of nanoparticles play a key role in understanding their conductive, photoelectric, sensor, catalytic, dielectric, and magnetic properties, as well as their role in plasmonic applications. The physics of charge carriers is essential in the processes of photon absorption, emission, and scattering by nanostructured materials. Various properties of nanoparticles, such as the way in which they absorb and scatter electromagnetic waves, the structure and frequency of plasmons, and their dielectric characteristics, depend on the distribution of the charge density over their entire volume. On the other hand, the conductivity within the nanoparticle system depends mainly on the concentration of conduction electrons at the surface. In addition, the sensor effect  the change in conductivity of the semiconductor nanoparticle thin film upon introduction of various environmental gases6,8−16  is also determined by the spatial distribution of charge carriers. Therefore, the quantitative description of the sensor effect requires knowledge of the concentration of the conduction electrons near the nanoparticle’s surface and its dependence on the concentration of the surface adsorbates. As in bulk semiconductors, the concentration of electron and hole charge carriers inside semiconductor nanoparticles depends on the temperature as well as the concentration of the dopants. In addition to the bulk effects, the surface of semiconductor nanoparticles is of great importance due to the presence of surface defects and oxygen adsorbates from air. The © 2015 American Chemical Society

amount of such electron traps might exceed that of the bulk donors, such as oxygen vacancies, resulting in significant changes in the concentration of the conduction electrons inside the nanoparticle. In particular, the atomic oxygen atoms, appearing upon dissociation of oxygen molecules adsorbed at the surface of the nanoparticle from the air, effectively capture electrons because the electronic levels of negative oxygen ions are usually deeper than the donor levels in the bulk of the material.13 These negative O− ions form a negatively charged layer on the nanoparticle surface, resulting in redistribution of the free electrons in the conduction band from the bulk to the surface and the appearance of uncompensated positive charge inside the nanoparticle in the amount equal to the total surface negative charge to maintain total electroneutrality within the system. The inhomogeneous effects become significant in the case of nanoparticles containing a substantial amount of conduction electrons. For example, such effects are insignificant in the case of the SnO2 oxide nanoparticles, which possess conduction electron concentration nc significantly smaller than the concentration of oxygen donor vacancies nd ≈ 1016 cm−3. A nanoparticle with diameter D = 100 nm contains (π/6)D3nc ≤ 1 electrons; therefore, there are not enough conduction electrons to form the negatively charged surface layer. In contrast, In2O3 oxide semiconductor nanoparticles17−19 have conduction electron concentrations in excess of 1019 cm−3 due to donor Received: February 10, 2015 Revised: June 1, 2015 Published: June 3, 2015 16286

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The Journal of Physical Chemistry C impurities.20,21 Therefore, an individual nanoparticle with diameter D = 100 nm might contain up to 104 electrons, and the inhomogeneous effects caused by the interaction of the conduction electrons with the O atoms adsorbed on the nanoparticle surface become significant. The redistribution of the bulk conduction electrons results in uncompensated positive charge, inhomogeneously distributed inside the nanoparticles. Such effects have already been considered in refs 7 and 11 under the assumption that the negative surface charge is determined solely by an unspecified density of the surface states, with the distribution of the positive charges inside a nanoparticle being considered homogeneous. As will be shown below, both of these assumptions are unfounded. In this paper, inhomogeneous spatial distributions of charge carriers within semiconductor oxide nanoparticles are investigated by taking into account the processes involving interactions of conduction electrons with oxygen donor vacancies in the bulk and oxygen adsorbates at the surface. The main characteristics of the semiconductor nanoparticles, such as the surface charge, the distributions of positive and negative charges in the bulk, and the temperature dependence of the concentration of conduction electrons, are determined self-consistently by taking into account the interaction of all charges within the system under the condition of complete thermodynamic equilibrium. The governing equations are obtained by minimizing the total free energy of the system of the charges. Their solution allows us to determine the net surface negative charge on O−, the spatial distributions of the conduction electrons, the positively charged ionized donors, and the electrical potential inside the nanoparticle as a function of the nanoparticle radius and the temperature.

A set of equations to determine the total number of conduction electrons and the radial charge distributions inside a spherical nanoparticle can be derived by minimizing the free energy of the system consisting of fixed charged donor centers and O− ions on the surface, as well as mobile electrons in the conduction band. The free energy F consists of several contributions: F = F1 + F2 + F3 + F4

(1)

where F1 is the free energy of the gas of free conduction electrons, F2 is the potential energy of interaction of all the positive and negative charges in the system, F3 is the free energy associated with the varying positive charges on ionized donors, and F4 is the free energy of the electrons bound to atomic oxygen atoms at the nanoparticle surface. The free energy expressions take into account the spatial inhomogeneity of both conduction electron and positive ionized donor densities by introducing the following radial distributions: the concentration of the conduction electrons nc(r) and the concentration of the positively charged ionized donors n+(r). In addition, the total number of oxygen negative ions (surface electron traps) N−O uniformly distributed over the surface of a nanoparticle is introduced. It is assumed that the nanoparticles are of spherical shape with uniform distribution of oxygen traps over its surface. This is a reasonable approximation, as these surface defects are formed upon collision of multiple, randomly growing crystalline faces at the surface of individual semiconductor nanoparticles. The total number of free electrons in the nanoparticle is equal to

■

Nc =

EQUILIBRIUM DISTRIBUTION OF THE CHARGES INSIDE SPHERICAL NANOPARTICLES Let us consider a semiconductor with a high concentration of conduction electrons, such as In2O3, where donors are the oxygen vacancies introduced to the material upon its synthesis and processing. Although the electron binding energy εd of a single oxygen vacancy donor is unknown, the estimate of εd can be obtained by using the experimental temperature dependence of the conductivity of a bulk sample of In2O3 measured in refs 22 and 23. This is because the conductivity of the semiconductor is approximately proportional to the concentration of conduction electrons nc ∼ exp{−εd/2kT}.15,16 The estimates based on the conductivity of bulk In2O3 give εd ≈ 0.2 eV, which is approximately the same as the value obtained using conductivity measurements for the nanostructured In2O3 film sample in vacuum, i.e. upon removal of the oxygen adsorbates.24 Using known values of the average diameter of the nanoparticles, and the conductivity of the sample, the donor concentration nd can be evaluated, nd = (1019 to 1020) cm−3.16 In this work, we use the following values εd = 0.2 eV and nd = 1020 cm−3. The total number of donor vacancies in the semiconductor nanoparticle with diameter D = (10−100) nm is Nd = (π/6) D3nd ≈ (102 to 105), and the total number of conduction electrons nc ∼ Nd is also high. The number of adsorbed molecules, oxygen atoms, and charged traps (O− ions) at the nanoparticle surface is also appreciable. Therefore, statistical mechanics can be applied to determine the inhomogeneous properties of basic physical quantities, such as charge distributions and their resulting electric field.

∫0

R

4πr 2nc(r )dr

(2)

We assume that the total number of donors Nd, i.e. oxygen vacancies, uniformly distributed over the volume of the nanoparticle is fixed by conditions of synthesis. The total number of ionized, positively charged vacancies is equal to N−O + Nc, and their inhomogeneous distribution is defined by n+(r) so that

∫0

NO− + Nc =

R

4πr 2n+(r )dr

(3)

The free energy of the gas of conduction electrons F1 associated with the kinetic energy of the equilibrium electron gas is F1 = 4π

∫0

R

drr 2Fkin(r )

(4)

where Fkin is the kinetic energy density of the electron gas Fkin(r ) = −

2 2 m*3/2 × 3π 2

∫0

∞

+ μnc(r )

25

ε 3/2 dε exp{(ε − μ)/kT } + 1 (5)

Here m* is the effective mass of the electron in the conduction band (atomic units e = ℏ = me = 1 are used throughout the paper), and the chemical potential μ is defined implicitly as a function of nc(r) and kT as nc(r ) =

2 π2

(m*kT )3/2 ×

∫0

∞

ε1/2 dε exp{(ε − μ)/kT } + 1 (6)

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brackets inside the integral in eq 10 is the local entropy of the ionized donors, while the expression in square brackets in eq 11 is the entropy of the oxygen negative ions O−. The term containing the kT ln 2 multiplier in eqs 10 and 11 corresponds to the two possible spin states of an electron at the donor and at an oxygen atom on the surface. The surface concentration of the oxygen atoms nO is determined by the equilibrium value of the surface concentration of the adsorbed molecules O2 at atmospheric pressure and by using the equilibrium condition for the association and dissociation reactions O2 ↔ 2O16

Equations 5 and 6, derived for both dense and rarefied Boltzmann gas, are applicable over a wide range of temperatures and electron densities. The free energy contribution F2 due to the interaction of all the positive and negative charges in the system is F2 =

χ 2

∫0

R

⎛ dφ ⎞ 2 r 2⎜ ⎟ dr ⎝ dr ⎠

(7)

where χ is the permittivity of the nanoparticle material and φ(r) is the total electrostatic potential of the system, which is determined by solving the Poisson equation 1 d ⎛ 2 dφ ⎞ 4π ⎜r ⎟ = − n(r ) 2 χ r dr ⎝ dr ⎠

⎧ Δ − εO ⎫ νO − OnOlim a 2 ⎬ nO = exp⎨− des ⎡ K 2 kT ⎭ a 2ν 1 + O2 (T ) ⎤ ⎩ ad O⎢ KO2(T )PO2 ⎥ ⎣ ⎦

(8)

Here n(r) is the total charge density inside the nanoparticle at r ≤ R, n(r ) = n+(r ) − nc(r )

Here Δ is the dissociation energy of an adsorbed oxygen molecule, εOa is the activation energy for the surface diffusion jumps of an O atom with a jump length a, νO−O is the oscillation frequency of O atoms in the adsorbed O2 molecule, νO is the oscillation frequency of an O atom in a local well on the surface, nOlim2 is the limiting surface concentration of adsorbed oxygen molecules, Kad O2(T) is the constant of oxygen molecular adsorption, PO2 is the pressure of oxygen in the air, and Kdes O2 ≈ νO2 exp{−εdes/kT} is the constant of desorption for the oxygen molecules, where νO2 is the frequency of an O2 molecule oscillating in an adsorption well. For numerical calculations, eq 12 can be rewritten as

(8a)

It is assumed that adsorbed oxygen atoms form a thin shell of thickness d = 2 au (or ∼1 Å) at the surface of the nanoparticle. Therefore, the charge density extends beyond the geometrical boundary of the nanoparticle and is determined by the charge density due to O− ions: n(r ) = −NO−/4πR2d

(8b)

for R < r < R + d, and d = 2. The boundary conditions for eq 8, φ(R + d) = 0,

dφ (r = 0) = 0 dr

(9)

⎧ Δ ⎫ nO = nÕ exp⎨− 1 ⎬ ⎩ kT ⎭ 1 +

reflect the facts that the potential is constant (zero) outside the particle with zero net charge, and the electric field is zero at the center of the nanoparticle. Assuming that the donor vacancies are uniformly distributed over the volume of the spherical nanoparticle with a fixed concentration nd = Nd/4/3πR3 and that the adsorbed O atoms, serving as electron traps, are uniformly distributed over the surface, the free energies F3 and F4 associated with the varying positive charges on ionized donors, as well as the electrons bound to atomic oxygen atoms at the nanoparticle surface, can be written as16

2 1/2 (νO−Onlim O2 /a νO) ,

∫

nd − n+(r ) ⎤ ⎥dr n+(r ) ⎦

1 B T

Δ

{ }

exp − kT2

(12a)

where ñ = Δ1 = (Δ − Δ2 = εdes, and B is the temperature-independent part of the ratio Kdes O2 (T)/ (T); see refs 15 and16 for details. Kad O2 It is worth noting that eqs 4 and 10, both involving the inhomogeneous charge distributions and taking into account the entropy of a nonequilibrium system,25 can be used to obtain the equilibrium distributions by minimizing the corresponding free energy. This free energy due to an inhomogeneous electron gas is similar to that used in the Thomas−Fermi method for calculating the electronic properties of a multielectron atom26−28 with only one exception: an electron gas is considered to be degenerate in the Thomas−Fermi method, whereas we treat the system at nonzero temperature. The equilibrium values Nc, N−O and functions nc(r), n+(r) are determined by minimizing the total free energy in eq 1 with respect to these variables. It is important to note that

F3 = −(Nd − Nc − NO−)(εd + kT ln 2) ⎡ R nd − kT 4πr 2⎢nd ln 0 nd − n+(r ) ⎣ + n+(r ) ln

(12)

(10)

and ⎡ NO F4 = − NO−(εO + kT ln 2) − kT ⎢NO ln NO − NO− ⎣

n+(r ) ≤ nd

εOa )/2,

(13)

(11)

and the equality n+(r) = nd is achieved upon complete ionization of donor vacancies. Thus, the sum N−O + Nc = ∫ R0 4πr2nd(r)dr = 4/3πR3nd becomes constant and should be used instead of eq 3.

Here, εO is the binding energy of an electron at the adsorbed O atom, and the total number of O atoms on the nanoparticle surface is NO = 4πR2nO, where nO is the surface concentration of oxygen atoms. The first terms in eqs 10 and 11 are the total binding energy of the electrons at the donors and the surface oxygen atoms, respectively. The expression in the square

FREE ENERGY MINIMIZATION TO CALCULATE INHOMOGENEOUSE CHARGE DISTRIBUTIONS The unknown functions nc(r) for the electron concentration, n(r), for the total charge and total numbers of the conduction electrons, Nc, and for negative oxygen ions, N−O, can be

+ NO− ln

NO − NO− ⎤ ⎥ NO− ⎦

■

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The Journal of Physical Chemistry C determined using standard methods of variational calculus29 to minimize the total free energy functional F under constraint of the normalization conditions of eqs 2, 3, and14: 4

∫0

R

πr 2n(r )dr = NO−

φ(r ) = μ(r ; T ) − μ(R ; T ) +

φ (r ) =

The main difficulty is a noncanonical form of the F 2 contribution in eq 1 due to its implicit nonlocal dependence on the unknown functions nc(r) and n(r) through the electrostatic potential φ(r), see Supporting Information. The resulting set of governing equations for nc(r) and n(r) are obtained upon minimizing the free energy functional:

{

1 + λ exp

μ(r ; T ) kT

(m*kT )

2 π 3/2

RESULTS AND DISCUSSIONS To study the inhomogeneous electrical properties of In2O3 semiconductor nanoparticles, the system of eqs 15−15b is solved iteratively by minimizing the total free energy in eq 17 using the following parameters: nd = 1020 cm−3; m* = 0.4m0, where m0 is the mass of free electron; εν = 0.2 eV; εO = 0.54 eV; χ = 1.5; ñO = 1014 cm−2; B = 2 × 103 (eV)1/2; Δ1 = 0.13 eV; and Δ2 = 0.35 eV. The results are shown in Figures 1−6 and discussed below.

(15a)

⎡ dμ(r ; T ) ⎤ =0 ⎢ ⎥ ⎣ dr ⎦(r = 0) ⎡ dμ(r ; T ) ⎤ N− = − O2 ⎢ ⎥ ⎣ dr ⎦(r = R) χR

(15b)

Function Lim(x) in eq 15a is the polylogarithm special function, m−1 which is defined as Lim(x) = −1/(Γ(m))∫ ∞ /(1 − x 0 dy[y exp{y})].30 In the system of eqs 15 and 15a, the unknown functions are the total charge n(r) and the chemical potential μ(r;T), with the latter being used instead of the conduction electron concentration nc(r), while both quantities are connected via eq 6. The numerical parameter λ in eq 15a can be found from the normalization condition of eq 2. The normalization condition of eq 14 is automatically satisfied in the boundary conditions of eq 15b, which can be verified by a single integration of eq 15. Equations 15−15b are solved numerically by the finite difference pseudoviscosity method that involves a stationary solution of the time-dependent differential equation, which is obtained by introducing a time derivative dμ/dt to the righthand side of eq 15. The stationary solution is obtained in the limit t → ∞.31 The time-dependent solution of eq 15 is sought by implementing a fully implicit scheme with linearization of the right-hand side and by using a sweep method at each time step.31 The iterative process is implemented to satisfy the normalization conditions involving the parameter λ. The entire calculation proceeds as follows. First, the functions n(r) and μ(r,T) are determined by solving eqs 15−15b at fixed values of the parameters Nc and N−O. Then, the conduction electron concentration nc and the concentration of ionized donors n+(r) are calculated using eqs 6 and 8a. To find the values of Nc and N−O, the total free energy F is minimized with respect to these parameters.

Figure 1. Top panel: The radial dependence of the electron nc(r) and ionized donor n+(r) concentrations normalized by the average concentrations, ⟨nc⟩ and ⟨nc+⟩; bottom panel: the electrostatic potential ϕ; all values are calculated for a specific case of a nanoparticle with average radius R = 20 nm and temperature T = 500 K.

Our calculations demonstrate a substantial spatial variation of the charge concentrations and electrostatic potential inside semiconductor nanoparticles in the subsurface region. Figure 1 shows radial dependence of the conduction electron nc and positively charged oxygen vacancy donor n+ concentrations, and the resulting electrostatic potential for a specific case of a nanoparticle with radius R = 20 nm and temperature 500 K. Although both electron nc and ionized donor n+ concentrations are almost constant in the interior of the spherical nanoparticle, a substantial decrease of nc and increase of n+ are observed in the layer of the material close to its surface. The mobile nature of conduction electrons is responsible for maintaining local charge neutrality in the core region of the nanoparticle, explaining almost constant values of nc, n+, and ϕ at r/R ≤ 0.8. One of the measures of inhomogeneity is the ratio of electron (ionized donor) concentrations n(r/R = 0)/n(r/R = 1) at the center r = 0 and at the surface r = R of the nanoparticle.

F(Nc , NO−) = ⎡⎣F1(nc(r )) + F2(Nc , NO− , nc(r ), n+(r )) + F3(Nc , NO− , n+(r )) + F4(NO−)⎤⎦

(17a)

■

(15)

}

Li3/2(exp{μ(r ; T )/kT )

(R + d)3 + 2R2(R + 3d) ⎤ ⎥ R+d ⎦

at R ≤ r ≤ R + d. Equations 17 and 17a are obtained from eqs 8 and 15 using the boundary conditions of eq 9 at r = 0.

3/2

+

(17)

NO− ⎡ r 3 + 2R2(R + 3d) ⎢ r 6χdR2 ⎣ −

nd

n(r ) =

6χR2(R + d)

at 0 ≤ r ≤ R, and

(14)

d ⎡ 2 dμ(r ; T ) ⎤ 4πr 2 n(r ) = 0 ⎢r ⎥+ ⎦ dr ⎣ dr χ

NO−d(3R − d)

(16)

Finally, the net electrostatic potential is calculated using the following expressions: 16289

DOI: 10.1021/acs.jpcc.5b01410 J. Phys. Chem. C 2015, 119, 16286−16292

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Figure 5. Radial dependence of the electrostatic potential inside the nanoparticle with 25 nm radius at different temperatures: 300, 350, 400, 450, 850, and 1000 K.

Figure 2. (a and b) Concentration ratios: (a) nc(0)/nc(R) and (b) n+(0)/n+(R), and (c) the total number of oxygen negative ions N−O as a function of nanoparticle radius R at temperature T = 500 K.

Figure 6. Radial distribution of the concentration of conduction electrons inside the nanoparticle calculated using the current model and that of refs 7 and 11, which assumes n+ = const.

asymptotically approaches the value R = ∞, which corresponds to a planar surface of the material. At large radii of nanoparticles, the total number of negative oxygen ions N−O exhibits quadratic dependence (see Figure 2c), as the surface concentration of the charges n−O = N−O(R)/4πR2 is almost constant and is close to the equilibrium charge concentration at the planar surface of the bulk semiconductor. The temperature dependence of the ratio of concentrations nc(r/R = 0)/nc(r/R = 1) and n+(r/R = 0)/n+(r/R = 1) is nonmonotonic, with their respective maxima and minima being achieved at Tmax ≈ 400 K; see Figures 3a and 3b, respectively. Such a trend can be correlated with the nonmonotonic temperature dependence of surface charges N−O(T) (see Figure 3c), which exhibits a maximum due to the imbalance of positive and negative charges inside the particle and, therefore, exhibits the largest inhomogeneity in the charge distributions. The dependence of the total amount of conduction electrons on the temperature Nc(T) for a nanoparticle of radius R = 25 nm is shown in Figure 4. It does not display the exponential dependence characteristic of bulk semiconductors, as it substantially depends on the number of surface charges N−O in this case. As is seen in Figure 5, the electrostatic potential ϕ is constant within the interior of the nanoparticle, which is a consequence of the charge neutrality (n+ = nc). However, uncompensated positive charge near the surface layer of the nanoparticle produces a radial electric field, resulting in a corresponding drop in electrostatic potential; see Figure 5. The thickness of this surface layer is almost independent of nanoparticle radius and is equal to the thickness of the potential drop within the subsurface region of the bulk semiconductor. The net chemical potential of the electrons inside the nanoparticle, μtot = μ(nc(r)) − φ(r), is constant, as it should be at equilibrium, which demonstrates the accuracy of our calculations. Interestingly, μtot is almost independent of nanoparticle radius R. For example,

Figure 3. (a and b) Ratio of concentrations: (a) nc(0)/nc(R) and (b) n+(0)/n+(R), and (c) the total number of oxygen negative ions N−O as a function of temperature for a specific case of nanoparticles with average radius R = 20 nm.

Figure 4. Temperature dependence of the total number of the conduction electrons in the nanoparticle of radius R = 25 nm.

In particular, nc(r/R = 0)/nc(r/R = 1) ≈ 34 and n+(r/R = 0)/ n+(r/R = 1) ≈ 0.07 for the case shown in Figure 1. The ratios are mostly decreasing for nc and increasing for n+ as functions of the nanoparticle radius R, although nonmonotonic behavior is observed at large radii R (see Figure 2). In this case, the ratio 16290

DOI: 10.1021/acs.jpcc.5b01410 J. Phys. Chem. C 2015, 119, 16286−16292

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The Journal of Physical Chemistry C μtot increases by 0.013 eV (or ∼150 K) upon increase of R from 5 to 50 nm at 500 K. As the conduction electron concentration near the surface of the nanoparticle is also independent of its radius, the electron transfer between nanoparticles of different radii is negligible. But there are always mutual charging effects in the system of nanoparticles at thermal equilibrium associated with statistical redistribution of additional electrons over the nanoparticles to increase the entropy of the system.32−35 Such entropically driven charging is independent of the surface charges due to oxygen surface adsorbates O−. To demonstrate the critical influence of inhomogeneity in the distribution of positive and negative charges, we calculated the spatial distribution of the conduction electrons under the assumption of refs 7 and 11  constant concentration of positively charged ionized donors n+ homogeneously distributed over the volume of the nanoparticle. A specific example of a nanoparticle with radius R = 25 nm, total number of electrons in the surface traps N−O = 195, and the temperature T = 500 K is considered. Figure 6 displays the radial dependence of the concentration of the conduction electrons calculated using our model and that of refs 7 and 11, which assumes that n+ = const. At small radii (r/R ≤ 0.8), the electron concentration in our model is three times larger than that of refs 7 and 11. The difference becomes even bigger at large radii by 5 orders of magnitude: the relative concentration of the conduction electrons at the surface of the nanoparticle is 0.033 (our model) and 1.0 × 10−7 (model of refs 7 and 11). Such a large difference is not unexpected, as in our model there is a substantial pileup of positive charge at the surface of the nanoparticle (top panel of Figure 1), which causes an additional inflow of the electrons due to attractive Coulomb interaction between positive and negative charges.

δ≈

U (x ) =

(22)

■

CONCLUSION The theory of inhomogeneous charge distribution inside a quasi-spherical semiconductor nanoparticle has been developed. A fraction of the electrons captured by the surface traps produces the nonuniform distribution of positive and negative charges that causes the electric field inside the nanoparticle. The spatial distributions of charges and the resulting potential were obtained by solving equations obtained by minimizing the free energy of the system. This allowed us to investigate the charge distributions and resulting electrostatic potentials, as well as the negative surface charge, as functions of nanoparticle radius and temperature. Such inhomogeneous effects have a pronounced influence on various electrical and physicochemical properties of nanostructured films, including the electrical conductivity associated with the electron transfer between nanoparticles, the sensor properties of semiconductor nanoparticles, the absorption and scattering of electromagnetic waves, as well as dielectric properties, to mention a few.

(18)

where Nc is the total number of the conduction electrons inside the nanoparticle of radius R. Assuming constant concentration of positive charges (ionized oxygen vacancies), the thickness of the positively charged subsurface layer can be obtained by the condition that the uncompensated total positive charge of the sublayer is equal to the total number of surface electron traps 4πR2n−O:

■

ASSOCIATED CONTENT

S Supporting Information *

The derivation of the functional derivative of the free energy functional F2. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.jpcc.5b01410.

(19)

In most cases, the thickness δ of the subsurface layer is much smaller than the radius of the nanoparticle R,

δ = R − Rn ≪ R

1 2 n+x 2

The potential energy drops to zero at the surface of the nanoparticle. The potential barrier experienced by the electrons upon transfer from the interior to the nanoparticle surface is due to the formation of an electron depleted subsurface layer similar to the Schottky barrier considered in ref 7. However, due to inhomogeneous charge redistribution effects (see Figures 1 and 2), the magnitude of this barrier and its spatial dependence are drastically different from those of the Schottky barrier. In the case of nonspherical nanoparticles, a quasi-spherical charge-neutral area still forms inside the nanoparticle. The local thickness of the positive charge subsurface layer is proportional to the local surface concentration of negative charge. It is reasonable to assume that the concentration of adsorbed O atomic traps is proportional to the local curvature of the surface of the nanoparticle, because of the concentration of surface defects, and to assume that the concentration of the adsorption sites is also proportional to the curvature. Then, the thickness of the positively charged layer will be larger near surface areas with higher curvature. For a closed surface with a positive curvature, the surface sites with greater curvature are located farther from the center of the particle. This is also true for the quasi-spherical shape of the neutral zone.

QUALITATIVE DESCRIPTION OF THE CHARGE DISTRIBUTION AND THE POTENTIAL INSIDE THE NANOPARTICLES The general features of the electron distribution inside a spherical nanoparticle containing surface electron traps can be qualitatively described using a simple model. The electric field due to the surface charge is zero inside the nanoparticles, while the conduction electrons are redistributed to maintain charge neutrality in the core region of the nanoparticle. The radius of such a region, the so-called charge neutrality radius Rn, can then be defined as

1 3 (R − R n3)n+ = R2nO− 3

(21)

As the charge density n+ is constant in the region Rn ≤ r ≤ R, the electrostatic potential energy of the electron depends quadratically on the distance x from the charge-neutral core r ≤ Rn :

■

4π 3 R n nc = Nc 3

nO− n+

■

(20)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (I.I.O.).

therefore, it follows from eq 19 that 16291

DOI: 10.1021/acs.jpcc.5b01410 J. Phys. Chem. C 2015, 119, 16286−16292

Article

The Journal of Physical Chemistry C Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank V. L. Bodneva and M. I. Ikim for their help and useful discussions. This work was supported by the National Science Foundation (Grant No. CMMI-1030715), and the Russian Scientific Foundation under Grant Number 1419-00781.

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DOI: 10.1021/acs.jpcc.5b01410 J. Phys. Chem. C 2015, 119, 16286−16292