Effects of Localized Surface Charges on a Two-Dimensional Potential

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Effects of Localized Surface Charges on a Two-Dimensional Potential Distribution and Photovoltage at a Schottky Barrier of a Semiconductor Electrode Yoshihiro Nakato* The Institute of Scientific and Industrial Research (ISIR), Osaka University, 8-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan ABSTRACT: Although experimental studies have long been reporting the importance of a beneficial effect of a metal nanodot coating of a semiconductor electrode for solar energy conversion, little theoretical work has been done on quantitative analysis of the effect most probably because of mathematical difficulty in calculating a two-dimensional potential distribution in the space charge layer induced by localized surface modifications or charges. This paper reports a simple method for calculating such a two-dimensional potential distribution and clarifies how the Schottky-barrier height (or flat-band potential) is affected by localized surface charges of various sizes and separations. The results not only give an unambiguous theoretical basis to the beneficial effect of a nanodot coating but also provide effective information for analysis of kinetics in semiconductor devices with nanostructures.

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INTRODUCTION Photoelectrochemical or photocatalytic solar energy conversion by the use of a semiconductor electrode (or semiconductor/ electrolyte interface) has been attracting growing attention recently from the point of view of development of renewable energy resources. Experimental studies have gradually revealed that most naked semiconductor electrodes lack sufficient catalytic activity for reactions suitable for solar to chemical conversion, such as hydrogen evolution, oxygen evolution, and carbon dioxide reduction, and thus surface modification with certain catalytic materials is indispensable for achieving high efficiencies. An interesting finding in this respect is that sparse deposition of nanosized metals or other catalytic materials on semiconductor surfaces leads to the generation of very high photovoltages, in sharp contrast to continuous homogeneous deposition, and can be used for efficient solar energy conversion. Such a beneficial effect of nanodot coating is clearly demonstrated in n-1−3 and p-type4,5 silicon (n-Si and pSi) electrodes modified with metal nanodots. The effect can also give a reasonable explanation to many of reported highefficiency photoelectrochemical (PEC) solar cells6−8 and photocatalysts.9−11 Very recently, another interesting example was reported by Abe et al.,12−14 who showed that efficient and stable water splitting into hydrogen and oxygen by visible (solar) light can be achieved with n-type tantalum oxynitride (n-TaON) electrodes modified with nanosized iridium- or cobalt-oxide particles. A Schottky barrier at the semiconductor/electrolyte interface plays a key role in photoelectrochemical and photocatalytic energy conversion.15,16 The modification of a semiconductor surface with metal dots generates an inhomogeneous distribution of surface charges, which in turn causes an inhomogeneous modulation of electric potential in the space © 2012 American Chemical Society

charge layer of a Schottky barrier and strongly affects the conversion efficiency. In spite of many experimental studies on the beneficial effect of metal nanodot coating,1−14 little theoretical work has been done on quantitative analysis of the effect, except for our previously reported qualitative model.1−5 In fact, the analysis of a potential distribution in a Schottky barrier has, in most cases including solid-state devices,17 been carried out by use of a one-dimensional model under the assumption that the surface is homogeneous. The present paper reports a simple method for calculating a two-dimensional potential distribution in the space charge layer of a semiconductor electrode and clarifies how a Schottky barrier is modulated by localized surface charges and how the modulation affects the conversion efficiency.

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CALCULATION METHOD In general, a potential distribution for a given charge distribution is obtained by solving a Poisson equation in electrostatics. However, the application of this method to a semiconductor electrode with inhomogeneous localized surface charges leads to a complicated mathematical solution for a potential in the space charge layer, which is difficult to interpret. Therefore we in the present paper adopt another method, namely, we calculate the potential, using the principle that potentials due to various charges can be superposed. First, let us consider a semiconductor electrode with a homogeneous surface charge distribution, to which a onedimensional potential model can be applied, as a preliminary study. Surface band energies (or the flat-band potential, Ufb) for Received: October 22, 2012 Revised: December 6, 2012 Published: December 10, 2012 427

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surface band energies (or Ufb) shift with pH at the slope of −(kT/q) ln 10 (= −0.059 V at 300 K) if the term, ln(γSO−/ γSOH){θ/(1 − θ)}, is regarded as constant, in agreement with experimental results. This condition is met when the density of surface OH group, N, is enough high, say, 1015 cm−2, and θ is not close to 0 or 1.0, as explained below. The Helmholtz layer can be regarded as a parallel plate condenser and hence ΔϕH,i can also be expressed as follows:

a semiconductor electrode are in general determined by the electron affinity (or work function) of the semiconductor and a potential drop, ΔϕH, in the Helmholtz layer, the latter of which is determined by various surface charges such as charges of adsorbed ions, electronic charges at surface states, and surface bond dipoles. Figure 1A schematically illustrates a charge

ΔϕH,i = −qθN /C H

where CH is the capacitance of the Helmholtz layer. As CH can be taken to be about 1.0 × 10−5 F cm−2, if N is 1015 cm−2, a change in ΔϕH,i of the order of (kT/q) ln 10 is attained by only a small change of a few tenth % in θ, which causes little changes in ln(γSO−/γSOH){θ/(1 − θ)}. Further details are discussed elsewhere.18,19 In case a semiconductor electrode has a (chemically stable) surface state (SS) as well as surface OH group and is in contact with an aqueous electrolyte containing a redox couple composed of an oxidant (Ox) and a reductant (Ox−), the surface band energies (or Ufb) are determined by the redox potential of the redox couple through electronic equilibrium between SS and redox couple.17,20,21

Figure 1. Charge distributions (A) and a corresponding potential distribution (B) for an n-type semiconductor electrode with a surface state as well as surface OH group and in contact with an aqueous redox electrolyte. Panel C is a potential distribution in the absence of a surface state.

SS + Ox → SS+ + Ox −

μSS ̅ + μOx ̅ = μSS ̅ + + μOx ̅ −

(6)

which gives the potential drop, ΔϕH,SS, between the position (or plane) of SS or SS+ and the electrolyte, expressed as follows: 0 0 ΔϕH,SS = (1/q)[{μOx − μOx − + kT ln(aOx / a Ox −)}

+ {μSS0 − μSS0+ + kT ln(γSS/γSS+)(1 − θ )/θ }] 0 = Uredox − USS + (kT /q) ln(γSS/γSS+){(1 − θ )/θ }

(7)

(1)

where θ is the ratio of the density of SS (NSS ) against the total density of a surface state (N), i.e., NSS+ = θN, γSS+ and γSS are the activity coefficients for SS+ and SS, and aOx and aOx− are the activity for Ox and Ox−. Uredox and USS are redox potentials of a redox couple in solution and a surface sate, respectively. Note here that Uredox in eq 7 is kept nearly constant when a semiconductor is brought into contact with a redox electrolyte, because the density of an oxidant in the electrolyte, say, 6 × 1019 cm−3 (0.1 M), is much higher than the density of SS+, which is 1015 cm−2 at most, and the term (aOx/aOx−) is little affected by the progress of reaction 5. Thus we can see from eq 7 that ΔϕH,SS and hence the surface band energies (or Ufb) in this case are determined by Uredox and shift in parallel with an alteration in Uredox if the term, ln(γSS/γSS+){(1 − θ)/θ}, can be regarded as constant. It can be shown by the same argument as earlier that this condition is met if the total density of a surface state (N) is enough high. The above conclusion means that the surface band energies (or Ufb) in this case are pinned by a +

where S−OH refers to surface OH group. This is because the equilibrium determines the potential drop, ΔϕH,i, between the position (or plane) of S−OH or S−O− and the electrolyte by the following equilibrium condition μSOH = μSO (2) ̅ ̅ − + μH ̅ + Here, the quantity, μ̅α, refers to the electrochemical potential of a species, α. Equation 2 can be rewritten as follows: 0 0 ΔϕH,i = (1/q)[(μSO + μH0+ ) − kT (ln 10)pH − − μ SOH

+ kT ln(γSO−/γSOH){θ /(1 − θ )}]

(5)

In fact, experiments show that the surface band energies (or Ufb) for an n-GaP electrode in an aqueous electrolyte containing a Fe(CN)63‑/Fe(CN)64‑ couple are determined by the redox potential of the redox couple, independent of pH, though they shift with the solution pH at the slope of −0.059 V/pH in the absence of the redox couple.21 This fact can be explained by taking into account the equilibrium condition

distribution in a Schottky barrier of an n-type semiconductor electrode, which has a (chemically stable) surface state as well as surface OH group and is in contact with an aqueous electrolyte containing a redox couple. Figure 1B represents a corresponding potential distribution, while (C) does a potential distribution in the absence of a surface state, for reference. The summation of all of the charges including space charges and charges at the outer Helmholtz layer is zero by electroneutrality. The potential drops in the Helmholtz layer, ΔϕH,i and ΔϕH,SS, shown in Figure 1B,C are determined as follows. Experiments show that surface band energies (or Ufb) for semiconductor electrodes with surface OH group, such as Ge, GaAs, GaP, InP, ZnO, and TiO2, shift with the solution pH at the slope of −0.059 V/pH at 300 K. The shift can be explained as due to ion adsorption equilibrium15,16 such as S − OH → S − O− + Haq +

(4)

(3)

where q is the elementary charge (q > 0), k the Boltzmann constant, T the temperature, μ the chemical potential, θ the coverage of S−O−, γSOH and γSO− are the activity coefficients for S−OH and S−O−, and the superscript “0” refers to the standard state. Equation 3 indicates that ΔϕH,i and hence 428

+

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surface state, which is in electronic equilibrium with a redox couple in the electrolyte. Now, let us consider a semiconductor electrode with an inhomogeneous surface charge distribution. Figure 2A

Figure 3. Inhomogeneous surface charge distribution in which a surface state-free area is localized in a small area.

distribution such as Figure 2C or 3C causes the modulation of potential in the space charge layer, as mentioned earlier. The modulation can be calculated by using a diagram of Figure 4.

Figure 2. Charge distribution (A) for a semiconductor electrode with a surface state in a small localized area. Charge distribution (A) can be expressed by the superposition of three partial distributions (B−D).

schematically displays a charge distribution for a semiconductor electrode of the same type as Figure 1A, except that a surface state is present only in a small localized area. Space charges and corresponding counter charges at the outer Helmholtz layer are not shown for simplicity. It is clear from Figure 1B,C that ΔϕH and hence the surface band energies for such an electrode are modulated by the presence of a localized surface state, thus resulting in the modulation of potential in the space charge layer. A two-dimensional potential distribution in the space charge layer thus formed can be calculated by expressing the charge distribution in Figure 2A as the superposition of three partial charge distributions; 2(B), (C), and (D). Charge distribution (B) is formed by ion adsorption equilibrium in surface OH group, while (C) is generated by electronic equilibrium between a localized surface state and a redox couple in the electrolyte. Charge distribution (D) is a small correction to charge distributions (B) and (C), which arises from the fact that both ΔϕH,i in a surface state-free area and ΔϕH,SS in a surface state-covered area are kept constant along the surface, because they are determined by ion adsorption equilibrium and electronic equilibrium, respectively, independently of each other. If the density of surface OH group and that of a surface state are both sufficiently high, the area of charge distribution (D) is very narrow, less than 0.5 nm, as shown in the figure, because charge distribution (D) is of a similar type to one in an accumulation layer of a semiconductor22 or a Gouy layer of a high-concentration electrolyte. Note also that the charge density in (D) is much lower than that in (B) and (C). Figure 3 illustrates another type of inhomogeneous surface charge distribution, in which almost all parts of the semiconductor surface are covered with a surface state, contrary to the case of Figure 2, and a surface state-free area is localized in a small area. The surface band energies for such an electrode are also modulated by the presence of a surface state. The induced modulation of a potential in the space charge layer (or a twodimensional potential distribution in it) can be calculated by expressing the charge distribution in Figure 3A as the superposition of three partial charge distributions (B−D), in a similar way to Figure 2. The charge distribution in Figure 2B or 3B is of a conventional type, i.e. of the same type as in a parallel plate condenser and only gives a constant potential in the space charge layer. On the other hand, an inhomogeneous charge

Figure 4. Diagram for calculating a potential in the space charge layer, generated by surface charge (C) of Figure 2 or 3.

Let us assume that the area in which a surface state is present in Figure 2C or a surface state is absent in Figure 3C has the circular form of the radius R and that the plane of a surface state is separated from that of adsorbed ions by a distance, d (d ≤ 0.3 nm). The effect of such an inhomogeneous charge distribution is largest on the line, OP, which is normal to the plane of a surface state and passes the center, O, of the above-mentioned circular area where a surface state is present or absent. Therefore we calculate the potential at a point on this line. In the case of Figure 2C, the potential at a point, P, located at a distance, x, from the center, O, is, under the approximation of x ≫ d, given as follows: ϕ2C(x , R , d) = (σ /4πε0ε)

∫0

R

[(x 2 + r 2)−1/2

− {(x + d)2 + r 2}−1/2 ]2πr dr ≅ (σd /2ε0ε)

∫0

R

rx(x 2 + r 2)−3/2 dr

(8)

where σ is the areal density of charges of a surface state (C cm−2), ε0 the permittivity of vacuum, ε the dielectric constant of the semiconductor, and r is a variable representing the radius of a circular belt (see Figure 4). The integration can be performed and gives ϕ2C(x , R , d) = (σd /2ε0ε)[1 − {1 + (R /x)2 }−1/2 ]

(9)

Similarly, in the case of Figure 3C, the potential at a point, P, is obtained by performing the integration in eq 8 in the range from R to ∞. 429

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∫R

∞

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in ϕ3C(x, R, d) is very sharp at small R, e.g., at R = 2 nm, but becomes quickly gentle as R gets larger. Let us now consider the effect of localized charge distributions such as Figures 2C and 3C on the height of a Schottky barrier and photovoltage at a semiconductor electrode, the latter of which is in proportion to the conversion efficiency. As discussed earlier, the surface band energies or ΔϕH for a surface state-free or a surface state-covered area are determined by ion adsorption equilibrium or electronic equilibrium, respectively, and fixed against the potential of the electrolyte. Thus, the main issue is how the potential or energy bands in the space charge layer are distributed under such a fixed modulation of surface band energies or ΔϕH. Figure 7 schematically illustrates potential distributions and

[(x 2 + r 2)−1/2

− {(x + d)2 + r 2}−1/2 ]2πr dr ≅ (σd /2ε0ε){1 + (R /x)2 }−1/2

(10)

A charge distribution in Figure 2D or 3D is a small correction to charge distributions (B) and (C) and the charge density in the former is much lower than that in the latter, as mentioned earlier. Therefore we in the present work neglect a contribution of charge distribution (D) to the potential modulation.

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RESULTS AND DISCUSSION Figure 5 plots the integrand of eq 8, i.e., rx (x2 + r2)−3/2, as a function of x, with r taken as the parameter. Interestingly, the

Figure 5. Plots of the integrand of eq 8, i.e., rx (x2 + r2)−3/2, as a function of x, with r taken as the parameter.

integrand shows the peak at x = r/√2, though it becomes rapidly lower and broader as r gets larger. Figure 6 shows the result of integration in eq 8, that is, ϕ2C(x, R, d)/(σd/2εε0) or [1 − {1 + (R/x)2}−1/2] in eq 9. In contrast

Figure 7. Potential distributions (A) and band bending (B) in surface state-free (solid line) and surface state-covered (broken line) areas for an n-type semiconductor electrode of the type of Figure 2. Ec refers to the bottom of the conduction band. See text for other details.

band bending in surface state-free (solid line) and surface statecovered (broken line) areas for an n-type semiconductor electrode of the type of Figure 2. The width of the space charge layer in general ranges from a few hundreds to 1000 nm and thus the potential due to space charges very gradually changes with x and can be regarded as nearly constant in the range of small x. On the other hand, the potential ϕ2C(x, R, d), generated by a localized surface state, for small R sharply decays with x toward the interior of the semiconductor. For example, for R = 2 nm, ϕ2C(x, R, d) at x = 2R (= 4 nm) is only one tenth of the value at x = 0. Thus, the “effective” energy barrier, qϕBeff, for an n-type semiconductor electrode of the type of Figure 2 with small R is nearly the same as the energy barrier, qϕB, for a surface state-free electrode, as shown in Figure 7. This means that the photovoltage and hence the conversion efficiency are hardly affected by a localized surface state with small R. Quantitatively, the open-circuit photovoltage (Voc) for a photoelectrochemical solar cell with an n-type semiconductor electrode is given by1,2,17

Figure 6. Plots of ϕ2C(x, R, d)/(σd/2εε0), i.e., [1 − {1 + (R/x)2}−1/2] as a function of x, with R taken as the parameter.

to the integrand of eq 8 shown in Figure 5, the potential ϕ2C(x, R, d) only monotonously decays with x. In addition, the slope of decay in ϕ2C(x, R, d) is very sharp at small R, e.g. at R = 2 nm, much sharper than that of the integrand of eq 8. However, the slope becomes quickly gentle as R gets larger, similar to the integrand of eq 8, finally becoming flat at R = ∞. This result is reasonable because the potential outside a parallel plate condenser consisting of infinitely extending plates is constant. Note also that eq 9 indicates that the value of ϕ2C(x, R, d) at R = ∞ is (σd/2ε0ε), which is half the potential difference in a parallel plate condenser, as expected. The behavior of ϕ3C(x, R, d) or (σd/2ε0ε) {1 + (R/x)2}−1/2 in eq 10 can be understood easily from Figure 6, which plots the function, [1 − {1 + (R/x)2}−1/2]. The potential ϕ3C(x, R, d) monotonously increases with x and reaches the value of (σd/ 2ε0ε) at x = ∞. Naturally it also reaches (σd/2ε0ε) at R = 0, i.e., in the absence of a surface state-free area. The slope of increase

Voc = (kT /q) ln(jp /j0 + 1) ≅ (kT /q) ln(jp /j0 )

(11)

where jp is the photocurrent density and j0 the saturation current density. The j0 is composed of the majority-carrier saturation current density j0n and the minority-carrier one j0p, i.e., j0 = j0n + j0p. In general, for a Schottky-barrier solar cell, j0n is much higher than j0p and thus VOC is mainly determined by j0n.1,2 Also, the j0n for a Schottky barrier can be well described by thermoionic emission current density 430

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height Δϕ (and hence the lowering of VOC) by a localized surface state for ϕm = 0.3 V is of the order of magnitude of kT/ q and negligibly small when the areas of a localized surface state are very narrow, say, 4 to 5 nm in radius, and sparsely distributed, though it increases with R and ϕm. On the other hand, if the areas of a localized surface state are very narrow but densely distributed, namely, if not only surface state-covered areas but also surface state-free ones are very narrow, the modulated potential in both areas sharply decays toward the interior of the semiconductor, as can be seen from the behavior of ϕ2C(x, R, d) and ϕ3C(x, R, d). Thus the potential drop in the Helmholtz layer takes a value between ΔϕH,SS and ΔϕH,i, depending on the ratio of sizes of these areas. The effective barrier height (or Ufb) in this case is then considerably affected by the presence of a surface state even though its areas are very narrow. If both surface state-covered and surface state-free areas are sufficiently wide, e.g. as wide as or wider than the width of the space charge layer, the potential modulation by a localized surface state decays very gradually toward the interior of the semiconductor and is distributed over the whole space charge layer. Thus, such a semiconductor electrode behaves like a mixture of surface state-covered and surface state-free electrodes. Finally, note that the foregoing discussions for a semiconductor electrode with a localized surface state equally apply to one modified with localized metal dots. The surface band energies for a semiconductor electrode of the latter type are also modulated by the presence of metal dots, because the surface band energies in a naked part of the semiconductor surface are determined, e.g. by ion adsorption equilibrium while those in a metal-coated part are determined by the Fermi level of metal dots, which are in electronic equilibrium with a redox couple in the electrolyte, and the barrier height at the semiconductor/metal contact, independently of each other. Figure 9A schematically illustrates an inhomogeneous surface

(12)

where A* is the effective Richardson constant and qϕB the barrier height.17 Thus, we obtain Voc = ϕB + const

(13)

This means that the lowering of qϕB by a localized surface state directly leads to the lowering of VOC and hence the conversion efficiency. The lowering of qϕB by a localized surface state can be estimated as follows. As can be seen from Figure 7, in which the potential at the plane of adsorbed ions is taken to be zero, the potential in the space charge layer for a surface-state covered area can be expressed by the sum of the potential due to space charges and ϕ2C(x, R, d). ϕ = (qND/2ε0εs)(2Wx − x 2) + ϕm[1 − {1 + (R /x)2 }−1/2 ]

(14)

where ND is the donor density of the n-type semiconductor, εs the dielectric constant of the semiconductor, W the width of the space charge layer, and ϕm the potential due to charges in a surface state at the semiconductor surface. Note that the potential ϕ takes the minimum at x = x′ and the ϕ value at the minimum gives the lowering of the barrier height (qΔϕ = qϕB − qϕBeff). The value of x′ can be calculated from the condition, dϕ/dx = 0. Under the approximation of x/W ≪ 1, we obtain x′ = R[p − 1]1/2

(15)

p = (ε0εsϕm /qNDWR )2/3

(16)

Thus the lowering of the barrier height qΔϕ by a localized surface state is given as follows: Δϕ = (ϕB − ϕBeff ) = ϕ(x = x′) = (qNDW /ε0εs)R [p − 1]1/2 + ϕm[1 − (1 − 1/p)1/2 ]

(17)

−3

For n-Si with ND = 10 cm and εs = 12 in the presence of band bending ΔU of 0.4 V (a value expected for the maximum power point of a solar cell), W [= {(2ε0εs/qND)(ΔU − kT/ q)}1/2] is calculated to be 730 nm at 300 K, and hence, Δϕ is represented as 15

Δϕ = 1.10 × 10−3R[93.43(ϕm /R )2/3 − 1]1/2 + ϕm[1 − {1 − 0.011(R /ϕm)2/3 }1/2 ]

(18)

where the unit of Δϕ and ϕm is V and that of R is nm. Figure 8 shows Δϕ calculated by eq 18 as a function of R for ϕm = 0.3 and 0.5 V. The figure indicates that the lowering of the barrier

Figure 9. Charge distribution (A) expected for a semiconductor electrode modified with small metal dots. Charge distribution (A) can be expressed by the superposition of partial distributions (B−D).

charge distribution expected for such an electrode. The modulation of a potential (or a two-dimensional potential distribution) in the space charge layer can be calculated by expressing the charge distribution in Figure 9A as the superposition of three partial charge distributions (B−D), in a similar manner to Figure 2 or 3. Charge distribution (C) is an imagined double layer introduced for deleting charges in (B) in this part, namely, the sum of charge distributions (B and C) gives a double layer similar to a charge distribution of Figure 3C. In charge distribution (D), charges at the semiconductor/ metal interface, which may arise from the presence of a surface state at this interface, are neglected for simplicity because they

Figure 8. Lowering of the barrier height Δϕ calculated for n-Si with ND = 1015 cm−3 and band bending ΔU = 0.4 V as a function of R, ϕm being taken as a parameter. 431

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(10) Kudo, A.; Kato, H.; Tsuji, I. Chem. Lett. 2004, 33, 1534−1539. (11) Maeda, K.; Teramura, K.; Lu, D.; Takata, T.; Saito, N.; Inoue, Y.; Domen, K. Nature 2006, 440, 295. (12) Abe, R.; Higashi, M.; Domen, K. J. Am. Chem. Soc. 2010, 132, 11828−11829. (13) Higashi, M.; Domen, K.; Abe, R. Energy Environ. Sci. 2011, 4, 4138−4147. (14) Higashi, M.; Domen, K.; Abe, R. J. Am. Chem. Soc. 2012, 134, 6968−6971. (15) Gerischer, H. Semiconductor Electrochemistry. In Physical Chemistry: An Advanced Treatise; Eyring, H., Henderson, D., Jost, W., Eds.; Academic Press: New York, 1970; Vol. 9A, Chapter 5, pp 463− 542. (16) Memming, R. Semiconductor Electrochemistry: Wiley-VCH: Weinheim, Germany, 2002; pp 81−111. (17) Sze, S. M. Physics of Semiconductor Devices: 2nd ed., John Wiley & Sons: New York, 1981; pp 245−311. (18) Gerischer, H. Electrochim. Acta 1989, 34, 1005−1009. (19) Nakato, Y. Chem. Lett. in press. (20) Bard, A. J.; Bocarsly, A. B.; Fan, F. F.; Walton, E. G.; Wrighton, M. S. J. Am. Chem. Soc. 1980, 102, 3671−3677. (21) Nakato, Y.; Tsumura, A.; Tsubomura, H. J. Electrochem. Soc. 1981, 128, 1300−1304. (22) Morrison, S. R. The Chemical Physics of Surfaces; Prenum Press: New York, 1977; pp 35−39.

have the same form of distribution as (C) and can be neglected (or included into (C)) without losing the generality of discussion. Charges at the metal/electrolyte interface have no effect on the potential in the space charge layer and can also be excluded from consideration because they are screened by the metal having the constant potential within it. The remaining charges in (D) are low-density ones, which arise from the fact that the surface band energies in naked and metal-coated parts of the semiconductor surface are kept constant along the surface in each part, and can also be neglected, as discussed earlier. Accordingly, the modulation of potential in the space charge layer for a semiconductor electrode modified with small metal dots is mainly caused by charge distribution (C) and can be treated in the same way as for an electrode of the type of Figure 2, because the (imagined) double layer in Figure 9C is separated only by a negligibly small distance (about 0.3 nm) from the double layer formed by charges at a surface state and adsorbed ions in Figure 2C. As mentioned in the Introduction, semiconductor electrodes modified with small metal dots generate very high photovoltages and yield high efficiencies,1−14 providing a promising approach to high-efficiency solar energy conversion. For such an electrode, the size and density of metal dots can be fairly well controlled experimentally.1−3,14 Thus, the analytical (or conceptual) understanding of the lowering of the barrier height (and photovoltage) by deposited metal dots as functions of various properties of electrodes, such as given in eq 17 and Figure 8, will be highly useful for development of efficient solar cells.

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CONCLUSIONS The present work has proposed a simple effective method for calculating a two-dimensional potential distribution in the space charge layer of a semiconductor electrode and clarified how the Schottky-barrier height (or Ufb) is affected by a localized surface state or metal dots of various sizes and separations. The results not only give an unambiguous theoretical basis to the experimentally observed beneficial effect of nanodot coating but also provide effective information for analysis of kinetics and operations in various semiconductor devices with nanostructures including solid-state ones.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

REFERENCES

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