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electrolyte interface with high density of grain boundaries
Thin Solid Films 405 (2002) 55–63
Potential distribution at the semiconductor thin filmyelectrolyte interface with high density of grain boundaries Carlos Alcober, Sara A. Bilmes* ´ ´ ´ ´ ´ INQUIMAE, Departamento de Quımica Inorganica, Analıtica y Quımica-Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pab. II, 1428 Buenos Aires, Argentina Received 27 February 2001; received in revised form 29 October 2001; accepted 1 December 2001
Abstract The electronic structure of an n-CdSyelectrolyte interface was analyzed by in situ surface conductance for CdS thin films obtained by chemical precipitation onto a conducting substrate. The films are compact, composed of crystallites and exhibit grain boundaries that affect the potential distribution on the semiconductor space charge layer. We propose an explicit evaluation of the charge carrier density, nL, taking into account the relation between the space charge layer length, LSC , and the crystallite length, L. Using this model, values of the interfacial and electronic parameters, such as the flat band potential, Ufb , the carrier mobility, NmM, and the electron density, n0 , were obtained from in situ surface conductance data. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Surface conductivity; Cadmium sulfide; Grain boundary; Electrochemistry
1. Introduction For charge transfer processes, such as those occurring in photoelectrochemical cells and in photocorrosion, the kinetics are driven by the properties of the interfacial layer formed at the semiconductoryelectrolyte junction. In particular, the space charge layer dimensions, Fermi level position, mobility and density of charge carriers are key parameters for the description of photoelectrochemical processes at compact, thin-film semiconductor electrodes. Although these parameters can be well determined for single crystals, their estimation for polycrystalline films enables the knowledge of boundaries, carrier traps and surface state density in the material. The contact between a single-crystal semiconductor electrode and an electrolyte gives rise to a space charge layer inside the electrode at the solidyliquid interface w1,2x. For polycrystalline film electrodes, this is not necessarily true, and depends on the material morphology and electronic properties. From a general point of view, two different situations can be distinguished. In * Corresponding author. Fax: q54-11-45763341. E-mail address: [email protected] (S.A. Bilmes).
porous electrodes, the semiconductor network is interpenetrated by the electrolyte, and interfacial properties are largely determined those of nanometer-sized structural units and by the very large fractal solidysolution interface w3–6x. Then, porous electrodes can be viewed as a collection of interconnected micro- or nanoelectrodes surrounded by the electrolyte, which, except for highly doped semiconductors, can develop a space charge layer only when free carriers accumulate at the interface w7,8x. Compact polycrystalline electrodes, on the other hand, can be thought of as single crystals considering the interface located at the macroscopic limit between the semiconductor and the solution. In this case, a space charge layer develops both in depletion and accumulation regimes, and it is possible to define a flat band potential, Ufb, when the net surface charge is zero w9–11x. However, the description of the surface space-charge layer and its effect on transport properties requires the material heterogeneity to be taken into account. Several experimental methods have been developed for the description of semiconductoryliquid junctions. Electrochemical impedance spectroscopy is the most widely employed, and experimental data for single
0040-6090/02/$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 1 . 0 1 7 5 1 - 5
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crystals are usually interpreted by the Mott–Schottky model w1,12x. Other techniques employing either optical or electrical perturbations have been successfully used for the study of single crystals w13–15x, but they involve significant uncertainties andyor simplifying approximations for the description of thin film semiconductors. A powerful method for the study of semiconductor filmy electrolyte interface is in situ surface conductance, which has been successfully used for Ge w16x and Si w17x single-crystal electrodes, although it has limited use for thin semiconductor films w5,18x. In previous work, we demonstrated that it is a suitable tool for the study of compact semiconductor thin filmyelectrolyte interfaces w19,20x, even for materials with fairly high resistivity. In situ surface conductance is proportional to the carrier mobility, m, and the surface free carrier density excess, nL: (1)
DsssysfbsemnL
where sfb is the surface conductance for nLs0. For compact materials, sfb corresponds to the surface conductance at flat band, and Ds gives a measure of the surface potential barrier VS. For porous semiconductors, the physical meaning of Ds is not straightforward, and is probably related to both ion and electron excess at the surface. In this work, we analyzed the CdS filmyaqueous electrolyte interface by in situ surface conductance. In order to interpret surface conductance data for CdS films electrodes in a wide potential range, we propose an approach for the explicit evaluation of nL and m as a function of VS for compact semiconductor films. The material heterogeneity introduces intercrystallite barriers and trapped charges, which are taken into account from the relation between the space-charge layer length, LSC, and the crystallite length, L. 2. The space charge layer for a polycrystalline semiconductor For a compact electrode in contact with an electrolyte, r , is calculated by solving the the internal potential, V ™
Ž.
three-dimensional Poisson equation: r = VŽ™ r. sy 2
Ž™r.
´r´0
(2)
with ´r the relative permittivity and the charge density, r™ r , given by:
Ž. rŽ™ r. seN
V
¯ .qeNSŽr,V ¯ .qeNLŽr,V ¯ . Žr,V
(3)
In Eq. (3), eNL and eNV are the free and bounded (in gap states) charge densities inside each crystallite and eNS is the localized charge density at the interface between the boundaries. It is important to stress that
this last term influences both the potential distribution due to the surface barrier and the control of charge transport through the film. Due to the inhomogeneous nature of polycrystalline materials, estimation of r ™ r
Ž.
requires detailed knowledge of the spatial distribution of crystallite boundaries and gap state density, which are generally unknown, except for simple systems, such as bicrystals w21x. However, by considering the relation between the space charge layer and crystallite length, nL can be evaluated for two limiting situations: LSC4L and LSC-L. 2.1. LSC4L (continuous model) In this case, it is possible to assume that the spatial distribution of electronic states, arising from both boundaries and the bulk of each crystallite, is uniform, and r no longer depends on spatial co-ordinates. Then, nL in Eq. (1) can be calculated as in a single crystal, with an appropriated density of gap states, DS(E). It should be noted that in this description, it is not necessary to take into account the particular transport mechanism, and m represents a phenomenological macroscopic parameter. Although m may depend on the surface potential VS w22,23x, this dependence can usually be neglected for potentials in the depletion region, because charge carriers move outside the space charge layer, and the surface barrier only modifies the effective thickness of the film. Then, in the continuous model, nL –VS plots give the surface conductance variation in the depletion region. On the other hand, in accumulation, it is necessary to take into account the dependence of m on VS, owing to the fact that carriers move inside the space charge region. In order to give an explicit dependence of nL on VS, we briefly review two defined situations for a nonnegligible density of states in the gap: (i) a single trap level; and (ii) a uniform density of traps. Complete calculations, both analytical and numerical, can be found in the literature w9,22–25x. The simplest case is for an n-type semiconductor in a semi-infinite (planar) geometry, with a parabolic density of states in the conduction band, for which nL is obtained by solving Eq. (1) in one dimension, with: rseNNdMye2py1y2NCF1y2Žvywb. NNtM qe 1qŽ1y2.exp wŽEtyEF.ykTxyv
µ
(4)
∂
for a monoenergetic trap level at energy Et, or rseNNdMye2py1y2NCF1y2Žvywb. EC NNtMdE qe Ei 1q Ž 1y2. exp w ŽEyEF. ykTxyv
|
µ
∂
(5)
for a continuous density of trap levels in the gap. In
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57
particular case of a single trap level, this deviation takes place for surface potentials (VS-y0.2 V in Fig. 1a), for which the trap energy, Et, is above the Fermi level at the surface. 2.2. LSC-L (localized model)
Fig. 1. Excess surface carrier density of CdS for different trap density in the gap: (a) single trap level, with Ets0.2 eV below conduction band; (b) uniform trap density. Calculations were carried out using ´rs8.7 and n0s2.08=1020 my3 with the trap density given in the figure.
Eqs. (4) and (5), wbs(EFyEC)ykT, vseVykT, NCs 3y2 2Ž2pŽm*nkTyh2.. , Nd is the donor density, F1y2 is the 1y2 Fermi integral; m*n is the electron effective mass; and e, k, T and h have their usual meaning. It should be noted that Nt in Eq. (4) is the density (per unit volume) of trap states, while in Eq. (5), Nt is the uniform density (per unit volume and energy) of trap states in the gap. From integration of the Poisson equation with the boundary condition Vs0 at x™`, the surface free-carrier excess density, nL, is w23,24x: 0
|
nLsLD
vS
2py1y2NCF1y2Žvywb.yn0 .FŽwb,v.
dv
In this case, only those crystallites that are in contact with the electrolyte need to be considered. Then, r has two contributions: free charges (eNL) and those trapped in bulk states (eNV). On the other hand, charges trapped at boundaries, eNS, give rise to interfacial barriers, which in turn give new boundary conditions for integration of the Poisson equation. The presence of these interfacial barriers introduces several effects. First, the barriers are depleted of mobile carriers, reducing the effective crosssectional area in which carriers can move. Another not so obvious effect can be observed with the help of a sketch of a bidimensional square grain in contact with an electrolyte under depletion conditions (Fig. 2). The region labeled ‘a’ is that occupied by the surface barrier, and those labeled ‘b’ and ‘c’ by the lateral and the back interfacial barrier, respectively. Applying the superposition principle, it can be shown that in depletion, for LD
(7)
where a is the fraction of grain length (along the xaxis) free of interfacial barriers was1y(2WyL), with W the width of the interfacial barriersx, and n*L is the surface carrier density excess in the same crystallite free of interfacial barriers, with carrier density n0 . n*L can be
(6)
where LDs(´r´0kTye 2n0)1y2; the minus and plus sign holds in depletion and accumulation respectively, and dv .FŽwb,v.sLD . dx Fig. 1 shows the nL –VS plots for (a) a single trap level and (b) uniform trap density obtained from Eq. (6), using parameters for CdS. In both cases, as the trap density increases, the density of free carriers removed from the surface decreases, and the dependence of nL on VS deviates from the simple parabolic dependence that represents the behavior for Nts0 w23,24x. In the
Fig. 2. Schematic barrier distribution in a square crystallite, with LSC-L. The semiconductoryelectrolyte interface is located along the x-axis. Region a is that occupied by the surface barrier, with b and c by the interfacial barriers.
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developed a model in which the conductivity for a crystal of length L can be written as a function of the microscopic electronic parameters: ks
e 2 n 0L expŽyEBykT. Ž2pmn*kT.1y2
(8)
with EB the interfacial barrier height. Under the depletion approximation, EB can be approximated by: E Bs
Fig. 3. (D) Excess surface carrier density for a 60=60-nm2 square crystallite and (—) that calculated using Eq. (7). Calculations were carried out using ´rs8.7 and n0s1.23=1024 my3. See the text for details.
e2L2Ž1ya.2n0 8´r´0
Taking into account the reduction in both the effective cross-sectional and surface area, the change in surface conductance for a film of thickness d composed of cubic crystallites of side L can be expressed in terms of two parameters, n0 and EB: sŽVS.ysfbsŽkayn0.nL* ŽVS.
calculated using the appropriate DS(E) that takes into account bulk gap states, using the expressions derived in the previous section (for a tridimensional cubic crystallite, nLsa2n*L). Fig. 3 shows a plot of nL –VS for CdS using Eq. (7) (solid line). It was calculated, as previously stated, using n0s1.23=1024 my3 and as 0.5. For the sake of comparison, nL values from numerical calculations for a bidimensional 60=60-nm2 square crystal taking into account interfacial barriers are included (symbols). The parameters for this calculations (n0s 1.23=1024 my3 and density of interfacial traps Nts 7.4=1016 my2) also correspond to as0.5. In both cases, it was assumed that trap density in the bulk is negligible. Comparison of both curves shows that Eq. (7) is a good approximation in depletion, whereas in accumulation, it predicts a lower surface carrier density than that calculated taking into account the interfacial barriers. This is because in accumulation, a is no longer constant, and the space charge layer expands, increasing a. This expansion can be observed through the equipotential lines inside the crystallite, as depicted in Fig. 4 for two VS values. As VS increases (i.e. increasing accumulation), the space charge layer expands more, increasing a, and hence the free-carrier density excess at the surface. For a complete description of Ds on VS, it is only necessary to express the microscopic mobility. Transport properties in polycrystalline CdS films w26–30x and other semiconductor materials w28,31–35x have been widely studied using Hall measurements in solid-state devices. Contrary to single-crystal materials, the charge transport in compact polycrystalline materials is limited by the presence of potential barriers located at the interface between crystallites, and it is mainly controlled by thermionic emission w28,32–35x (although tunneling is important in heavily doped semiconductors). For this particular case, Seto w33x and Baccarani et al. w34x
(9)
(10)
with sfbskda2. As in the continuous approximation, k is expected to be constant in depletion, and the dependence of Ds on VS is that given by nL* .
Fig. 4. Potential distribution in a square crystallite for two different surface potentials under accumulation. Surface potential in (a) is more positive than in (b). Calculations were carried out with the same conditions as in Fig. 3.
C. Alcober, S.A. Bilmes / Thin Solid Films 405 (2002) 55–63
Fig. 5. RX diffractogram for a 400-nm CdS film over glass. Miller indexes are indicated for the cubic (C) and hexagonal (H) crystallographic faces.
For the intermediate case, LSC(L, it is necessary to consider the interaction between the space charge layer and the intercrystallite barriers. This interaction is not merely an overlap between the surface barrier and the intercrystallite barriers, but a rather complex phenomenon; the surface barrier changes the trapped charge in the interface boundaries and modifies the height and extension of these barriers w21x. It is reasonable to assume that the extension of the barriers is also comparable to the grain size, L, and the grains are completely depleted of carriers. For this particular situation, the Fermi level is very close to the trap levels w34x, and chargeydischarge of trap levels is important. 3. Methods and materials All measurements were carried out with a homemade potentiostat in a standard three-electrode configuration. The counter electrode was a Pt plate. The reference electrode was either HgyHgSO4 (saturated K2SO4) or HgyHgO (0.1 M NaOH). All reagents were analytical grade or better. Solutions were prepared with Milli-Q water and all measurements were made in N2-saturated electrolytes. Potential values are quoted against the saturated calomel electrode (SCE).
59
Briefly, CdS films were prepared from an aqueous solution of 0.1 M Cd(NO3)2 and 0.1 M thiourea, which was maintained at 60 8C under vigorous stirring. The substrates were immersed in the solution and the reaction was initiated by slow addition of concentrated NH4OH up to a final concentration of 0.6 M. After 10 min, the reaction was stopped and the films cleaned in acetone in an ultrasonic bath. Thin layers of ca. 60 nm were successively deposited in order to achieve a mean thickness of ;0.40 mm. Film thickness was estimated by the Cd content in a known area of the film determined by atomic absorption spectroscopy. Between successive deposits, the samples were carefully cleaned with acetone in an ultrasonic bath to remove small particles that remain adhered to the surface. The conductivity of the samples was improved by annealing in air at 200 8C. Samples where then stored in a dry box. 3.2. Morphologic characterization Samples were characterized using RX diffractograms (Siemens diffractometer with CuKa radiation) and tunnel microscopy (Nanoscope III scanning tunnel microscope with a Pt–Ir tip). The RX diffractograms (Fig. 5) show that the films have a high degree of crystallinity, with a preferential growth in the direction of the cubic (1 1 1) or the hexagonal (0 0 2) planes. Analysis of the peak widths shows a mean grain radius of 10"2 nm. STM images were taken ex situ in the dark (Nanoscope III scanning tunnel microscope with a Pt–Ir tip, working at constant current under feedback control). All the samples show an inhomogeneous surface (see Fig. 6) and integration of the area in the STM images gave a roughness factor 3FRF4.
3.1. Sample preparation For conductance measurements, a strip ;30 mm wide of conducting substrate was removed by etching, leaving two conducting areas electrically isolated. The CdS was then deposited on the substrate, and contacts were made with silver paint at the edges of the conducting substrate w19x. In this arrangement, indium–tin oxide (ITO) provides the electrical contact to the film. The silver contacts and any exposed wire were masked with galvanoplastic lacquer. Thin CdS films were chemically deposited onto ITO glass (Balzers, 200 V h, 0.7=1.4 cm2), by the same procedure previously described w19x.
Fig. 6. STM image of a 400-nm CdS film on ITO. The image was taken with a bias of y3.18 V and a current set point of y1 nA. The x and y scales are 50 nm per division, while the z scale is 40 nm per division.
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3.3. Surface conductance measurements The experimental set-up was the same as described in a previous paper w19,20x. Briefly, the current Im was measured at each potential U, fixed by the potentiostat, for applied bias between sample contacts of Vas 0, q 5 mV (or q10 mV), y5 mV (or y10 mV). Each Im value was obtained 30 s after changing Vm in order to allow relaxation of the system to the steady state. As was previously shown, the conductance measured at each potential U, (1yR)m(U), has two contributions: the true sample conductance, (1yR)s(U), and that arising from Faradaic currents across the semiconductoryelectrolyte interface, (1yR)e(U): wI
Ž1yR.msx
m
ŽU,qVa.yImŽU,yVa. z
2Va sŽ1yR. SqŽ1yR. C y
| ~
(11)
with Im(U,"Va) the current measured at potential U and applied bias "Va. In the ideal case, i.e. in the absence of faradaic currents, (1yR)m(U)s(1yR)s (U). In the presence of Faradaic currents, their contribution is taken into account by considering: wI
Ž1yR.esx
m
ŽUqVa,0.yImŽUyVa,0. z
y
2Va
| ~
(12)
with Im(U"Va,0) the current measured at potential U"Va and zero applied bias. In order to minimize the contribution of the Faradic current, the inactive part of the film (i.e. that deposited over the conducting part of the substrate) was masked as much as possible to minimize this contribution. In most cases, this contribution is negligible, and when not it is lower than 10% of the overall current. Finally, for a sample with length L and width W, the surface conductance, s, is related to the conductance measured by s(U)s(1yR)(U)LyW w23x. 4. Results and discussion The surface potential, VS, is related to the applied potential, U, by: dUsdVHqdVS
(13)
where VH is the potential drop in the Helmholtz layer on the solution side. Taking into account that dUsdQ yC, and that the Helmholtz layer charge, dQH, equals the charge at the semiconductor surface, integration of the above equation between the flat band potential, Ufb, and U gives: UyUfbsŽ ye nyCH.yVS
(14)
where CH is the capacity of the Helmholtz layer, which is assumed constant, and n is the excess surface carrier
Fig. 7. Surface conductance data, s, for two different thin CdS films in 0.1 M Na2SO4. Arrows indicate the characteristic potential U*.
density (free and bounded), which includes the contribution from both trapped and free carriers. In what follows, we assume that in depletion, VH
C. Alcober, S.A. Bilmes / Thin Solid Films 405 (2002) 55–63
Fig. 8. Surface conductance variation, Ds, for two different thin CdS films in (a) 0.1 M Na2SO4 and (b) 0.5 M NaOH: (s) experimental data, (—) calculated curve.
Ufb)1y2 relationship is a good indication that the contribution of gap states is negligible. Then, assuming that mobility in the depletion region is constant, n*L can be calculated by Eq. (7) with Nts0. Fig. 8 shows experimental data of surface conductance variation, Ds(U), along with the best fit using n0, EB and Ufb as adjustable parameters. The dependence of Ds on applied potential is that predicted for an n-type semiconductor without gap states. This does not necessarily mean that the material is free of localized states in the gap; these are deep enough, and as EF is always above Et, their charge is constant. In addition, this confirms that the assumption that carrier mobility is independent of U is correct, at least in depletion. A confidence interval of ;5% and ;10% was obtained for n0 and EB, respectively. For Ufb, an uncertainty of ca. "20 mV was found for all samples. For all samples, n0 lies in the range 1024;4=1024 my3, in good agreement with other values reported for polycrystalline CdS films w26–30,36x. LSCF10 nm for all n0 values, in agreement with the assumptions. Then, the Fermi level is estimated to be at 50 meV below the bottom of the conduction band w37x. The small dispersion in n0 values indicates a high reproducibility between samples, even those arising from different batches, even sfb values span nearly two orders of magnitude
61
(2=10y9;4=10y7 S). Then, differences between films mainly arise from carrier transport through boundaries, and heat treatments modify the crystallite boundaries, with little effect on carrier density. EB values fall in the range 0.17;0.29 eV, in agreement with those found in the literature w26,38x. This gives, from Eq. (9), 0.1FaF0.3, corresponding to a crystallite mostly occupied by barriers. With n0, Ufb and Eb, Ds is calculated in accumulation, assuming that the microscopic mobility and a are independent of the applied surface barrier potential. Because of the closeness of the Fermi level to the bottom of the conduction band, n*L was calculated using Eq. (6), taking into account surface degeneration. In this calculation, F(wb,v) was numerically calculated assuming the absence of trap states in the gap (i.e. Nts 0) w21,22x. As in accumulation, the surface charge density can be comparable to the net charge in the Helmholtz layer, the potential distribution at the interface was calculated using Eq. (14), with CHs0.1 F my2 w39x. Fig. 9 shows the experimental and calculated Ds values in accumulation for two representative samples. The model systematically underestimates Ds, and can be addressed to several facts that increase either nL or m: (i) the barrier height becomes lower or (ii) narrower; and (iii) the Fermi level enters in the conduction band. The lowering of barrier heights due to an induced
Fig. 9. Surface conductance variation, Ds, in accumulation for the same films as in Fig. 8: (s) experimental data, (—) calculated curve.
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crystallites, the volume free of intergrain barriers of which changes with the applied potential. On this basis, the electronic parameters of CdS films in contact with electrolytes (charge carrier density and intergrain barrier height) are derived from surface conductance experiments by considering the interaction between the potential drop at the interface and that at the intergrain boundaries, and that charge transport is controlled by thermionic emission through the barriers. The main conclusion of this work is that for polycrystalline materials, there is a wide interval of bias for which positive and negative excess of free carriers are simultaneously present at different points of the surface. Then, the meaning of ‘flat band potential’ derived from experiments is that at which the surface charge equals zero. Acknowledgments Fig. 10. Schematic surface potential profile under various degrees of accumulation. EB is the effective interfacial barrier height and accumulation increases going from a to d. The meaning of VS,C and VS,B is explained in the text.
electric field at the surface has been invoked for the mobility increase in metal-oxide semiconductor (MOS) structures using CdS w38x and other semiconductors w35,40,41x. This effect is related to the potential profile in each crystallite (see Fig. 4), then changing with U, as schematized in Fig. 10. Then, carriers may accumulate at the crystallite center (the region labeled VS)0 in Fig. 4), while close to the interface between crystallites, the surface is still depleted. This inhomogeneous surface-charge distribution implies that close to the boundaries, the potential drops inside the semiconductor, and ±DU±(±DVS,B±, whereas in those regions where the charge accumulate, an appreciable fraction of the potential drops in the Helmholtz layer, and ±DU±)±DVS,C±. As accumulation increases (going from curve a to d in Fig. 10), the difference between VS,B and VS,C becomes progressively smaller, reducing EB at the surface. This expansion of the crystallite effective surface also leads to narrower barriers, and tunneling becomes an additional channel for transport through it. In addition, nL is also underestimated due to expansion of the surface space charge layer. Then, the observed deviation of the proposed model from the experimental Ds in the accumulation region can be interpreted as a combined effect of both a lowering of the interfacial barriers at the surface and an expansion of the space charge layer, which are not taken into account in the model. 5. Conclusions CdS films prepared from solution chemistry exhibit a microscopic heterogeneity that affects the potential distribution at the filmyelectrolyte interface. The surface of this material can be thought of as a collection of
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C. Alcober, S.A. Bilmes / Thin Solid Films 405 (2002) 55–63 w22x H.C. De Graaff, in: G. Harbeke (Ed.), Polycrystalline Semiconductors, Springer-Verlag, New York, 1985, p. 170. w23x A. Manny, Surface Science, I, International Atomic Energy Agency, Vienna, 1975, p. 447. w24x S.M. Sze, Physics of Semiconductor Devices, 2nd, John Wiley & Sons, New York, 1981, p. 362. w25x R. Kingston, S.F. Neustadter, J. Appl. Phys. 26 (1955) 718. w26x J.W. Orton, B.J. Goldsmith, M.J. Powell, J.A. Chapman, Appl. Phys. Lett. 37 (1980) 557. w27x S.Y. Kim, D.S. Kim, B.T. Ahn, H.B. Im, Thin Solid Films 229 (1993) 227. w28x J. von Windheim, H. Wynands, M. Cocivera, J. Electrochem. Soc. 138 (1991) 3435. w29x Y.Y. Ma, R.H. Bube, J. Electrochem. Soc. 124 (1977) 1430. w30x C. Wu, R.H. Bube, J. Appl. Phys. 45 (1974) 648. w31x D. He, N. Okada, C.M. Fortman, I. Shimizu, J. Appl. Phys. 76 (1994) 4728.
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w32x Y. Wu, in: S.H. Choh, S.K. Min ŽEds.., Proceedings Seoul International Symposium on the Physics of Semiconductors and Applications, Seoul, South Korea, July 1982, 1982, p. 87. w33x J.Y.W. Seto, J. Appl. Phys. 46 (1975) 5247. w34x G. Baccarani, B. Ricco, ` G. Spadini, J. Appl. Phys. 49 (1978) 5565. w35x L. Dawar, A.D. Sen, H.K Dewan, O.P. Taneja, P.C. Mathur, J. Appl. Phys. 53 (1982) 3308. w36x S. Baranski, M.S. Bennett, W.R. Fawcett, J. Appl. Phys. 54 (1983) 6390. w37x J.P. McKelvey, Solid State and Semiconductor Physics, Harper & Row, New York, 1966. w38x A. Waxman, V.E. Henrich, F.V. Shallcross, H. Borkan, P.K. Weimer, J. Appl. Phys. 36 (1965) 168. w39x J.C. Liu, C.P. Huang, Langmuir 8 (1992) 1851. w40x A. Mercier, M. El Koosy, A. Le Glaunec, E. Le Tiran, J. Phys. C 143 (1982) 369, In French.. w41x Y.N. Kumishima, M. Miyayama, H. Yanagida, J. Electrochem. Soc. 143 (1996) 1334.
Potential distribution at the semiconductor thin filmyelectrolyte interface with high density of grain boundaries Carlos Alcober, Sara A. Bilmes* ´ ´ ´ ´ ´ INQUIMAE, Departamento de Quımica Inorganica, Analıtica y Quımica-Fısica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pab. II, 1428 Buenos Aires, Argentina Received 27 February 2001; received in revised form 29 October 2001; accepted 1 December 2001
Abstract The electronic structure of an n-CdSyelectrolyte interface was analyzed by in situ surface conductance for CdS thin films obtained by chemical precipitation onto a conducting substrate. The films are compact, composed of crystallites and exhibit grain boundaries that affect the potential distribution on the semiconductor space charge layer. We propose an explicit evaluation of the charge carrier density, nL, taking into account the relation between the space charge layer length, LSC , and the crystallite length, L. Using this model, values of the interfacial and electronic parameters, such as the flat band potential, Ufb , the carrier mobility, NmM, and the electron density, n0 , were obtained from in situ surface conductance data. 䊚 2002 Elsevier Science B.V. All rights reserved. Keywords: Surface conductivity; Cadmium sulfide; Grain boundary; Electrochemistry
1. Introduction For charge transfer processes, such as those occurring in photoelectrochemical cells and in photocorrosion, the kinetics are driven by the properties of the interfacial layer formed at the semiconductoryelectrolyte junction. In particular, the space charge layer dimensions, Fermi level position, mobility and density of charge carriers are key parameters for the description of photoelectrochemical processes at compact, thin-film semiconductor electrodes. Although these parameters can be well determined for single crystals, their estimation for polycrystalline films enables the knowledge of boundaries, carrier traps and surface state density in the material. The contact between a single-crystal semiconductor electrode and an electrolyte gives rise to a space charge layer inside the electrode at the solidyliquid interface w1,2x. For polycrystalline film electrodes, this is not necessarily true, and depends on the material morphology and electronic properties. From a general point of view, two different situations can be distinguished. In * Corresponding author. Fax: q54-11-45763341. E-mail address: [email protected] (S.A. Bilmes).
porous electrodes, the semiconductor network is interpenetrated by the electrolyte, and interfacial properties are largely determined those of nanometer-sized structural units and by the very large fractal solidysolution interface w3–6x. Then, porous electrodes can be viewed as a collection of interconnected micro- or nanoelectrodes surrounded by the electrolyte, which, except for highly doped semiconductors, can develop a space charge layer only when free carriers accumulate at the interface w7,8x. Compact polycrystalline electrodes, on the other hand, can be thought of as single crystals considering the interface located at the macroscopic limit between the semiconductor and the solution. In this case, a space charge layer develops both in depletion and accumulation regimes, and it is possible to define a flat band potential, Ufb, when the net surface charge is zero w9–11x. However, the description of the surface space-charge layer and its effect on transport properties requires the material heterogeneity to be taken into account. Several experimental methods have been developed for the description of semiconductoryliquid junctions. Electrochemical impedance spectroscopy is the most widely employed, and experimental data for single
0040-6090/02/$ - see front matter 䊚 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 0 - 6 0 9 0 Ž 0 1 . 0 1 7 5 1 - 5
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56
crystals are usually interpreted by the Mott–Schottky model w1,12x. Other techniques employing either optical or electrical perturbations have been successfully used for the study of single crystals w13–15x, but they involve significant uncertainties andyor simplifying approximations for the description of thin film semiconductors. A powerful method for the study of semiconductor filmy electrolyte interface is in situ surface conductance, which has been successfully used for Ge w16x and Si w17x single-crystal electrodes, although it has limited use for thin semiconductor films w5,18x. In previous work, we demonstrated that it is a suitable tool for the study of compact semiconductor thin filmyelectrolyte interfaces w19,20x, even for materials with fairly high resistivity. In situ surface conductance is proportional to the carrier mobility, m, and the surface free carrier density excess, nL: (1)
DsssysfbsemnL
where sfb is the surface conductance for nLs0. For compact materials, sfb corresponds to the surface conductance at flat band, and Ds gives a measure of the surface potential barrier VS. For porous semiconductors, the physical meaning of Ds is not straightforward, and is probably related to both ion and electron excess at the surface. In this work, we analyzed the CdS filmyaqueous electrolyte interface by in situ surface conductance. In order to interpret surface conductance data for CdS films electrodes in a wide potential range, we propose an approach for the explicit evaluation of nL and m as a function of VS for compact semiconductor films. The material heterogeneity introduces intercrystallite barriers and trapped charges, which are taken into account from the relation between the space-charge layer length, LSC, and the crystallite length, L. 2. The space charge layer for a polycrystalline semiconductor For a compact electrode in contact with an electrolyte, r , is calculated by solving the the internal potential, V ™
Ž.
three-dimensional Poisson equation: r = VŽ™ r. sy 2
Ž™r.
´r´0
(2)
with ´r the relative permittivity and the charge density, r™ r , given by:
Ž. rŽ™ r. seN
V
¯ .qeNSŽr,V ¯ .qeNLŽr,V ¯ . Žr,V
(3)
In Eq. (3), eNL and eNV are the free and bounded (in gap states) charge densities inside each crystallite and eNS is the localized charge density at the interface between the boundaries. It is important to stress that
this last term influences both the potential distribution due to the surface barrier and the control of charge transport through the film. Due to the inhomogeneous nature of polycrystalline materials, estimation of r ™ r
Ž.
requires detailed knowledge of the spatial distribution of crystallite boundaries and gap state density, which are generally unknown, except for simple systems, such as bicrystals w21x. However, by considering the relation between the space charge layer and crystallite length, nL can be evaluated for two limiting situations: LSC4L and LSC-L. 2.1. LSC4L (continuous model) In this case, it is possible to assume that the spatial distribution of electronic states, arising from both boundaries and the bulk of each crystallite, is uniform, and r no longer depends on spatial co-ordinates. Then, nL in Eq. (1) can be calculated as in a single crystal, with an appropriated density of gap states, DS(E). It should be noted that in this description, it is not necessary to take into account the particular transport mechanism, and m represents a phenomenological macroscopic parameter. Although m may depend on the surface potential VS w22,23x, this dependence can usually be neglected for potentials in the depletion region, because charge carriers move outside the space charge layer, and the surface barrier only modifies the effective thickness of the film. Then, in the continuous model, nL –VS plots give the surface conductance variation in the depletion region. On the other hand, in accumulation, it is necessary to take into account the dependence of m on VS, owing to the fact that carriers move inside the space charge region. In order to give an explicit dependence of nL on VS, we briefly review two defined situations for a nonnegligible density of states in the gap: (i) a single trap level; and (ii) a uniform density of traps. Complete calculations, both analytical and numerical, can be found in the literature w9,22–25x. The simplest case is for an n-type semiconductor in a semi-infinite (planar) geometry, with a parabolic density of states in the conduction band, for which nL is obtained by solving Eq. (1) in one dimension, with: rseNNdMye2py1y2NCF1y2Žvywb. NNtM qe 1qŽ1y2.exp wŽEtyEF.ykTxyv
µ
(4)
∂
for a monoenergetic trap level at energy Et, or rseNNdMye2py1y2NCF1y2Žvywb. EC NNtMdE qe Ei 1q Ž 1y2. exp w ŽEyEF. ykTxyv
|
µ
∂
(5)
for a continuous density of trap levels in the gap. In
C. Alcober, S.A. Bilmes / Thin Solid Films 405 (2002) 55–63
57
particular case of a single trap level, this deviation takes place for surface potentials (VS-y0.2 V in Fig. 1a), for which the trap energy, Et, is above the Fermi level at the surface. 2.2. LSC-L (localized model)
Fig. 1. Excess surface carrier density of CdS for different trap density in the gap: (a) single trap level, with Ets0.2 eV below conduction band; (b) uniform trap density. Calculations were carried out using ´rs8.7 and n0s2.08=1020 my3 with the trap density given in the figure.
Eqs. (4) and (5), wbs(EFyEC)ykT, vseVykT, NCs 3y2 2Ž2pŽm*nkTyh2.. , Nd is the donor density, F1y2 is the 1y2 Fermi integral; m*n is the electron effective mass; and e, k, T and h have their usual meaning. It should be noted that Nt in Eq. (4) is the density (per unit volume) of trap states, while in Eq. (5), Nt is the uniform density (per unit volume and energy) of trap states in the gap. From integration of the Poisson equation with the boundary condition Vs0 at x™`, the surface free-carrier excess density, nL, is w23,24x: 0
|
nLsLD
vS
2py1y2NCF1y2Žvywb.yn0 .FŽwb,v.
dv
In this case, only those crystallites that are in contact with the electrolyte need to be considered. Then, r has two contributions: free charges (eNL) and those trapped in bulk states (eNV). On the other hand, charges trapped at boundaries, eNS, give rise to interfacial barriers, which in turn give new boundary conditions for integration of the Poisson equation. The presence of these interfacial barriers introduces several effects. First, the barriers are depleted of mobile carriers, reducing the effective crosssectional area in which carriers can move. Another not so obvious effect can be observed with the help of a sketch of a bidimensional square grain in contact with an electrolyte under depletion conditions (Fig. 2). The region labeled ‘a’ is that occupied by the surface barrier, and those labeled ‘b’ and ‘c’ by the lateral and the back interfacial barrier, respectively. Applying the superposition principle, it can be shown that in depletion, for LD
(7)
where a is the fraction of grain length (along the xaxis) free of interfacial barriers was1y(2WyL), with W the width of the interfacial barriersx, and n*L is the surface carrier density excess in the same crystallite free of interfacial barriers, with carrier density n0 . n*L can be
(6)
where LDs(´r´0kTye 2n0)1y2; the minus and plus sign holds in depletion and accumulation respectively, and dv .FŽwb,v.sLD . dx Fig. 1 shows the nL –VS plots for (a) a single trap level and (b) uniform trap density obtained from Eq. (6), using parameters for CdS. In both cases, as the trap density increases, the density of free carriers removed from the surface decreases, and the dependence of nL on VS deviates from the simple parabolic dependence that represents the behavior for Nts0 w23,24x. In the
Fig. 2. Schematic barrier distribution in a square crystallite, with LSC-L. The semiconductoryelectrolyte interface is located along the x-axis. Region a is that occupied by the surface barrier, with b and c by the interfacial barriers.
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C. Alcober, S.A. Bilmes / Thin Solid Films 405 (2002) 55–63
developed a model in which the conductivity for a crystal of length L can be written as a function of the microscopic electronic parameters: ks
e 2 n 0L expŽyEBykT. Ž2pmn*kT.1y2
(8)
with EB the interfacial barrier height. Under the depletion approximation, EB can be approximated by: E Bs
Fig. 3. (D) Excess surface carrier density for a 60=60-nm2 square crystallite and (—) that calculated using Eq. (7). Calculations were carried out using ´rs8.7 and n0s1.23=1024 my3. See the text for details.
e2L2Ž1ya.2n0 8´r´0
Taking into account the reduction in both the effective cross-sectional and surface area, the change in surface conductance for a film of thickness d composed of cubic crystallites of side L can be expressed in terms of two parameters, n0 and EB: sŽVS.ysfbsŽkayn0.nL* ŽVS.
calculated using the appropriate DS(E) that takes into account bulk gap states, using the expressions derived in the previous section (for a tridimensional cubic crystallite, nLsa2n*L). Fig. 3 shows a plot of nL –VS for CdS using Eq. (7) (solid line). It was calculated, as previously stated, using n0s1.23=1024 my3 and as 0.5. For the sake of comparison, nL values from numerical calculations for a bidimensional 60=60-nm2 square crystal taking into account interfacial barriers are included (symbols). The parameters for this calculations (n0s 1.23=1024 my3 and density of interfacial traps Nts 7.4=1016 my2) also correspond to as0.5. In both cases, it was assumed that trap density in the bulk is negligible. Comparison of both curves shows that Eq. (7) is a good approximation in depletion, whereas in accumulation, it predicts a lower surface carrier density than that calculated taking into account the interfacial barriers. This is because in accumulation, a is no longer constant, and the space charge layer expands, increasing a. This expansion can be observed through the equipotential lines inside the crystallite, as depicted in Fig. 4 for two VS values. As VS increases (i.e. increasing accumulation), the space charge layer expands more, increasing a, and hence the free-carrier density excess at the surface. For a complete description of Ds on VS, it is only necessary to express the microscopic mobility. Transport properties in polycrystalline CdS films w26–30x and other semiconductor materials w28,31–35x have been widely studied using Hall measurements in solid-state devices. Contrary to single-crystal materials, the charge transport in compact polycrystalline materials is limited by the presence of potential barriers located at the interface between crystallites, and it is mainly controlled by thermionic emission w28,32–35x (although tunneling is important in heavily doped semiconductors). For this particular case, Seto w33x and Baccarani et al. w34x
(9)
(10)
with sfbskda2. As in the continuous approximation, k is expected to be constant in depletion, and the dependence of Ds on VS is that given by nL* .
Fig. 4. Potential distribution in a square crystallite for two different surface potentials under accumulation. Surface potential in (a) is more positive than in (b). Calculations were carried out with the same conditions as in Fig. 3.
C. Alcober, S.A. Bilmes / Thin Solid Films 405 (2002) 55–63
Fig. 5. RX diffractogram for a 400-nm CdS film over glass. Miller indexes are indicated for the cubic (C) and hexagonal (H) crystallographic faces.
For the intermediate case, LSC(L, it is necessary to consider the interaction between the space charge layer and the intercrystallite barriers. This interaction is not merely an overlap between the surface barrier and the intercrystallite barriers, but a rather complex phenomenon; the surface barrier changes the trapped charge in the interface boundaries and modifies the height and extension of these barriers w21x. It is reasonable to assume that the extension of the barriers is also comparable to the grain size, L, and the grains are completely depleted of carriers. For this particular situation, the Fermi level is very close to the trap levels w34x, and chargeydischarge of trap levels is important. 3. Methods and materials All measurements were carried out with a homemade potentiostat in a standard three-electrode configuration. The counter electrode was a Pt plate. The reference electrode was either HgyHgSO4 (saturated K2SO4) or HgyHgO (0.1 M NaOH). All reagents were analytical grade or better. Solutions were prepared with Milli-Q water and all measurements were made in N2-saturated electrolytes. Potential values are quoted against the saturated calomel electrode (SCE).
59
Briefly, CdS films were prepared from an aqueous solution of 0.1 M Cd(NO3)2 and 0.1 M thiourea, which was maintained at 60 8C under vigorous stirring. The substrates were immersed in the solution and the reaction was initiated by slow addition of concentrated NH4OH up to a final concentration of 0.6 M. After 10 min, the reaction was stopped and the films cleaned in acetone in an ultrasonic bath. Thin layers of ca. 60 nm were successively deposited in order to achieve a mean thickness of ;0.40 mm. Film thickness was estimated by the Cd content in a known area of the film determined by atomic absorption spectroscopy. Between successive deposits, the samples were carefully cleaned with acetone in an ultrasonic bath to remove small particles that remain adhered to the surface. The conductivity of the samples was improved by annealing in air at 200 8C. Samples where then stored in a dry box. 3.2. Morphologic characterization Samples were characterized using RX diffractograms (Siemens diffractometer with CuKa radiation) and tunnel microscopy (Nanoscope III scanning tunnel microscope with a Pt–Ir tip). The RX diffractograms (Fig. 5) show that the films have a high degree of crystallinity, with a preferential growth in the direction of the cubic (1 1 1) or the hexagonal (0 0 2) planes. Analysis of the peak widths shows a mean grain radius of 10"2 nm. STM images were taken ex situ in the dark (Nanoscope III scanning tunnel microscope with a Pt–Ir tip, working at constant current under feedback control). All the samples show an inhomogeneous surface (see Fig. 6) and integration of the area in the STM images gave a roughness factor 3FRF4.
3.1. Sample preparation For conductance measurements, a strip ;30 mm wide of conducting substrate was removed by etching, leaving two conducting areas electrically isolated. The CdS was then deposited on the substrate, and contacts were made with silver paint at the edges of the conducting substrate w19x. In this arrangement, indium–tin oxide (ITO) provides the electrical contact to the film. The silver contacts and any exposed wire were masked with galvanoplastic lacquer. Thin CdS films were chemically deposited onto ITO glass (Balzers, 200 V h, 0.7=1.4 cm2), by the same procedure previously described w19x.
Fig. 6. STM image of a 400-nm CdS film on ITO. The image was taken with a bias of y3.18 V and a current set point of y1 nA. The x and y scales are 50 nm per division, while the z scale is 40 nm per division.
C. Alcober, S.A. Bilmes / Thin Solid Films 405 (2002) 55–63
60
3.3. Surface conductance measurements The experimental set-up was the same as described in a previous paper w19,20x. Briefly, the current Im was measured at each potential U, fixed by the potentiostat, for applied bias between sample contacts of Vas 0, q 5 mV (or q10 mV), y5 mV (or y10 mV). Each Im value was obtained 30 s after changing Vm in order to allow relaxation of the system to the steady state. As was previously shown, the conductance measured at each potential U, (1yR)m(U), has two contributions: the true sample conductance, (1yR)s(U), and that arising from Faradaic currents across the semiconductoryelectrolyte interface, (1yR)e(U): wI
Ž1yR.msx
m
ŽU,qVa.yImŽU,yVa. z
2Va sŽ1yR. SqŽ1yR. C y
| ~
(11)
with Im(U,"Va) the current measured at potential U and applied bias "Va. In the ideal case, i.e. in the absence of faradaic currents, (1yR)m(U)s(1yR)s (U). In the presence of Faradaic currents, their contribution is taken into account by considering: wI
Ž1yR.esx
m
ŽUqVa,0.yImŽUyVa,0. z
y
2Va
| ~
(12)
with Im(U"Va,0) the current measured at potential U"Va and zero applied bias. In order to minimize the contribution of the Faradic current, the inactive part of the film (i.e. that deposited over the conducting part of the substrate) was masked as much as possible to minimize this contribution. In most cases, this contribution is negligible, and when not it is lower than 10% of the overall current. Finally, for a sample with length L and width W, the surface conductance, s, is related to the conductance measured by s(U)s(1yR)(U)LyW w23x. 4. Results and discussion The surface potential, VS, is related to the applied potential, U, by: dUsdVHqdVS
(13)
where VH is the potential drop in the Helmholtz layer on the solution side. Taking into account that dUsdQ yC, and that the Helmholtz layer charge, dQH, equals the charge at the semiconductor surface, integration of the above equation between the flat band potential, Ufb, and U gives: UyUfbsŽ ye nyCH.yVS
(14)
where CH is the capacity of the Helmholtz layer, which is assumed constant, and n is the excess surface carrier
Fig. 7. Surface conductance data, s, for two different thin CdS films in 0.1 M Na2SO4. Arrows indicate the characteristic potential U*.
density (free and bounded), which includes the contribution from both trapped and free carriers. In what follows, we assume that in depletion, VH
C. Alcober, S.A. Bilmes / Thin Solid Films 405 (2002) 55–63
Fig. 8. Surface conductance variation, Ds, for two different thin CdS films in (a) 0.1 M Na2SO4 and (b) 0.5 M NaOH: (s) experimental data, (—) calculated curve.
Ufb)1y2 relationship is a good indication that the contribution of gap states is negligible. Then, assuming that mobility in the depletion region is constant, n*L can be calculated by Eq. (7) with Nts0. Fig. 8 shows experimental data of surface conductance variation, Ds(U), along with the best fit using n0, EB and Ufb as adjustable parameters. The dependence of Ds on applied potential is that predicted for an n-type semiconductor without gap states. This does not necessarily mean that the material is free of localized states in the gap; these are deep enough, and as EF is always above Et, their charge is constant. In addition, this confirms that the assumption that carrier mobility is independent of U is correct, at least in depletion. A confidence interval of ;5% and ;10% was obtained for n0 and EB, respectively. For Ufb, an uncertainty of ca. "20 mV was found for all samples. For all samples, n0 lies in the range 1024;4=1024 my3, in good agreement with other values reported for polycrystalline CdS films w26–30,36x. LSCF10 nm for all n0 values, in agreement with the assumptions. Then, the Fermi level is estimated to be at 50 meV below the bottom of the conduction band w37x. The small dispersion in n0 values indicates a high reproducibility between samples, even those arising from different batches, even sfb values span nearly two orders of magnitude
61
(2=10y9;4=10y7 S). Then, differences between films mainly arise from carrier transport through boundaries, and heat treatments modify the crystallite boundaries, with little effect on carrier density. EB values fall in the range 0.17;0.29 eV, in agreement with those found in the literature w26,38x. This gives, from Eq. (9), 0.1FaF0.3, corresponding to a crystallite mostly occupied by barriers. With n0, Ufb and Eb, Ds is calculated in accumulation, assuming that the microscopic mobility and a are independent of the applied surface barrier potential. Because of the closeness of the Fermi level to the bottom of the conduction band, n*L was calculated using Eq. (6), taking into account surface degeneration. In this calculation, F(wb,v) was numerically calculated assuming the absence of trap states in the gap (i.e. Nts 0) w21,22x. As in accumulation, the surface charge density can be comparable to the net charge in the Helmholtz layer, the potential distribution at the interface was calculated using Eq. (14), with CHs0.1 F my2 w39x. Fig. 9 shows the experimental and calculated Ds values in accumulation for two representative samples. The model systematically underestimates Ds, and can be addressed to several facts that increase either nL or m: (i) the barrier height becomes lower or (ii) narrower; and (iii) the Fermi level enters in the conduction band. The lowering of barrier heights due to an induced
Fig. 9. Surface conductance variation, Ds, in accumulation for the same films as in Fig. 8: (s) experimental data, (—) calculated curve.
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C. Alcober, S.A. Bilmes / Thin Solid Films 405 (2002) 55–63
crystallites, the volume free of intergrain barriers of which changes with the applied potential. On this basis, the electronic parameters of CdS films in contact with electrolytes (charge carrier density and intergrain barrier height) are derived from surface conductance experiments by considering the interaction between the potential drop at the interface and that at the intergrain boundaries, and that charge transport is controlled by thermionic emission through the barriers. The main conclusion of this work is that for polycrystalline materials, there is a wide interval of bias for which positive and negative excess of free carriers are simultaneously present at different points of the surface. Then, the meaning of ‘flat band potential’ derived from experiments is that at which the surface charge equals zero. Acknowledgments Fig. 10. Schematic surface potential profile under various degrees of accumulation. EB is the effective interfacial barrier height and accumulation increases going from a to d. The meaning of VS,C and VS,B is explained in the text.
electric field at the surface has been invoked for the mobility increase in metal-oxide semiconductor (MOS) structures using CdS w38x and other semiconductors w35,40,41x. This effect is related to the potential profile in each crystallite (see Fig. 4), then changing with U, as schematized in Fig. 10. Then, carriers may accumulate at the crystallite center (the region labeled VS)0 in Fig. 4), while close to the interface between crystallites, the surface is still depleted. This inhomogeneous surface-charge distribution implies that close to the boundaries, the potential drops inside the semiconductor, and ±DU±(±DVS,B±, whereas in those regions where the charge accumulate, an appreciable fraction of the potential drops in the Helmholtz layer, and ±DU±)±DVS,C±. As accumulation increases (going from curve a to d in Fig. 10), the difference between VS,B and VS,C becomes progressively smaller, reducing EB at the surface. This expansion of the crystallite effective surface also leads to narrower barriers, and tunneling becomes an additional channel for transport through it. In addition, nL is also underestimated due to expansion of the surface space charge layer. Then, the observed deviation of the proposed model from the experimental Ds in the accumulation region can be interpreted as a combined effect of both a lowering of the interfacial barriers at the surface and an expansion of the space charge layer, which are not taken into account in the model. 5. Conclusions CdS films prepared from solution chemistry exhibit a microscopic heterogeneity that affects the potential distribution at the filmyelectrolyte interface. The surface of this material can be thought of as a collection of
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