In the Laboratory
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Flat-Band Potential of a Semiconductor: Using the Mott–Schottky Equation K. Gelderman, L. Lee, and S. W. Donne* Discipline of Chemistry, University of Newcastle, Callaghan, NSW 2308, Australia; *
[email protected] It has long been known that metals are good conductors of electricity. However, the discovery of semiconductors and the transistor effect by Bardeen, Shockley, and Brattain in 1948 (summarized in ref 1) generated considerable interest in the electronic properties of all materials and paved the way for the development of the myriad of electronic devices we have today. Of particular relevance to this article are photovoltaic cells, which can be used to convert light into electrical energy. With the current movement away from fossil fuel-based energy towards more environmentally friendly, renewable energy sources, research in this area is again gaining momentum. Previously, it was at a peak during the “energy crisis” of the early 1980s, when there was a global deficiency of fossil fuels. At the present time, however, the motivation is the inherent realization that fossils fuels are a finite resource, as well as being detrimental to the environment when combusted. In this article we describe an experiment to determine one of the fundamental properties of any semiconductor–electrolyte system; namely, its flat-band potential. To gain a more thorough understanding of this semiconductor property, which will be of significance to both senior undergraduate and graduate students, we begin by describing the nature of the semiconductor–electrolyte interface, together with the Mott–Schottky equation for determining the flat-band potential.
A
B
The Semiconductor–Electrolyte Interface Of primary importance in the development of electrochemical photovoltaic cells is understanding the relationship between semiconductor and electrolyte energy levels (2–5). An energy-level diagram for both an n-type semiconductor and a redox couple in an electrolyte solution is shown in Figure 1A. For the semiconductor we have identified the valenceand conduction-band edges (VB and CB, respectively), the band-gap energy (EG), and the Fermi level (EF), which is the energy at which the probability of an electronic state being occupied is 0.5. These bands are dependent on the semiconductor potential, φ, changing as ᎑eφ where e is the charge on an electron. The energy levels for redox-active species in solution arise by virtue of the donors (Red) and acceptors (Ox) in solution; that is, (1) Ox + e− Red The energies of the solution states depend on whether the state is occupied (Red) or vacant (Ox), owing to the different solvent-sheath energies, λ, around the Red and Ox species. Since solvent molecule exchange between the coordination sphere of the redox-active species and the bulk electrolyte is a dynamic process leading to a range of solventsheath energies, the density of redox states is best described in terms of separate Gaussian distributions (Figure 1A). The redox Fermi level, EF(redox), is again the energy at which the probability of a state being occupied by an electron is 0.5. www.JCE.DivCHED.org
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C
Figure 1. (A) Schematic of an n-type semiconductor showing the valence and conduction bands (VB and CB, respectively), Fermi level (EF), band-gap energy (EG), and the redox states in solution (Ox and Red), with their corresponding Fermi level (EF(redox)) and solvent-reorganization energy (λ). (B) Electronic equilibrium between the n-type semiconductor and redox couple in solution. (C) Situation when the semiconductor is at its flat-band potential Vfb.
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Here C and A are the interfacial capacitance and area, respectively, ND the number of donors, V the applied voltage, kB is Boltzmann’s constant, T the absolute temperature, and e is the electronic charge. Therefore, a plot of 1兾C 2 against V should yield a straight line from which Vfb can be determined from the intercept on the V axis. The value of ND can also be conveniently found from the slope knowing ε and A. Experimental
Figure 2. Schematic of the assembled electrode.
When an n-type semiconductor and a redox couple come into contact, where EF is higher in energy compared to EF(redox), equilibrium can be achieved through the transfer of electrons from the semiconductor to Ox so that the Fermi levels for both phases are equal, as in Figure 1B. This has the effect of charging the semiconductor positively, and since semiconductor carrier densities are much lower than those in solution, the diffuse charge in the semiconductor (space charge region) is counterbalanced essentially by a sheet of charge in the electrolyte. Changing the voltage of the semiconductor artificially through the use of a potentiostat causes the semiconductor and redox couple Fermi levels to separate, and hence the level of band bending owing to electron depletion in the semiconductor will change depending on the applied voltage. When the applied voltage is such that there is no band bending, or charge depletion (Figure 1C), then the semiconductor is at its flat-band potential, Vfb .
Electrode Preparation A schematic of the semiconductor electrode used in this work is shown in Figure 2. Essentially a compacted, sintered disk of polycrystalline ZnO was mounted using chemically resistant epoxy into a polypropylene tube. Before being encapsulated, a contact wire was attached to the back of the ZnO disk using Ag-loaded epoxy. The surface of the ZnO was polished, thoroughly washed with ultra-pure water, and then patted dry prior to use. Electrochemical Protocol The ZnO electrode was immersed in an aqueous solution of 7 × 10᎑4 M K3[Fe(CN)6] in 1 M KCl, together with a saturated calomel reference electrode (SCE) and a Pt counter electrode. Previously the electrolyte solution had been degassed of oxygen by purging with nitrogen. The basis of an electrochemical impedance spectroscopy (EIS) experiment is to apply a small amplitude sinusoidal ac voltage, V(t), and then measure the amplitude and phase angle (relative to the applied voltage) of the resulting current, I(t). From this the impedance, Z(ω), can be determined using Ohm’s law (5):
The Mott–Schottky Equation Under the circumstances shown in Figure 1A, that is, where EF > EF(redox), the Mott–Schottky equation can be used to determine the flat-band potential of the semiconductor. Understanding its derivation is essential for this experiment because it reinforces many key concepts associated with the semiconductor–electrolyte interface. For the complete derivation the reader is referred to the Supplemental MaterialW for this experiment. However, in short, the starting point for the derivation is Poisson’s equation in one dimension that describes the relationship between charge density and potential difference, φ, in a phase, d2 φ dx
2
= −
ρ ε ε0
(2)
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(4a)
I (t ) = I 0 + I m sin (ω t + θ)
(4b)
Z (ω ) =
V (t ) I (t )
(4c)
Here V0 and I0 are the dc bias potential and the steady-state current flowing through the electrode, respectively, when the impedance experiment is conducted, Vm and Im are the maximum voltage and current of the supplied sinusoidal signal, respectively, and θ is the phase angle of the resultant current. An alternative, more convenient description is to express Z(ω) in terms of orthogonal axes rather than polar coordinates:
where ρ corresponds to the charge density at a position x away from the semiconductor surface, ε is the dielectric constant of the semiconductor, and ε0 is the permittivity of free space. Using the Boltzmann distribution to describe the distribution of electrons in the space charge region and Gauss’ law relating the electric field through the interface to the charge contained within that region, Poisson’s equation can be solved to give the Mott–Schottky equation: k T 1 2 V − Vf b − B = 2 2 e C ε ε 0 A e ND (3)
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V (t ) = V0 + Vm sin (ω t )
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Z (ω ) = Z ′ + j Z ′ ′
(5a)
Z ′ = Z (ω ) cos (θ)
(5b)
Z ′ ′ = Z (ω ) sin (θ)
(5c)
Z (ω ) =
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Vm Im
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(5d)
In the Laboratory
Here j is the imaginary number ( j = √᎑1). A range of frequencies, ω, can be examined to generate an impedance spectrum. In this experiment, the impedance of the ZnO electrode was measured at bias potentials ranging from +0.8 to ᎑0.5 V (versus SCE) in 50-mV increments, with 15 minutes allowed for equilibration at each new potential. The frequency range was from 20 kHz to 0.1 Hz, with Vm set at 5 mV. Clearly this is a long experiment and so it is highly preferable to use an automated system that can control the experiment without the need for manual input. Specialized Equipment
Figure 3. Typical EIS response for ZnO immersed in 7 × 10᎑4 M K3[Fe(CN)6] (+0.8 V versus SCE).
• Pellet press and hydraulic ram • High-temperature oven capable of at least 1200 ⬚C • Equipment to carry out EIS; e.g., gain-phase analyzer (Solartron 1253), potentiostat (Princeton Applied Research EG&G 273A), controlling software (ZPlot by Schlumberger) Figure 4. Modified Randles circuit used to model the ZnO electrode interface. Terms are defined in the text.
Hazards K3[Fe(CN)6] is toxic if swallowed or by skin contact; however, the quantities used in this experiment are small. ZnO and KCl do not pose a serious hazard in this experiment. Both the Ag-epoxy and chemically resistant epoxy can be hazardous if in contact with the skin. In terms of techniques, using a high-temperature furnace can be a considerable hazard. Any user should wear appropriate personal protective equipment such as a face mask, lab coat, and thermally insulating gloves, as well as use long tongs when placing in or extracting samples from the furnace. When using electrochemical apparatus, the user should always ensure correct electrical contacts between the equipment and cell. Furthermore, equipment compliance should be evaluated using a dummy cell. Results and Discussion
EIS Data and Analysis A typical EIS result is shown in Figure 3. The first notable feature is the depressed semicircular response to changes in frequency. Interpretation of the EIS data was carried out by considering the possible faradaic and non-faradaic processes that can occur at the ZnO surface and then relating those back to a modified Randles circuit (6). The only possible faradaic process involves charge transfer in the [Fe(CN)6]3−兾[Fe(CN)6]4− redox couple. This is probably one of the more well-defined and reversible redox couples and so is ideal for this study. Charge transfer in this redox couple is represented by a resistance (RCT) in the Randles circuit (Figure 4). In parallel with RCT is the non-faradaic electrode capacitance caused by the build up of charge at the ZnO electrode surface. In the Randles circuit we have represented this by a constant-phase element (CPE) to take into account any non-homogeneity of the ZnO surface; for example, surface roughness. Surface non-homogeneity is indicated by the depressed semi-
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circle seen in Figure 3. The impedance of a CPE in an ac circuit, ZCPE, is Z CPE = σ ω−m cos
mπ 2
− j sin
mπ 2
(6)
where σ is the CPE prefactor, ω is the angular frequency (ω = 2πf ), m is the CPE exponent (0 ≤ m ≤ 1), and j is the imaginary number ( j = √᎑1). Note that if m = 1 then ZCPE represents an ideal capacitor, ZC. RCT and ZCPE are in parallel with each other because they represent alternate charge paths at the electrode surface. Also included in series with RCT is a Warburg impedance, ZW, which takes into account diffusion of electroactive species towards the electrode, which is most significant at low frequencies. ZW is essentially the same as ZCPE, but with m = 0.5 in eq 6. The final component is another resistance (RS) representing the voltage drop in the electrolyte owing to the passage of current between the surface of the ZnO electrode and the reference electrode. EIS data collected in this work were then modeled by complex nonlinear least-squares regression (7, 8) using the total impedance of the modified Randles circuit (Figure 4). From the extracted parameters, the interfacial capacitance was determined. Mott–Schottky Plot To establish a Mott–Schottky plot as described above, the interfacial capacitance, C, can be determined directly from σ (eq 6) if m = 1. However, m was substantially less than unity over the entire voltage range considered owing most probably to surface non-homogeneity. Nevertheless, m was
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slope of the Mott–Schottky plot, ND = 2 × 1024 m᎑3, which is comparable to previously reported values (6 × 1024 m᎑3; ref 2 ). The deviation is most probably due to the action of surface states in the polycrystalline electrode capturing and immobilizing the carriers. Summary and Conclusions
Figure 5. Mott–Schottky plot for ZnO in 7 × 10᎑4 M K3[Fe(CN)6] (1 M KCl).
In this experiment, suitable for fourth-year undergraduate and graduate students, we have explored the nature of semiconductor materials through determination of the flat-band potential using the Mott–Schottky equation. Experimentally, a technique was developed for preparing a suitable polycrystalline ZnO electrode for study. Note that a similar approach could be used for other semiconductor electrodes. Electrochemical impedance spectroscopy was then employed to examine the semiconductor–electrolyte, 7 × 10᎑4 M K3[Fe(CN)6] (1 M KCl), interface as a function of applied voltage. To achieve this a modified Randle’s circuit was developed to interpret the impedance data, from which a value for the interfacial capacitance was determined. A Mott–Schottky plot was then constructed that allowed a flat-band potential of ᎑0.316 ± 0.033 V versus SCE to be determined. The number of carriers, ND, was also determined: ND = 2 × 1024 m᎑3. Both results were comparable to literature data emphasizing the soundness of the technique. W
constant over the entire voltage range, having an average value of 0.57 ± 0.02 with no apparent trend in the data. Therefore, assuming that the electrode capacitance can be represented directly by 1兾σ, a Mott–Schottky plot was constructed (Figure 5). According to eq 3, the flat-band potential of ZnO was ᎑0.316 ± 0.033 V versus SCE in 7 × 10᎑4 M K3[Fe(CN)6] (1 M KCl). The steps apparent in Figure 5 most likely originate from the equivalent circuit fitting procedure applied to data with some low frequency noise (Figure 3). The resultant σ values when squared then tend to vary only slightly, as seen in the Mott–Schottky plot. In comparison with previous works, Freund and Morrison (9) reported ᎑0.41 V versus SCE for a similar system. However, in this example a single crystal of ZnO was used, allowing for a well-defined crystal plane, [001], to be exposed to the electrolyte. The polycrystalline ZnO electrode used here means that many crystal planes would be exposed, suggesting that the different Vfb values arise as a result of conductivity differences along different crystallographic planes. To determine ND from the Mott–Schottky equation (slope in Figure 5), Morrison (2) has quoted the use of a dielectric constant of 8.5 for this system. Therefore, from the
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Supplemental Material
Instructions for the students, including pre- and postlab questions and the complete derivation of the Mott– Schottky equation, and notes for the instructor are available in this issue of JCE Online. Literature Cited 1. Shockley, W. B. Proc. Electrochem. Soc. 1998, 98–1, 26. 2. Morrison, S. R. Electrochemistry at Semiconductor and Oxidized Metal Electrodes; Plenum Press: New York, 1980. 3. West, A. R. Solid State Chemistry and Its Applications; John Wiley and Sons: Chichester, United Kingdom, 1984. 4. Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry: Plenum Press: New York, 1970; Vols. 1 and 2. 5. Impedance Spectroscopy: Emphasizing Solid Materials and Systems; Macdonald, J. R., Ed.; John Wiley and Sons: New York, 1987. 6. Randles, J. E. B. Discuss. Faraday Soc. 1947, 1, 11. 7. Boukamp, B. A. Solid State Ionics 1986, 18, 136. 8. Boukamp, B. A. Solid State Ionics 1986, 20, 30. 9. Freund, T.; Morrison, S. R. Surface Science 1968, 9, 119.
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