Band-gap determination from diffuse reflectance ...

ARTICLE IN PRESS

Solar Energy Materials & Solar Cells 91 (2007) 1326–1337 www.elsevier.com/locate/solmat

Band-gap determination from diffuse reflectance measurements of semiconductor films, and application to photoelectrochemical water-splitting A.B. Murphya,b, a

CSIRO Industrial Physics, P.O. Box 218, Lindfield, NSW 2070, Australia CSIRO Energy Transformed National Research Flagship, PO Box 330, Newcastle, NSW 2300, Australia

b

Received 16 March 2007; received in revised form 4 May 2007; accepted 4 May 2007 Available online 20 June 2007

Abstract Measurements of the diffuse reflectance of TiO2 semiconductor coatings, such as are used for water splitting, are analysed using the Kubelka–Munk radiative transfer model. The widely used practice of determining the band gap of the coating directly from the diffuse reflectance is found to be inaccurate, since the diffuse reflectance depends on parameters such as the thickness, refractive index and surface roughness of the coating. However, it is shown that the absorption coefficient can be derived from the diffuse reflectance using an inversion method; the band gap can then be obtained from the absorption coefficient. Finally, the diffuse reflectance of carbon-doped TiO2 presented by Khan et al. [Science 297 (2002) 2243-2245] is analysed; it is found that while the band-gap wavelength is extended into the visible region, it is overestimated. Moreover, light at visible wavelengths is only very weakly absorbed, and is expected to make only a minor contribution to the water-splitting efficiency. Crown Copyright r 2007 Published by Elsevier B.V. All rights reserved. Keywords: Photocatalysis; Absorption coefficient; Tauc plot; Band gap; Solar hydrogen production

1. Introduction In photoelectrochemical water-splitting, hydrogen is produced in an electrochemical cell by the action of light on a photoelectrode, typically a metal-oxide semiconductor thin film on a conducting substrate. The semiconductor absorbs photons at wavelengths below its band-gap wavelength, producing electron–hole pairs. These charge carriers diffuse to the water and the conducting substrate, driving the water-splitting reaction to produce hydrogen and oxygen. Since the diffusion length of the charge carriers is small (200 nm or less in titanium dioxide), it is important that the photons are absorbed near the surface; i.e., that the semiconductor has a large absorption coefficient for sub-band-gap wavelengths. Further, so that Corresponding author at: CSIRO Industrial Physics, P.O. Box 218, Lindfield, NSW 2070, Australia. Tel.: +61 294137150; fax: +16 294137200. E-mail address: [email protected]

as large a proportion as possible of the solar spectrum is utilized, it is important that the band gap of the semiconductor is as close as possible to the 2 eV required for the charge carriers to have sufficient energy to split water [1–3]. Measurement of diffuse reflectance with a UV-visible spectrophotometer is a standard technique in the determination of the absorption properties of materials. In the case of semiconductors for water splitting, the properties that can potentially be estimated from the diffuse reflectance are the band-gap energy (also referred to as the band gap) and the absorption coefficient. Determination of the band gap from the measurement of the diffuse reflectance of a powder sample is a standard technique [4,5]. The powder sample has to be sufficiently thick that all incident light is absorbed or scattered before reaching the back surface of the sample; typically a thickness of 1–3 mm is required. It may also be reasonable to apply this method to coatings that are sufficiently thick to absorb or scatter all the incident light. Alternatively, for non-opaque coatings and

0927-0248/$ - see front matter Crown Copyright r 2007 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.solmat.2007.05.005

ARTICLE IN PRESS A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

substrates with optically smooth surfaces, a combination of reflectance and transmittance measurements allows the optical properties of the coating to be determined by fitting of the data to standard optical equations that take into account reflection and transmission at each interface [6–10]. All these standard methods allow the determination of the absorption coefficient as a function of wavelength, or at least a variable proportional to the absorption coefficient; it is then possible to use standard techniques to find the band gap. The absorption coefficient also allows the depth distribution of light absorption in the semiconductor to be determined. Spectroscopic ellipsometry is also widely used to find the optical properties of thin film coatings with optically smooth surfaces, but it generally does not allow accurate determination of the absorption coefficient for wavelengths at which the coating does not absorb strongly [11,12]. In many cases of interest for water splitting, however, the above techniques cannot be applied. The semiconductor coating may be too thin to completely absorb or scatter the incident radiation, and further the surfaces of the coating and substrate may be optically rough, or the substrate may be opaque. For example, many workers have investigated titanium dioxide coatings on a titanium substrate, formed variously by thermal oxidation of a titanium sheet [13–19], anodic electrodeposition on titanium sheet [13], ‘microplasma’ oxidation of titanium sheet [20], magnetron sputtering on a titanium mesh substrate [21], spin coating of TiO2 nanopowder onto titanium sheet [22], and growth of arrays of TiO2 nanotubes on a titanium substrate [23–25]. Examples of TiO2 coatings on different opaque substrates include chemical vapour deposition of TiO2 onto semiconductors [26], dip coating of TiO2 dip onto a gold foil substrate [27], and electrophoretic and spray coating of TiO2 powder onto stainless steel as well as titanium [28]. Other semiconductor coatings include WO3 on a tungsten substrate, formed by both oxidation of tungsten sheet and sputtering of WO3 powder onto the tungsten sheet [29], and coatings of CdTe and CdS on titanium formed by cathodic electrodeposition, and CdS on titanium formed by chemical bath deposition [30]. In some of these cases, authors have attempted to derive band-gap energies and draw conclusions about absorption from measurements of diffuse reflectance, without calculating the absorption coefficient. Khan et al. [14] used measurements of the diffuse reflectance of carbon-doped rutile TiO2, formed by oxidizing titanium sheet in a methane flame, to deduce a band-gap wavelength of 535 nm, much larger than that derived for undoped rutile, and to hypothesize that the TiO2 absorbed strongly up to this wavelength. So et al. [31] used diffuse reflectance measurements of CdS–TiO2 particulate films to examine the influence of TiO2 particles on the band gap. Shangguan et al. [32] measured the diffuse reflectance of 0.3% Pt/TiO2 films on glass, and commented that the thinnest film had a larger band gap than the other films, even though its photocatalytic activity was the same. He et al. [21] derived

1327

the band gap of TiO2 sputtered onto indium–tin-oxide conducting glass from diffuse reflectance measurements. Shankar et al. [23] used diffuse reflectance measurements of TiO2 nanotubes on a titanium substrate to determine the band gap, finding an decrease in the band gap for nitrogendoped TiO2. Chen et al. [33] derived ‘optical absorption thresholds’ of TiO2 and nitrogen-doped TiO2 films, deposited on an aluminium substrate by reactive magnetron sputtering, from diffuse reflectance measurements, finding an increase in the threshold wavelength for the doped films. One purpose of this paper is to investigate the validity of such attempts to derive band gaps of semiconductor coatings from measurements of diffuse reflectance. For this purpose, I use a two-flux (Kubelka–Munk) radiative transfer model to investigate the absorption and scattering of the radiation in the semiconductor coating. The Kubelka–Munk model is the basis for measurements of the band gap of thick powder samples. To apply the model to thin coatings, reflections at the semiconductor–air and semiconductor–substrate interfaces are taken into account in a manner that is applicable to both optically smooth and optically rough surfaces. The dependence of the calculated diffuse reflectance on parameters such as the absorption coefficient, scattering coefficient, film thickness and surface roughness is investigated. The implications for efforts to determine the band gap are examined. The second purpose of the paper is to demonstrate a method by which the band gap can be derived from measurements of diffuse reflectance. To this end, measurements of the diffuse reflectance of titanium dioxide films of different thicknesses on a titanium substrate are reported. The measurements are first used to check the calculated dependence of the diffuse reflectance on film thickness. An inversion method, described elsewhere [34], is used to determine the absorption coefficient, and the band gap is then derived using standard methods. The inversion model is also applied to the measurements of Khan et al. [14] to estimate the absorption coefficient and band gap of their carbon-doped TiO2 photoelectrode. From this, the spectral dependence of the depth of absorption of light in the semiconductor is calculated, and the visible light contribution to the water-splitting efficiency of the photoelectrode is estimated. The implications for visible water-splitting efficiency of anion-doped TiO2 photoelectrodes in general are discussed. The paper is arranged as follows. Theoretical and experimental methods are described in Section 2. In Section 2.1, the Kubelka–Munk radiative transfer model and its application to diffuse reflectance measurements are outlined. Reflection coefficients from optically rough surfaces, required in the Kubelka–Munk expressions for diffuse reflectance, are presented in Section 2.2. The method used to produce the TiO2 coatings, and the properties of these coatings, are given in Section 2.3. The measurements of diffuse reflectance are described in Section 2.4. Theoretical results obtained by application of the Kubelka–Munk model to TiO2 coatings on a titanium

ARTICLE IN PRESS A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

1328

substrate are presented in Section 3. The dependence of diffuse reflectance on coating properties is examined in Section 3.1, and the influence of the coating thickness on the apparent band gap of the coating, derived directly from the wavelength dependence of the diffuse reflectance, is considered in Section 3.2. Diffuse reflectance measurements of the TiO2 coatings are presented and discussed in Section 4. The measurements are used to examine two methods of determination of the band gap; directly from the diffuse reflectance in Section 4.1, and via inversion of the diffuse reflectance to obtain the absorption coefficient in Section 4.2. In Section 5, the implications of these results for water splitting using doped electrodes that absorb visible light are considered, using the particular example of the diffuse reflectance measurements of carbon-doped TiO2 electrodes presented by Khan et al. [14]. Conclusions are presented in Section 6. 2. Methods 2.1. Kubelka–Munk model The Kubelka–Munk model [35,36] allows calculation of reflectance from a layer that both scatters and absorbs light. It is a ‘two-flux’ model, which means that only diffuse light is considered. This is appropriate where strong scattering occurs, where the incident light is diffuse, or where surfaces are optically rough [37]. In cases where the light is not fully diffuse, four-flux models [38], which treat both collimated and diffuse light, are more appropriate. In standard spectrophotometers, such as that used here, the incident light is collimated, and in many coatings, scattering is weak. However, for coatings produced by oxidation of metals, by sol-gel techniques, and by some other means, the surface of the coating is optically rough, and it is reasonable to use the Kubelka–Munk model. The Kubelka–Munk model uses an effective scattering coefficient S and an effective absorption coefficient K to describe the optical properties of the coating. The effective scattering coefficient is related to the usual scattering coefficient s by S ¼ 2ð1  zÞs,

(1)

where the forward scattering ratio z is defined as the ratio of the energy scattered by a particle in the forward hemisphere to the total scattered energy. For Rayleigh scattering, z ¼ 12, while for Mie scattering, 12ozo1. The effective absorption coefficient is related to the usual absorption coefficient k by K ¼ k,

(2) 

að z Þ ¼ K

where the average crossing parameter e is defined such that the average path length travelled by diffuse light crossing a length dz is e dz. For collimated light, e ¼ 1, while for semiisotropic (i.e., isotropic in the direction of propagation) diffuse light, e ¼ 2 [39]. It is usual in applying the Kubelka–Munk model to write z ¼ 12 and e ¼ 2, so that S ¼ s and K ¼ 2k. In applying the Kubelka–Munk model to real coatings, it is important to take into account reflection from the surfaces and interfaces. Here a coating on an opaque substrate under collimated illumination is considered. The light reflected from the front surface may have both collimated and diffuse components, but it is assumed that the light transmitted into the coating is diffuse. As noted above, this is in most cases reasonable in the case of a rough surface at the air–coating interface. Note that the term scattering is sometimes used to describe what is here called diffuse reflection. It is also important to distinguish between reflectance, which refers to the reflection of light from the complete coating–substrate system, and reflection coefficients, which refer to reflection from a single surface or interface. I will use R to denote reflectance, and r to denote a reflection coefficient. The following reflection coefficients have to be considered: rfcc and rfcd , which denote coefficients for the reflection of collimated light from the front of the air–coating interface, as collimated light and as diffuse light, respectively; and rbdd and rsdd , which denote coefficients for the reflection of diffuse light as diffuse light, from the back of the air–coating interface, and from the front of the coating–substrate interface, respectively. The diffuse reflectance in terms of the reflection coefficients from the different interfaces, the coating thickness, and the scattering and absorption coefficients is then given by [37]    1  rfcd  rfcc 1  rbdd RKM f Rcd ¼ rcd þ , (3) 1  rbdd RKM where RKM ¼

1  rsdd ½a  b cothðbShÞ . a þ b cothðbShÞ  rsdd

a ¼ ðS þ K Þ=S, b¼

pffiffiffiffiffiffiffiffiffiffiffiffiffi a2  1 ,

(4) (5) (6)

and h is the thickness of the coating. The collimated reflectance is given simply by Rcc ¼ rfcc .

(7)

The depth distribution of the absorbed power, normalized to the input light flux, is given by [34]

      1  rfcc  rfcd b 1 þ brsdd coshðSbh  SbzÞ þ ð1 þ aÞ 1  rsdd sinhðSbh  SbzÞ     , b 1  rbdd rsdd coshðSbhÞ þ a  rbdd  rsdd þ arbdd rsdd sinhðSbhÞ

(8)

ARTICLE IN PRESS A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

where z is the distance from the surface of the coating; a(z) has dimension m1. The absorbed photon flux per unit distance (photons m3 s1) is then Z 1 I l al ðzÞ dl, (9) J z ð zÞ ¼ 0

where Il (photons m2 s1 nm1) is the incident spectral photon flux in the wavelength band from l to l+dl, and where a(z) is written al(z) to denote its value in the wavelength band from l to l+dl. The cumulative absorbed photon flux (photons m2 s1) to depth d is given by Z dZ 1 J C ðd Þ ¼ I l al ðzÞ dl dz. (10) 0

0

2.2. Calculation of reflection coefficients for an opticallyrough surface

1329

rate data given by Dechamps and Lehr [44], and the values of s0 and t of the surface, measured with an atomic force microscope, are given in Table 1. While s0 and t were measured for only two of the samples, it is reasonable to assume that they will have similar values for the other sample. The TiO2 coatings produced appeared to be opaque. 2.4. Diffuse reflectance measurements The diffuse reflectance of the samples was measured using a Cary 5 spectrophotometer fitted with a diffuse reflectance attachment. The incident beam is collimated, and reflected light is captured by an integrating sphere. A matt Teflon reference was used to provide a nominal 100% reflectance measurement. 3. Kubelka–Munk results

Expressions for the reflection coefficients from the front surface rfcc and rfcd were derived for perpendicular incidence of collimated light, assuming that the height distribution of the surface is Gaussian and spatially isotropic, and therefore can be described by the rms surface roughness s0 and the autocorrelation length t [34,37]. The diffuse– diffuse reflection coefficients rbdd and rsdd were calculated using an average over all angles of incidence of the Fresnel reflection coefficient [34,37]. All the reflection coefficients depend on the complex refractive indices n+ik of the materials on either side of an interface. Values of the refractive index n and the extinction coefficient k for TiO2 were taken from Cardona and Harbeke [40] and Devore [41] as reported by Ribarsky [42]. Values of n and k for titanium were taken from Lynch and Hunter [43]. Note that the absorption coefficient is related to k by k ¼ 4pk=l,

(11)

where l is the wavelength of the light. 2.3. Preparation of TiO2 coatings on titanium substrates Rutile TiO2 coatings were produced by oxidizing a piece of titanium sheet (Sigma Aldrich, 99.7%, 0.25 mm thick) in oxygen at 1 bar in a tube furnace. The titanium sheet was etched in Kroll’s solution (one part 40% HF, one part 70% HNO3 and three parts water) for 10 s prior to oxidation, to give a rough substrate surface. The oxidation time and temperature, the thickness h estimated using the oxidation Table 1 Oxidation conditions and properties of rutile TiO2 coatings on Ti Sample Oxidation time (Min)

Temperature (1C)

h (mm)

s0 (mm)

t (mm)

1 2 3

750 750 850

0.5 1.0 2.0

0.583

6.49

0.571

6.48

4.8 19 10

3.1. Dependence of reflectance on coating properties Fig. 1 shows the diffuse reflectance Rcd , and the reflection coefficients rfcc , rfcd , rbdd and rsdd , calculated for the parameters h, s0 and t of sample 3. Also shown are the literature values of the refractive index n and the extinction coefficient k for rutile TiO2 that were used in the calculations. It is assumed in this instance that the scattering coefficient S is negligible. The expression used for Rcd is given in Eq. (3); full details of the calculations were given in Ref. [37]. The reflection coefficients follow the general shape of the refractive index, with a peak at 320 nm. The diffuse reflectance also follows this general shape, but shows an additional feature, a dip at wavelengths around 390 nm, corresponding to the wavelength at which the extinction coefficient, and therefore the absorption coefficient, approaches zero. The dip is a consequence of the transition from Rcd  rfcd at wavelengths for which absorption is strong, to Rcd 4rfcd at wavelengths for which the absorption is weak. It is this dip in the diffuse reflectance that provides information about the band gap of the TiO2 coating. It is instructive to examine the influence of the various parameters on the diffuse reflectance. Fig. 2 shows the effect of varying the refractive index n, the extinction coefficient k and the average crossing parameter e of the coating. The reflectance is strongly dependent on the refractive index at all wavelengths; this is a result of the dependence of the reflection coefficients on n. The dependence on the extinction coefficient occurs in two wavelength regions. At wavelengths shorter than about 350 nm, where kX0:05, changes in k affect the reflection coefficients. At wavelengths around 400 nm, where k is approaching zero, changes in k affect the absorption of light in the coating, and therefore the reflectance. The average crossing parameter e also affects the reflectance in this region; a decrease in e is equivalent to a decrease in k according to Eq. (2).

ARTICLE IN PRESS 1330

A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

Fig. 1. Reflection coefficients and diffuse reflectance, calculated using the complex refractive index of TiO2 (shown) on titanium, for a coating of thickness 1 mm and surface roughness parameters of sample 3.

Fig. 2. Effect of varying (a) refractive index n, (b) extinction coefficient k and (c) average crossing parameter e on the diffuse reflectance Rcd for a 1 um thick TiO2 coating on titanium with the surface roughness of sample 3.

Fig. 3 shows the effect of varying the scattering coefficient, the TiO2 coating thickness and the rms surface roughness on the reflectance. The scattering coefficient S is calculated using Eq. (1), with values of s and z calculated

Fig. 3. (a) Effect of varying scattering coefficient S on the diffuse reflectance Rcd; results are given for S corresponding to close-packed spheres of different radii. (b) Effect of varying TiO2 layer thickness h on the diffuse reflectance Rcd. (c) Effect of varying surface roughness on the diffuse reflectance Rcd, collimated reflectance Rcc, and total reflectance Rcd+Rcc. Standard conditions in all cases are a 1 um thick TiO2 coating on titanium with the surface roughness of sample 3 and negligible scattering coefficient.

using the Mie scattering computer code BHMIE [45]. The calculation assumes spherical particles with the complex refractive index of rutile TiO2, closely packed in an air environment. The maximum value of S under this

ARTICLE IN PRESS A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

assumption is of order 107 m1. For comparison, Kb107 m1 for lp340 nm and Kb107 m1 for lX380 nm. In real coatings, the scattering coefficient, and hence the effect of scattering, will be smaller, since the particles will not be separate; in some coatings, including the ones studied in the experiments reported here, scattering is negligible. Nevertheless, it is clear that scattering can have a significant influence on the reflectance for lX350 nm. Fig. 3b shows the effect of altering the thickness h of the TiO2 coating. Increasing h shifts the dip in the reflectance that occurs around 400 nm to longer wavelengths; as noted earlier, this dip is associated with the decrease in the absorption coefficient to zero. This effect will be examined in detail in Section 3.2. The surface roughness of the TiO2 coating can also influence the reflectance, as shown in Fig. 3c. Decreasing the rms roughness s0 to 10 nm for t/s010 leads to a decrease in the diffuse reflectance, and an increase in the collimated reflectance Rcc; the total reflectance remains approximately the same. For smaller values of t/s0, the total reflectance can decrease significantly [37]. Overall, it is clear from Figs. 2 and Fig. 3 that the diffuse reflectance depends strongly on a large number of parameters. In the region around 400 nm where the extinction coefficient decreases to zero, the refractive index, the average crossing parameter, the scattering coefficient, the coating thickness and the surface roughness can all have an effect on the reflectance as significant as that of the extinction coefficient (and hence the absorption coeffi-

1331

cient). In the next subsection, using the example of the coating thickness, I will examine the consequences for attempts to determine the band gap from measurements of diffuse reflectance. 3.2. Calculation of the band gap from the diffuse reflectance Many workers [14,21,23,32,33] have attempted to obtain the band gap of a semiconductor coating on an opaque substrate directly from diffuse reflectance measurements. They fitted a line to the long-wavelength edge of the dip in the diffuse reflectance (see Fig. 1); the band-gap wavelength was taken to be the intersection of this line with a horizontal line corresponding to the maximum reflectance. In practice, this was done by calculating the absorbance A ¼ Rmax  Rcd ,

(12)

where Rmax is the maximum value of Rcd for wavelengths longer than that of the dip in the diffuse reflectance. This sets the minimum absorbance to zero; the band-gap wavelength is obtained by extrapolating the long-wavelength edge of the peak in absorbance to this zero line. Fig. 4a, taken from Khan et al. [14], illustrates this method. Khan et al. obtained a band-gap wavelength of 414 nm (3.0 eV) for the undoped TiO2, and 535 nm (2.32 eV) and 440 nm (2.82 eV) for the carbon-doped TiO2. Fig. 4b shows the application of the same method to calculated absorbance curves, obtained using Eqs. (3) and (12), for TiO2 coatings of different thicknesses. It is clear that the band-gap wavelength obtained using the method

Fig. 4. Determination of the band-gap wavelength from diffuse reflectance measurements. (a) Method used to fit measurements of absorbance by Khan et al. [14]. (b) Fitting by the same method of the calculated absorbance of TiO2 coatings of different thicknesses.

ARTICLE IN PRESS 1332

A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

depends strongly on the thickness of the coating, increasing from 390 nm (3.18 eV) to 419 nm (2.95 eV) as the thickness increases from 100 nm to 10 mm. It is clear from Fig. 2 and Fig. 3 and the discussion in Section 3.1 that the band gap calculated using this method will also depend on other parameters, such as scattering coefficient, refractive index and average crossing parameter. It is interesting to note that the wavelength dependence of the diffuse reflectance of gallium arsenide has been investigated by Johnson and Tiedje [46] for the application of optical band-gap thermometry. They found that the apparent band gap derived from the diffuse reflectance depended on thickness of the gallium arsenide and its surface texture, necessitating compensation if the temperature was to be accurately measured. 4. Measurements and discussion 4.1. Diffuse reflectance and band-gap determination of TiO2 samples In order to examine the validity of the calculations, and to investigate the possibility of determining the band gap and absorption coefficient from diffuse reflectance measurements, three samples with TiO2 coatings of different thicknesses were prepared as described in Section 2.3. The diffuse reflectance of the samples is shown in Fig. 5a. The band gap is obtained using the method described in Section 3.2; this is shown in Fig. 5b.

Fig. 6 shows the band gaps obtained for the three samples. These are compared to the band gap obtained from absorbances calculated using the Kubelka–Munk model, such as those shown in Fig. 4b, for a range of coating thicknesses. The band gaps of the samples are a little lower than the calculations predict, and the dependence on the coating thickness is not as strong. This is mainly a consequence of the variations in the refractive index of the TiO2 coatings with thickness. It has been shown in an earlier publication that the refractive index of the thickest sample, sample 3, is closest to the value used in the calculations [34]; this is the sample for which best agreement with the calculations is obtained. The refractive index for the other samples is significantly lower than that used in the calculations [34]. This may be due to the presence of a suboxide layer that is expected to be formed close to the substrate for all coatings [44,47], and which will be of more relative significance for the thinner coatings. It can be seen from Fig. 2a that decreasing the refractive index leads to a decrease in the apparent band gap. The measurements confirm that the apparent band gap obtained by extrapolating the peak in the absorbance to the zero line depends on the thickness of the sample. The question remains whether it is possible to determine the real band gap from measurements of the diffuse reflectance. This is addressed in the next subsection. The results obtained by Shangguan et al. [32], who examined the diffuse reflectance of 0.3% Pt/TiO2 films on glass, and found that the thinnest film had a lower

Fig. 5. (a) Measured diffuse reflectance for three samples with TiO2 coatings of different thicknesses. (b) Absorbance of TiO2 coatings, showing linear extrapolations to obtain the band-gap wavelength.

ARTICLE IN PRESS A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

1333

Fig. 6. Dependence of apparent band gap energy, obtained by extrapolation of the peak in the calculated absorbance, on TiO2 coating thickness, and band gaps for the three samples obtained by the same method from the measured absorbance.

Fig. 7. a) Absorption coefficient of the three samples derived from the diffuse reflectance measurements, and (b) Tauc plots showing possible fits to obtain the band gap.

band-gap wavelength (i.e., higher band-gap energy) than the other thicker films, even though its photocatalytic activity was the same, is consistent with the trend shown in Fig. 6. It is hence likely that the decreased band-gap wavelength for the thinnest film is apparent rather than real. 4.2. Determination of band gap from absorption coefficient It has been shown in an earlier publication [34] that it is possible to derive the absorption coefficient from measurements of the diffuse reflectance of the samples. This was done using the spectral projected gradient method to invert

the Kubelka–Munk expression for the diffuse reflectance (Eq. (3)), thereby obtaining values of n and k (or equivalently, k) that give the best fit to the measurements. The surface roughness parameters s0 and t, and the thickness h were fixed, and the results were consistent with a negligibly small scattering coefficient. Fig. 7a shows the best fit values of the absorption coefficient for the three samples. For crystalline solids with an indirect band gap, such as rutile TiO2, the dependence of the absorption coefficient k on the frequency n can be approximated as  2 (13) khn ¼ A hn  E g ,

ARTICLE IN PRESS 1334

A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

where A is a constant [48]. It is clear from Eq. (13) that the band gap Eg can be obtained by extrapolating to zero a linear fit to a plot of (khv)1/2against hv (often referred to as a Tauc plot). Note that some authors exclude the hv factor on the left-hand side of Eq. (13) [49,50]; this has only a minor influence on the value of Eg obtained. Fig. 7b shows Tauc plots for the three samples. A straight line fits the sample 3 data well, yielding a band gap of 3.0070.02 eV. The fits for samples 2 and 3 are not as good, so the uncertainties in the band gap are larger; the band gaps are estimated to be in the range 3.070.1 eV. These results indicate that it is possible to determine the absorption coefficient, and hence the band gap, from inversion of the measurements of diffuse reflectance, although precise determination of the band gap may not always be possible. 5. Implications for water splitting by doped TiO2 electrodes In this section, I apply the methods that were used to determine the absorption coefficient of the oxidized titanium samples to the diffuse reflectance measurements of Khan et al. [14]. Unfortunately the measurements, reproduced in Fig. 4a, are in arbitrary units and no zero is given. The following procedure was adopted to circumvent this problem. First, the diffuse reflectance was obtained from the absorbance using Eq. (12) with Rmax ¼ 0. The undoped TiO2 reflectance curve was then scaled and shifted vertically so that it matched the diffuse reflectance of sample 3 at 389 nm (the wavelength at which the reflectance of sample 3 is minimum) and 430 nm (the maximum wavelength for which Khan et al. presented data). Fig. 8a

shows the resulting diffuse reflectance curves. The undoped TiO2 curve matches that for sample 3 closely, giving some confidence in the procedure. The absorption coefficient was then obtained for the carbon-doped TiO2 case using the inversion method. The thickness of the coating was set to 2 mm, as for sample 3. The result is shown in Fig. 8b. For wavelengths below about 400 nm, the absorption coefficient of the carbondoped TiO2 is similar to that of sample 3, reaching about 108 m1. For longer wavelengths, the carbon-doped TiO2 shows only weak absorption, with absorption coefficient between 104 and 105 m1 up to a wavelength of about 500 nm. Using Eq. (13), it is possible to derive two band gaps, as shown in Fig. 9, corresponding to the main (sub400 nm) section of the absorption coefficient against wavelength curve and the high-wavelength (400–510 nm) section. The band-gap energies are 3.1370.02 and 2.4470.02 eV, respectively. Both band gaps are larger than the values of 2.82 and 2.32 eV, respectively, obtained by Khan et al. [14] by fitting a line directly to the absorbance. In solar water-splitting applications, only charge carriers produced by absorption of photons within a relatively small distance of the surface will reach the electrolyte before recombining. This distance is material dependent, and depends on the depth of the depletion layer and the diffusion length of the charge carriers. For a flame-oxidized TiO2 coating, it was measured to be about 200 nm [3]. It is important, therefore, to determine the depth profile of photon absorption for the carbon-doped TiO2, which can be calculated using Eq. (10), assuming illumination by the standard AM1.5 global solar spectrum [51]. The results are shown in Fig. 10. Results are given for the full solar

Fig. 8. a) Diffuse reflectance of undoped and carbon-doped TiO2 coatings, modified from the data given by Khan et al. [14] so that the undoped TiO2 reflectance agrees with that of sample 3. (b) Absorption coefficients of the carbon-doped TiO2 coating and sample 3, obtained by inversion of the reflectance in (a).

ARTICLE IN PRESS A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

1335

Fig. 9. Tauc plot of data of Khan et al. [14], showing fits to obtain the band gaps.

Fig. 10. Cumulative absorbed photon flux JC(d) versus depth d for the carbon-doped TiO2 coating of Khan et al. [14]. Results are given for the full AM1.5 global solar spectrum, and the sub-410 nm part of the spectrum. The ratio of the two results is also given.

spectrum, and the part of the spectrum below 410 nm (corresponding to the 3.0 eV band gap of rutile TiO2). About 35% of the photons absorbed in the full 2mm depth of the coating are of wavelength longer than 410 nm, indicating that additional absorption due to the carbondoping of the TiO2 is significant. However, only 10% of the photons absorbed in the top 200 nm of the coating are of wavelength longer than 410 nm, indicating that the carbon doping of the TiO2 will only make a small contribution to water splitting. Allowing a larger distance of 500 nm increases the percentage of absorbed photons that have wavelength longer than 410 nm, but only to 17%. It is emphasized that the calculations on which these figures are based take into account the increased optical path length due to the surface roughness of the electrode and multiple reflections; in the Kubelka–Munk model, it is assumed that the light is diffuse, and reflections from the air–coating and coating–substrate interfaces are taken into account. Also,

they provide only an upper limit to the contribution of the visible light, since absorbed photons do not necessarily result in water-splitting reaction, due to for example recombination of electron–hole pairs at trap states [52]. Khan et al. [14] claimed that their carbon-doped electrode had a water-splitting efficiency of 8.35% under illumination by a xenon lamp with spectrum similar to AM1.5, compared with 1.08% for the undoped electrode. Murphy et al. [3] and Ha¨gglund et al. [53] have shown that the 8.35% efficiency is not possible for the band gap of 2.32 eV (535 nm), even with 100% utilization of all incident photons with energies greater than 2.32 eV in water-splitting reactions. Here it is shown that only a small increase in the efficiency is expected due to carbondoping. This is consistent with the results obtained by other workers [18,19,52] who have been unable to reproduce Khan et al.’s efficiencies for carbon-doped TiO2 electrodes.

ARTICLE IN PRESS 1336

A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337

It should be noted that because Khan et al. [14] provided neither a scale for their absorbance data nor thicknesses of their coatings, the above calculations cannot be precise. Nevertheless, the conclusions that the absorption coefficient in the visible light region is orders of magnitude smaller than in the ultraviolet region, and that the absorption of visible light will make only a minor contribution to the water-splitting efficiency, are robust. This is a consequence of the fact that a small decrease in absorbance corresponds to a decrease of orders of magnitude in the absorption coefficient, and the requirement that photons be absorbed within about 200 nm of the surface to be effective in water splitting. These considerations can also be applied to the diffuse reflectance measurements of other workers. For example, Shankar et al. [23] found an increased absorbance at visible wavelengths for nitrogen-doped TiO2 nanotubes compared to undoped TiO2 nanotubes. However, the absorbance at visible wavelengths is 50% or less of that at ultraviolet wavelengths, so the absorption coefficient will be greatly decreased. It should be noted here that quantitative analysis is more difficult in this case because of the complicated geometry. It is possible to mitigate the effects of a low absorption coefficient by employing nanostructured electrodes, incorporating for example large-aspect ratio nanotubes or nanorods aligned perpendicular to the substrate. Using such structures, electrode thicknesses parallel to the incident light can be increased significantly, while photons are still absorbed close to the surface. Nevertheless, as noted above, Fig. 10 shows that even with a 2 mm electrode thickness, only 35% of the absorbed photons are at wavelengths longer than 410 nm.The theoretical maximum efficiency for a semiconductor absorbing all photons at wavelengths shorter than 410 nm is 2.2% [3], so the maximum efficiency for a 2 mm thick carbon-doped TiO2 nanostructured electrode with the absorption coefficient shown in Fig. 9b is 3.0%. This neglects losses due to reflection, the requirement for a bias voltage, etc. Recent experimental studies have investigated watersplitting efficiencies for carbon-doped TiO2 nanotube arrays [24,25]. One of the studies showed an increase in efficiency with nanotube length for undoped nanotubes [25], and both demonstrated significant visible activity for doped nanotubes, although overall efficiencies were still below 1%, in accordance with the calculation above. In one case, incident photon conversion efficiency (IPCE) measurements were used to measure the band gap using a Tauc plot [25]. This method, which is widely used, requires the assumption that the absorption coefficient for interband transitions is proportional to the photocurrent density, but does have the advantage that absorption due to transitions not related to the band gap, such as the d–d transitions that occur in transition-metal-doped oxides, are ignored. A number of other recent studies have found visible photocatalytic activity using carbon-doped TiO2 in the

destruction of organic molecules [54–56]. However, these studies have used suspensions of doped TiO2 powder, in which case very large optical depths are obtainable. In such applications, the weak visible absorption coefficient is far less important than in photoelectrochemical water-splitting, in which a solid photoelectrode is required. 6. Conclusions The Kubelka–Munk radiative transfer model, modified to treat optically rough surfaces, has been applied to TiO2 coatings on a titanium substrate, as used for solar water splitting. It has been shown that the diffuse reflectance depends on parameters including the coating thickness, refractive index, scattering coefficient and surface roughness, as well as the absorption coefficient. Hence, derivation of the absorption coefficient and band gap of the coating has to take into account all these parameters. The effect of the coating thickness has been examined in detail, and it has been shown that the widely used practice of determining the band gap of the coating directly from the diffuse reflectance is likely to be inaccurate. However, it has been found that, by applying an inversion technique to the diffuse reflectance measurements, it is possible to derive the absorption coefficient, and hence the band gap, more reliably. This inversion technique has been also applied to the measurements of diffuse reflectance of carbon-doped TiO2 electrodes presented by Khan et al. [14], who measured an extension of the absorption into the visible light region due to carbon doping. It is found that the band gap estimated by Khan et al. is inaccurate, and further that the absorption coefficient in the visible region is orders of magnitude lower than in the ultraviolet region, with the consequence that only a small proportion of the visible photons are absorbed near the surface of the electrode. As a consequence, the visible photons are expected to make only a small contribution to the water-splitting efficiency. Use of electrodes incorporating large-aspect-ratio nanostructures, such as nanotubes or nanorods, may be a means of mitigating the effects of the low visible absorption coefficients. Acknowledgements I thank Dr Piers Barnes for providing rms roughness and autocorrelation length data for the TiO2 samples, and Dr Barnes, Ms Julie Glasscock and Dr Ian Plumb for useful discussions and comments. References [1] M.F. Weber, M.J. Dignam, J. Electrochem. Soc. 131 (1984) 1258–1265. [2] J.R. Bolton, S.J. Strickler, J.S. Connolly, Nature 316 (1985) 495–500. [3] A.B. Murphy, P.R.F. Barnes, L.K. Randeniya, I.C. Plumb, I.E. Grey, M.D. Horne, J.A. Glasscock, Int. J. Hydrogen Energy 31 (2006) 1999–2017.

ARTICLE IN PRESS A.B. Murphy / Solar Energy Materials & Solar Cells 91 (2007) 1326–1337 [4] B. Karvaly, I. Hevesi, Z. Naturforsch. 26a (1971) 245–249. [5] W.W. Wendlandt, H.G. Hecht, Reflectance Spectroscopy, Wiley Interscience, New York, 1966. [6] O.S. Heavens, Optical Properties of Thin Solid Films, Butterworths, London, 1955. [7] E. Shanti, E. Dutta, A. Banerjee, K.L. Chopra, J. Appl. Phys. 51 (1981) 6243–6251. [8] S.P. Lyashenko, V.K. Miloslavskii, Opt. Spectrosc. 16 (1964) 80–81. [9] R. Swanepoel, J. Phys. E: Sci. Instrum. 16 (1983) 1214–1222. [10] K. L. Eskins and K. W. Mitchell, in: Proceedings of the 18th IEEE Photovoltaic Specialists Conference, (IEEE, 1985), pp. 720–725. [11] D.E. Aspnes, A.A. Studna, Phys. Rev. B 27 (1983) 985–1009. [12] G.E. Jellison, L.A. Boatner Jr., J.D. Budai, B.-S. Jeong, D.P. Norton. J. Appl. Phys. 93 (2003) 9537–9541. [13] A. Fujishima, K. Kohayakawa, K. Honda, J. Electrochem. Soc. 122 (1975) 1487–1489. [14] S.U.M. Khan, M. Al-Shahry, W.B. Ingler Jr., Science 297 (2002) 2243–2245. [15] J. Akikusa, S.U.M. Khan, Int. J. Hydrogen Energy 22 (1997) 875–882. [16] S.-E. Lindquist, A. Lindgren, Y.-N. Zhu, J. Electrochem. Soc. 132 (1985) 623–631. [17] S.U.M. Khan, J. Akikusa, J. Electrochem. Soc. 145 (1998) 89–93. [18] P.R.F. Barnes, L.K. Randeniya, A.B. Murphy, P.B. Gwan, I.C. Plumb, J.A. Glasscock, I.E. Grey, C. Li, Dev. Chem. Eng. Mineral Process. 14 (2006) 51–70. [19] K. Noworyta, J. Augustynski, Electrochem. Solid State Lett. 7 (2004) E31–E33. [20] X. Wu, Z. Jiang, H. Liu, S. Xin, X. Hu, Thin Solid Films 441 (2003) 130–134. [21] C. He, X.Z. Li, N. Graham, Y. Wang, Appl. Catal. A: General 305 (2006) 54–63. [22] P.R. Mishra, P.K. Shukla, A.K. Singh, O.N. Srivastava, Int. J. Hydrogen Energy 28 (2003) 1089–1094. [23] K. Shankar, K.C. Tep, G.K. Mor, C.A. Grimes, J. Phys. D: Appl. Phys. 39 (2006) 2361–2366. [24] K. Shankar, M. Paulose, G.K. Mor, O.K. Varghese, C.A. Grimes, J. Phys. D.: Appl. Phys. (2005) 3543–3549. [25] J.H. Park, S. Kim, A.J. Bard, Nano Lett 6 (2006) 24–28. [26] P.A. Kohl, S.N. Frank, A.J. Bard, J. Electrochem. Soc. 124 (1977) 225–229. [27] W. Sun, C.R. Chenthamarakshan, K. Rajeshwar, J. Phys. Chem. B 106 (2002) 11531–11538. [28] J.A. Byrne, B.R. Eggins, N.M.D. Brown, B. McKinney, M. Rouse, Appl. Catal. B: Environ. 17 (1998) 25–36. [29] W. Gissler, R. Memming, J. Electrochem. Soc. 124 (1977) 1710–1714. [30] R.K. Pandey, S. Mishra, S. Tiwari, P. Sahu, B.P. Chandra, Solar Energy Mater. Solar Cells 60 (2000) 59–72.

1337

[31] W.-W. So, K.-J. Kim, S.-J. Moon, Int. J. Hydrogen Energy 29 (2004) 229–234. [32] W. Shangguan, A. Yoshida, M. Chen, Solar Energy Mater. Solar Cells 80 (2003) 433–441. [33] S.-Z. Chen, P.-Y. Zhang, D.-M. Zhuang, W.-P. Zhu, Catal. Commun. 5 (2004) 677–680. [34] A.B. Murphy, Appl. Opt., doc. 46 (2007) 3133–3143. [35] P. Kubelka, F. Munk, Z. Tech. Phys. 12 (1931) 593–601. [36] P. Kubelka, J. Opt. Soc. Am. 38 (1948) 448–457. [37] A.B. Murphy, J. Phys. D: Appl. Phys. 39 (2006) 3571–3581. [38] B. Maheu, J.N. Letoulouzan, G. Gouesbet, Appl. Opt. 23 (1984) 3353–3362. [39] G. Kortu¨m, Reflectance Spectroscopy, Springer, New York, 1969. [40] M. Cardona, G. Harbeke, Phys. Rev. 137 (1965) A1467–A1476. [41] J.R. Devore, J. Opt. Soc. Am. 41 (1951) 416–419. [42] M.W. Ribarsky, Titanium dioxide (TiO2) (rutile), in: E.D. Palik (Ed.), Handbook of Optical Constants, Academic, Orlando, Florida, 1985, pp. 795–800. [43] D.W. Lynch, W.R. Hunter, Introduction to the data for several metals, in: E.D. Palik (Ed.), Handbook of Optical Constants of Solids, vol. III, Academic, Orlando, Florida, 1998, pp. 233–286. [44] M. Dechamps, P. Lehr, J. Less-Common Metals 56 (1977) 193–207. [45] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley-Interscience, New York, 1983. [46] S.R. Johnson, T. Tiedje, J. Crystal Growth 175/176 (1997) 273–280. [47] P.R.F. Barnes, L.K. Randeniya, P.F. Vohralik, I.C. Plumb, J. Electrochem. Soc. 154 (2007) H249–H257. [48] E.J. Johnson, Absorption near the fundamental edge, in: R.K. Willardson, A.C. Beers (Eds.), Optical Properties of III-V Compounds, Semiconductors and Semimetals, vol. 3, Academic Press, New York, 1967, pp. 153–258. [49] J.I. Pankove, Optical Processes in Semiconductors, Dover, New York, 1971. [50] C.R. Aita, Y.-L. Liu, M.L. Kao, S.D. Hansen, J. Appl. Phys. 60 (1986) 749–753. [51] ASTM, Standard tables for reference solar spectral irradiance at air mass 1.5: Direct normal and hemispherical for a 371 tilted surface, American Society for Testing and Materials Standard G159-98, West Conshohocken, Pennsylvania, 1998. [52] B. Neumann, P. Bogdanoff, H. Tributsch, S. Sakthivel, H. Kisch, J. Phys. Chem. B 109 (2005) 16579–16586. [53] C. Ha¨gglund, M. Gra¨tzel, B. Kasemo, Science 301 (2003) 1673b. [54] S. Sakthivel, H. Kisch, Angew. Chem. Int. Ed. 42 (2003) 4908–4911. [55] H. Irie, Y. Watanabe, K. Hashimoto, Chem. Lett. 32 (2003) 772–773. [56] C. Xu, R. Killmeyer, M.L. Grey, S.U.M. Khan, Appl. Catal. B: Envir. 64 (2006) 312–317.