ASSESSING THE LIMITS OF ACCURACY FOR THE TAUC METHOD FOR OPTICAL BAND GAP DETERMINATION

Dunbar P. Birnie, III

Rutgers University, Department of Materials Science and Engineering New Brunswick, New Jersey, 08854-8065

ABSTRACT

Scientists and engineers working with nanotechnology and thin film optical devices often make use of "Tauc plots" to determine band gaps and evaluate the effect of processing conditions on the quality of coatings made for these applications. Broad-band optical data are easy to acquire and usually exhibit a region of reasonable transparency and then a sharp rise in absorption with increasing photon energy as the band-gap energy is exceeded. The shape of the onset of absorption is diagnostic of whether the band-gap is direct or indirect. Then, an appropriate linear regression can be used to extrapolate to the band gap value, though sometimes the extrapolation is quite far in absolute energy terms from the data used to make the extrapolation. This paper covers some of our recent work where we use known materials to standardize the fitting protocols and assess the accuracy of this simple method.

INTRODUCTION

In our earlier work with thin films (and for many studies in the literature that use the Tauc method) we've noticed that the distance of extrapolation in the fitting process may be relatively large, and the tail of sub-band-gap absorption can also be quite large. This raised the basic question about how accurate the Tauc method would be, and how to establish procedures that improve the accuracy of the fitting results[1]. We delved deeply into this problem by looking at ZnO thin films because they are an extremely well-studied material and ZnO is known to have a direct band gap. By looking closely at a population of over 120 thin film Tauc fits we found the band-gap results overall were consistent with a value of 3.27 +/- 0.05 eV, with evidence for two small outlier populations [1]. A subpopulation of higher gap values appeared to be caused by nanoparticle quantum confinement effects (not surprisingly), while a subpopulation of lower gap values appeared to be correlated with more defective samples. These were essentially cases that had stronger sub-band-gap absorption, which has the mathematical effect of shifting the intercept point somewhat to the left and making the confidence interval of the band-gap determination wider (less accurate). To quantify this effect and provide a figure of merit for identifying the more accurate samples, we introduced the "near-edge absorptivity ratio (NEAR)". And, when using the NEAR to focus on the more accurate data sets, we found that the Tauc method generally gave an experimental distribution of results with a standard deviation of only 0.033 eV, thus emphasizing the relatively high accuracy of the method in general.

We extend that work to the case of indirect band-gap materials and examine accuracy limits based on absorption coefficient values and coating thickness effects that can influence the signal-to-noise ratio of real optical absorption data. Indirect band-gaps are more difficult to characterize because their absorption intensities are characteristically weaker, which provides an added difficulty when most optical data are determined from thin film samples. We address this problem by working with single crystal data from silicon, probably the most well-characterized indirect band-gap material available.

BACKGROUND

The seminal work of Tauc, Grigorovici, and Vancu [2] presented a simple method that uses broad band absorption spectra and interpreted the shape of the absorption edge to arrive at a determination of the band gap, and its character. Their method was further developed in Davis and Mott's more general work on amorphous semiconductors [3, 4]. Together they've shown that the optical absorption strength depends on the difference between the photon energy and the band gap as shown in (Eq. 1):

where h is Planck's constant, v is the photon's frequency, a is the absorption coefficient, Eg is the band gap and A is a proportionality constant. The value of the exponent denotes the nature of the electronic transition, whether allowed or forbidden and whether direct or indirect:

Typically, the allowed transitions dominate the basic absorption processes, giving either n=1/2 or n=2, for direct and indirect transitions, respectively.

Thus, the basic procedure for a Tauc analysis is to acquire optical absorbance data for the sample in question that spans a range of energies from below the band gap transition to above it. Then, plotting the (a h v) with various test exponents versus photon energy allows the researcher to decide which of the exponents gives the most linear plot. Finally, with this exponent, the line is extrapolated down to intersect the X-axis, which will be the band-gap value (as can be interpreted from Equation 1). Of the four exponent choices listed, it is usually found that either the ½ and 2 exponents are most frequently used (being associated with the allowed transitions).

ANALYSIS OF DIRECT-GAP MATERIALS

Zinc oxide was a good candidate for evaluating the Tauc method because it has been widely studied for a number of useful applications [5-13]. Among these applications the band-gap plays a central and fundamental role as it controls many absorption and conductivity phenomena. Single crystal optical studies have found a direct band gap of 3.3 eV[14-16], though many of the papers surveyed in our thin film analysis were collected from very well crystallized films or even epitaxially grown layers[1]. ZnO was also attractive as a reference material because of its high level of stoichiometry. While every stoichiometric compound must thermodynamically have point defects at some level (and therefore by definition be non-stoichiometric), the phase of ZnO has been experimentally studied and found to have very little deviation from the ideal 1:1 ratio. For example, the early work of Allsopp and Roberts found a slight zinc excess, but less than 50 ppm [17]. This is much more stoichiometric than many phases and thus provided a good calibration test-case for the Tauc method.

Figure 1 gives one example Tauc plot for ZnO where the absorption coefficient times the photon energy to the second power is plotted versus the incident photon energy[18]. The second power was used as zinc oxide is well known to have a direct allowed transition. The characteristic features of Tauc plots are evident: at low photon energies the absorption approaches zero - the material is transparent; near the band-gap value the absorption gets stronger and shows a region of linearity in this squared-exponent plot. This linear region has been used to extrapolate to the X-axis intercept to find the band gap value (here about 3.28 eV).

Figure 1: Example Tauc Plot from UV-Vis analysis of a ZnO thin film, illustrating the method of fitting the linear region to extrapolate the band-gap at the X-axis intercept, here about 3.28 eV. Data replotted from ref.[18].

At even higher energies the absorption processes saturate and the curve again deviates from linear. To select and justify a linear region for extrapolation one must understand the reasons for these lower and upper deviations from linear behavior. On the low energy end, the deviation from linearity can be associated with defect absorption states that are near the band edge. This phenomenon has been investigated by Urbach [19] and in subsequent years, therefore, identified as an "Urbach Tail." These states are usually described by an exponential function, corresponding to a typical distribution of density of states, evident in the absorption behavior seen in the example Tauc plot (Figure 1). On the high energy end, saturation of available transition states can be responsible for a leveling out of absorption strength in most collected spectra.

The absorption data are rooted in the possible optical transitions within the electronic structure of the material. Figure 2 (next page) shows the band diagram for ZnO[20], showing that the material is direct and that the band-gap derives from states at, G, the center of the Brillouin Zone. A representative direct optical transition is shown for a photon energy slightly larger than the band-gap energy.

Selecting the "right" points to use for fitting from Figure 1 is largely subjective, but could also have a profound effect on the extrapolated value for the band-gap. In our earlier work on ZnO, we tried to develop a completely unbiased method for picking the linear portion of the plot and finding the band gap value[1]. Digital data were processed in a spreadsheet to achieve a series of linear regressions corresponding to incremental portions of the data set. We typically fitted using an 11-data-point window for evaluating the local linear regression (using +/- 5 datapoints on either side for any given local fit), and then we slid this fitting-window along and tested the fit at every possible location. The impact of fitting window width can be illustrated in Figure 3 where we plot the R2 value for each incremental linear regression fit for the data we extracted from the graph shown in Figure 1. Three different curves are presented that cover 5, 11, and 15 datapoint fitting windows, respectively. When fewer data points are used for fitting then better R2 results are generally obtained (as a mathematical certainty). However, if the actual linear region is relatively short then using a bigger span of datapoints will...