Optical band gap and the Burstein–Moss effect in iodine doped PbTe using diffuse reflectance infrared Fourier transform spectroscopy

Optical absorption edge measurements are performed on I doped PbTe using diffuse reflectance infrared Fourier transform spectroscopy. The Burstein–Moss shift, an increase in the absorption edge (optical band gap) with increasing doping level, is explored. The optical gap increases on the order of 0.1 eV for doping levels ranging from 3 × 1018 to 2 × 1020 cm−3, relevant doping levels for good thermoelectric materials. Chemical potential is estimated from transport measurements—specifically, Hall effect and Seebeck coefficient—using a single band Kane model. In heavily doped semiconductors, it is well-known that the band gap shrinks with increasing doping level. This effect, known as band gap renormalization, is fit here using an n1/3 scaling law which reflects an electron–electron exchange interaction. The renormalization effect in these samples is shown to be more than 0.1 eV, on the same order of magnitude as the band gap itself. Existing models do not explain such large relative changes in band gap and are not entirely self-consistent. An improved theory for the renormalization in narrow gap semiconductors is required.


Introduction
The maximum efficiency obtainable by a thermoelectric device is related to the dimensionless figure of merit, zT = S 2 σ κ T , where S is the thermopower (or Seebeck coefficient), σ is the electrical conductivity, κ is the thermal conductivity and T is the temperature. Each of the properties included in zT depends intimately upon the number of free carriers and electronic transport parameters of the material. These parameters arise from the materials' electronic band structure. Recent work on PbTe and other materials highlight the manipulation of these electronic states or 'band engineering' as a method for improving thermoelectric efficiency [1][2][3][4]. Band engineering in thermoelectric PbTe involves alloying and doping to modify the band structure in a way that improves the thermoelectric quality factor [4]. Of particular importance is the relative position of the primary and secondary transport bands that, as measured by the band gaps, can be modified with temperature, alloying and even doping.
PbTe alloys have some of the highest efficiencies for thermoelectric power generation, with zT values above 1 in the temperature range useful for waste heat recovery [1,[5][6][7]. In p-type PbTe a complex valence band structure exists that has been described with two valence bands: a lighter band at the L point (direct) and a heavier at the line in the Brillouin zone [1,2,6,[8][9][10][11][12][13][14][15][16][17]. While the qualitative results of density functional theory (DFT) calculations show excellent agreement, band gap values and transport properties are known to be dependent on the specific approximations made such as the specific exchange/correlation functional, how the alloy is accounted for (e.g. supercell or Korringa-Kohn-Rostoker coherent potential approximation (KKR-CPA)), assumptions about the relaxation time and the effect of temperature.
Measurement of the optical properties directly investigates the electronic structure of the material giving a wealth of information useful for engineering thermoelectric properties [18]. Optical measurements can be used to estimate the energy difference between the valence and conduction bands, the band gap (E g ), which can help determine the electronic and thermoelectric properties of materials. At high temperatures, when E g ∼ 3 or 4k B T , the Fermi distribution is sufficiently broad such that a significant minority carrier population can develop, which results in a cancelling of a portion of the Seebeck voltage. Peltier heat currents from different carrier types exist even at zero net electrical current thereby leading to the bipolar thermal conductivity [19]. The presence of minority carriers in thermoelectric materials is the primary reason for reduction of zT at high temperatures.
There are several experimental methods of measuring the band gap, both using electronic and optical properties. First, the band gap can be estimated from electronic resistivity for an intrinsic semiconductor (where the number of holes and electrons are approximately equal), but this simple calculation assumes that both electrons and holes have the same mobility and that it has a specific temperature dependence. Alternatively, the peak in the Seebeck coefficient that appears at high temperatures due to the influence of minority carriers can be used to estimate the band gap as E g = 2α max T [20], but again this requires similar assumptions about the relative mobility of holes and electrons. Because of the assumptions these methods can only give qualitative values (within 0.1 eV or ∼30% in PbTe). For a more direct measurement of band gap, optical methods have been preferred.
Due to their ease of heavy doping and high electrical conductivity, thermoelectric materials are generally small band gap semiconductors with gaps in the mid infrared (0.1-0.5 eV). Optical measurements in PbTe were first performed in the 1950 and 1960s and showed a band gap of around 0.3 eV with a temperature dependence of [21][22][23][24]. Prakash investigated the pressure dependence of the fundamental gap and determined that only about half of the change with temperature could be attributed to thermal expansion while the rest was probably due to interaction with phonons [25,26]. Nikolic and others have studied the optical band gaps in a variety of lead chalcogenide alloys [27][28][29][30]. Several studies of p-type PbTe found evidence of a second valence band using optical methods through inter-valence band transitions below the fundamental absorption energy [9,31]. Also, the formation of impurity states has been observed and their energies have been estimated using optical absorption for PbTe doped with In [32][33][34] and Tl [34,35]. Other thin film and bulk absorption measurements of alloys and quantum wells are useful for design of infrared lasers and detectors [36][37][38][39][40]. While there are many optical measurements of the band gap, many studies do not consider the effects of free carriers on the fundamental absorption edge.
Many of the previous reports of optical band gap measurements on PbTe have used single crystal samples measured in transmission [22, 23, 25, 29-31, 41, 42]. Transmission experiments have a few disadvantages. Firstly, optically transparent single crystals are often used to avoid internal reflections and light scattering at the grain boundaries-which are generally more difficult to prepare than polycrystalline samples. Secondly, the samples must be polished extremely thin since the absorbance scales exponentially with the absorption coefficient and sample thickness through Beer's law. This limits the highest absorption coefficient that can be measured for a given sample thickness. Lastly, in cases where the samples were sufficiently thin (i.e. thin film samples with thicknesses on the order of the light wavelength), oscillations can result in the measured absorbance. These Fabry-Perot fringes are related to interference during internal reflection. While the effect can be mitigated through mathematical models, additional analysis is required.
Other methods of measuring the optical properties in lead chalcogenides have also been used. PbTe, being a narrow gap semiconductor, has been rather thoroughly studied for uses in infrared lasers and photodetectors. Other studies have measured specular reflectivity of PbTe samples which requires Kramers-Kronig analysis in order to obtain the absorption coefficient-inevitably, some extrapolation of the dielectric function is necessary which can complicate the analysis [43]. Spectroscopic ellipsometry can bypass the necessity for the Kramers-Kronig analysis by measuring both the real and imaginary components of the dielectric function simultaneously, and some studies of lead chalcogenides as photodetectors or infrared lasers use these techniques [37,44]. Many of the optical measurements show the higher energy features (>1 eV) of PbTe which can be important to determining critical points in the joint density of states-a convenient way to compare with DFT calculations [18,45]. This work, however, will focus on the fundamental absorption edge.
Diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS) was used exclusively in this work for optical measurements due to its ease of sample preparation and data analysis. Diffuse reflectance can be obtained in a thin sample layer of a ground powder. The measurement requires only a small amount of polycrystalline sample (100 mg or less). Because of the Fourier transform analysis, light at all frequencies can be measured simultaneously-greatly reducing sampling time. Relating the diffuse reflectance to sample absorption can be performed using the Kubelka-Munk function. This report shows that DRIFTS is very sensitive to small changes in the energy of direct transitions across the band gap due to progressively higher doping levels. With proper extrapolation the small shifts associated with doping can be shown and understood according to existing optical analysis techniques.
Heavily doped semiconductors, including most good thermoelectric materials, have free carrier contributions to the optical absorption that can complicate the estimate of the band gap. Often the band gap is simply assumed to be the extrapolation of the absorption edge to zero absorption. In the case of degenerately doped semiconductors, this can cause large errors in the estimate of the true band gap on the order of the value of the electrochemical potential, ζ , relative to the band edge (which can be more than 0.1 eV in heavily doped samples). Since the early 1950s, it has been known that for semiconductors with small effective mass, the band gap tends to change with increasing doping [46,47]. These effects have been considered in lead chalcogenides by several authors [24,[48][49][50] using methods other than DRIFTS.
In the case of a degenerately doped n-type semiconductor, states near the conduction band edge have a nonzero occupancy. As a result, the photon energy required for excitation across a direct band gap becomes higher by an additional (1 + m c m v )ζ in the case of direct transitions (see figure 1) which results in an increase in the optical band gap, E g,opt , known as the Burstein-Moss shift. The true band gap (E g ) remains unaffected with doping if we consider the rigid band approximation which assumes the band gap and band parameters are independent of the electron filling. This often used approximation has proven effective in electronic transport modeling for many thermoelectric materials over a wide range of temperatures and doping levels. The thermal gap, E g,thermal , is relevant to transport properties and minority carrier excitation. The true band gap, E g , has been shown both experimentally through optical measurements [51][52][53] and theoretically [51][52][53] to be reduced as the carrier concentration increases in heavily doped semiconductors-known as band gap renormalization. Renormalization is thought to be related to Coulombic repulsion of the electrons and/or exchange interactions.
The difference between the optical and true band gaps have been studied in many III-V and other semiconductors and semiconductor devices. It is important to have an understanding of band gap renormalization for a variety of semiconductor devices where both the majority and minority carrier concentration and specific band positions can be necessary to optimize performance. In thermoelectric materials, though, the effects of heavy doping on the optical properties are generally not considered. In fact, there is often no distinction made between the thermal and optical gaps in doped materials [54,55]. While the differences can be circumvented in the case of an undoped material (PbTe undoped binary), some thermoelectric materials may include both shifts in the doping level and to the band structure itself. In this work, we attempt to quantify the Burstein-Moss shift on the optical gap, and gap renormalization in iodine doped PbTe using diffuse reflectance spectroscopy. We primarily use measured Seebeck coefficients to estimate the chemical potential which is an improvement to simply measuring the Hall carrier concentration. Further, we attempt to improve upon renormalization estimates by selfconsistently considering the band gap parameter.

Experimental and theoretical methods
PbTe samples were synthesized by melting and annealing. The ingots were hand-ground into a fine powder and hot pressed into 1/2 inch diameter discs using an induction hot press method described previously [56]. The particulars of their synthesis and electronic properties measurements are shown in a previous work [57]. Electronic properties measurements were used in conjunction with Boltzmann transport equation models and a single Kane type band which is outlined in detail by several references [1,10,58]. These models were used in order to obtain an estimate of the electrochemical potential ζ , which reduces to the chemical potential (ξ ) in the absence of an applied electric field, for each sample; this was accomplished in two ways. Firstly, using room temperature Seebeck coefficient measurements, the electrochemical potential could be estimated assuming a single Kane band model where acoustic phonon scattering is assumed as the primary electron scattering mechanism. The single Kane band model has been shown to be a good model for the conduction bands in PbTe [57]. The band gap parameter in the Kane model was assumed to have a constant value near that of the undoped PbTe, 0.295 eV, for this analysis. Secondly, the electrochemical potential could be estimated for a given carrier concentration using a constant effective mass; the effective mass was obtained from fitting the room temperature Seebeck relation with Hall carrier concentration (Pisarenko plot), m * d0 = 0.276 m e was a reasonable value for these PbTe samples. Alternatively, the chemical potential could be estimated from Seebeck coefficient measurements directly without assumptions about the effective mass. While a six-band model could be used to improve chemical potential estimates at high doping (>5 × 10 19 cm −3 ) [16], the Kane model shows good agreement with transport properties and was used instead.
Samples were measured in diffuse reflectance with a Praying Mantis attachment using a Nicolet 6700 Fourier transform infrared spectrophotometer with a deuterated triglycine sulfate (DTGS) detector and KBr beamsplitter. All samples were referenced to the provided alignment mirror, this was found to give nearly same results as when referenced to KBr powder without the added impurity features from KBr itself. The absorption coefficient was obtained using Kubelka Munk analysis where R is the fractional reflectance, α is the absorption coefficient and K is the scattering coefficient. For particle sizes greater than the light wavelengths measured (20-2 µm), the scattering coefficient is understood to be approximately independent of frequency [59].

Results
Each sample revealed an absorption edge near 0.3 eV (shown in figure 2(a)), but heavily doped samples also showed increasing absorption at low energy. For samples with high doping (>2 × 10 19 cm −3 ), there was an observable peak in absorption. This corresponds to the downward peak in reflection associated with the oscillating electron plasma in the material near the plasma frequency, ω p = N e 2 ∞ m * I [60], where N is the electron concentration, e is the fundamental electronic charge, ∞ is screened dielectric constant and m * I is the inertial effective mass. For metals, the plasma frequency is in the ultraviolet and visible, but in heavily doped semiconductors it can exist into the far infrared. The location of the reflectivity minimum can also be used as an estimate for the optical effective mass of the charge carriers, but this was neither the focus of this work nor will it be explored in detail here. In order to isolate interband transitions, these features were fit and subtracted with a power law (y = ax b + c). In almost all cases, the exponent, b, followed the expectation for Drude carriers of b = −2, but some deviations occurred for heavily doped samples which showed more clearly the plasma reflection minimum. This method was also compared to just normalizing the data to its minimum value, the resulting extrapolated band gap was consistently ∼5 meV lower for the fits with the free carrier contribution subtracted.
While the accepted view of PbTe is as a direct gap semiconductor, with evidence shown both experimentally [50,[61][62][63] and theoretically [14,15,64], some confusion as to whether PbTe was a direct or indirect gap semiconductor existed in early optical measurements of PbTe where some authors obtained good fits to α 1/2 versushω expected for indirect transitions [23,41]. Similar to these reports, the absorption spectra in this work have a linear α 1/2 versushω region where the extrapolated band-gap energy is slightly lower (∼30-60 meV) than those fit for direct transitions. Scanlon suggests that this difference can be attributed to indirect transitions where a phonon is absorbed [10,23], although single phonon energies are at most 12 meV or less based on a Debye temperature of 140 K [65] and neutron scattering experiments [66]. Prakash suggests Scanlon's observation could also be interpreted as merely a coincidental fit due to the Urbach tail (exponential decrease in density of states below the band edge) [25]. Although, we believe that Prakash's interpretation is correct in PbTe, many materials do exhibit both indirect and direct transitions with some energy separation. For example, in germanium, both direct and indirect transitions exist with the indirect gap occurring first at about 0.63 eV and direct transitions beginning at 0.81 eV [67]. The indirect portion involves a slow rise in absorption coefficient over a long energy range (∼100 cm −1 over ∼0.2 eV) while the direct absorption on the other hand gives a much steeper rise over a shorter period (∼10 4 cm −1 over ∼0.05 eV). As a result, the direct gap is much more easily observed particularly if both direct and indirect transitions occur at close to the same energies. The absorption coefficient of PbTe changes ∼10 4 cm −1 over ∼0.05 eV [49,68], suggesting it is a direct gap semiconductor. Recently, it has been suggested that the true gap value is convoluted by mid-gap defect states [18]. While these states may affect the measured optical band gap, we see no significant contribution from localized states in the transport properties which are characteristic of itinerant (delocalized) carriers giving metal-like transport behavior. Therefore, the measured absorption edge was assumed to be associated with the band edge in PbTe and not mid gap defect states, although defect states near the band edge could certainly play a role in band gap renormalization.
A series of PbTe samples doped with iodine (n-type) were measured using DRIFT spectroscopy, the normalized spectra are shown in figure 2(a). Free carrier absorption which scales as (hω) −2 in the classical limit is observed at low frequencies. As the doping level increases, the free carrier contribution to the absorption coefficient moves to higher and higher frequencies as expected from the Drude model, when ω τ −1 : α FC ∝ ne 2 n r cm * τ ω 2 where n is the number of free carriers, e is the elementary charge of an electron, n r is the real refractive index, c is the speed of light, τ is the Drude scattering time and ω is the photon frequency [69]. The free carrier absorption was subtracted for the analysis of the optical gaps. Optical band gaps were obtained using the Tauc method, where (αhω) 2 is extrapolated on a plot versus photon energy, hω, to zero absorption ( figure 2(b)) [60] which is based on direct transitions between parabolic bands. Other authors have plotted α 2 versushω for fitting the direct gap [23], but regardless of the method the result is similar (within the measurement error: ∼0.005 eV).
In heavily doped samples an optical band gap can be extracted in the same way, although it is not immediately clear the absorption should still have such a simple functional form. Several spectral features are observed near the absorption edge. First, many observers note an exponential increase in absorption just below the gap, known as an Urbach tail. The Urbach tail is thought to be related to the random distribution of impurity atoms in the material; we also observe an approximately exponential increase in absorption just below the band edge [60,70]. Next, as electrochemical potential moves deeper into the band with doping level, there is a decrease in the number of available states near the band edge for an electron to be excited into. This effect, known as the Burstein-Moss shift [46,48,60,71], means that a higher energy photon is required to produce the same amount of absorption-moving the absorption edge to higher energies. Lastly, the true band-gap E g has been shown to decrease with heavy doping, known as band gap renormalization [51,[72][73][74]. The Burstein-Moss shift and renormalization effects will be discussed further below. In order to relate the measured optical band gap to the thermal gap, a chemical potential estimate is required.
In many cases, the chemical potential is estimated by the 0 K limit of the parabolic band as an estimate of the Fermi energy: 0 K limit as E f =¯h 2 2m * (3π 2 n) 2/3 where m * is the band effective mass [75][76][77]. While this may be a good estimate for degenerate systems (at low temperature, high doping) for materials with parabolic band shapes, a better estimate involves solving the more general temperature dependent expression n = ∞ 0 D(E) f (E, ζ, T ) dE where D is the density of states and f is the Fermi distribution function which is a function of the electron energy E, chemical potential ζ and temperature T. It is well known that many semiconductors, including lead chalcogenides, deviate significantly from parabolic behavior. Ravich has developed an adaptation of the Kane band model for application to the lead chalcogenides which has shown excellent agreement to experimental transport data [1,10,57]. Using Ravich's solution of the Kane band model applied to lead chalcogenides, n can be calculated by numerically integrating equation (1) where m * d0 is the band edge density of states effective mass, ε is the dimensionless energy (= E k B T ), β is the non-parabolicity parameter (= k B T E g from Ravich) and η is the reduced electrochemical potential (= ζ k B T ). Calculation of the electrochemical potential as a function of the true carrier concentration is rather straightforward, but in order to compare to the Hall carrier concentration the Hall factor, r H , is needed. Equations for the Hall factor and other relevant model parameters were obtained assuming acoustic phonon scattering in a single band as described in detail by Ravich et al [10] and Wang et al [58] and relevant parameters for PbTe were used following LaLonde et al [1] and Pei et al [57].
Using the Boltzmann transport equation, it is possible to obtain an estimate for the electrochemical potential, ζ , directly from the measured Seebeck coefficient with no assumption  (figure 3) shows the calculated chemical potential for each iodine doped sample calculated using a single Kane band model both from the room temperature Seebeck measurements and using a constant band edge effective mass, m * d0 = 0.276m 0 (best fit of the Seebeck versus n H Pisarenko plot). Here, we can see the chemical potential for the most heavily doped samples increases to ∼0.15 eV from the band edge. It is important to note that the chemical potential estimate can vary depending on the particular band model. In the case of a single parabolic band, the chemical potential change to be as much as 50% higher for the same doping level. Once the electrochemical potential is known, it is possible to estimate the true gap from the optical gap measurement (figure 5(a)). As an alternative to the Tauc method, the gap can also be fit using the spectral Fermi distribution.

a modified version of the Fermi integrals from Wang et al
The Fermi distribution can be projected onto the unperturbed interband absorption as a multiplicative factor shown in equation (3). This technique and similar methods have been performed on a variety of materials, including lead chalcogenides [48,49] and other materials [46,51,78,79]: α 0 is the absorption for an unpopulated conduction band; at T = 0 K which for parabolic bands . f is the Fermi distribution, which is related to the photon energy as f (hω, ζ, The second term of equation (3) represents the electronic excitation probability based on the electron population from Fermi distribution. The electronic occupation probability is equal to 1 /2 at the chemical potential which can then be related to the photon energy,hω, assuming direct transitions and a fully occupied valence band (E g k B T ). The true band gap, calculated from the optical gaps by E g = E g,opt − 2ζ . A fit is performed using all of the data and the renormalization as in equation (4).
Using a proportionality factor and the band gap value as a fitting parameter, it is possible to obtain an estimate for the optical band gap using equation (3) from the measured absorption spectra for a given estimate of the chemical potential (as estimated from either room temperature Seebeck and/or n H measurements). The resulting fit is shown in figure 4. The optical and true gaps obtained by both the Tauc and Fermi spectra methods are shown in figures 5(a) and (b). Effectively, the Fermi projection method gives the optical band gap near the inflection point in the absorption (where the slope is the largest).
Two different methods have been used to calculate the band gap. The Tauc method takes the raw absorption data and extrapolates to estimate the optical gap while the Fermi spectra method projects the Fermi distribution onto the joint density of states which is fit to the measured absorption spectra to estimate the optical band gap. Figure 5(a) shows an increasing optical gap with doping level-a Burstein-Moss shift. As the doping level increases, the chemical potential in the conduction band moves upward which results in occupied states nearest the band edge which requires a higher energy photon for excitation of a carrier across the gap. The results from both the Tauc and Fermi spectra methods appear to agree within ±0.015 eV, although the broadening fits generally show a larger optical band gap. From figure 1, we can relate the optical gap to the true gap E g . Figure 5(b) shows the calculated true band gap with respect to doping level. In the case of direct transitions and similar valence and conduction band effective masses, the true band gap is related to the optical gap and the chemical potential as E g = E g,opt − 2ζ . The result shows a true gap that decreases significantly with doping level relative to the undoped sample. This is a result of the measured optical gap not increasing nearly as quickly as expected. Over the same doping range, the optical band gap is expected to increase by 2ζ , which is as much as 0.25 eV at the highest doping levels-see figure 3. The measured optical band gap only increases by about 40% of the electrochemical potential value or ∼0.07 eV at higher doping levels as shown in figure 5/table 1. The relatively small change of the optical band gap can be explained by the renormalization effect which is an estimate of band gap shrinkage with doping. The optical band gap increases according to equation (4) as described the schematic in figure 1 [71,75,76]: where E g0 is the band gap of the undoped material, RN is the band gap renormalization, m * c and m * v are the conduction and valence band effective masses respectively, and ζ is the electrochemical relative to the conduction band edge. This equation again assumes direct transitions which is the reason that a factor greater than 1 is multiplying the electrochemical potential. The true band gap in equation (4) can be represented by E g = E g0 − RN which accounts for a shrinking band gap with increasing doping level as shown in figure 5(b).
Both experimental and theoretical work has been done to determine how the band gap should change due to renormalization with increasing electron concentration in many semiconductors including: Si [51,53,74,79,80], Ge [80,81], GaAs [73,76], InP [52], ZnO [82] and other materials. The effect has proven important in semiconductor device applications where the material's band gap determines many device characteristics and careful engineering is required to optimize performance. Gap narrowing is thought to be due to combination of effects including electron-electron exchange interactions, electron-donor interactions and band tailing [73,75,80,83]. Electron-electron exchange has been shown to scale as n 1/3 in a weakly interacting free electron gas [80,82]; this model is often used empirically, although the specific form may vary with crystal and energy band structure. The effect can be thought of as Coulombic repulsion between electrons in the material which scales as 1/r where r is the mean distance between electrons. Experimentally, attempts have been made to empirically fit the theoretical models. Most use a combination of power laws whose prefactors can in theory be calculated, but are most often used as fitting parameters [75,80,83]. Drabkin et al [49], for example, suggest that for PbTe a shift on the order of 10 meV is reasonable for doping levels on the order of 1 × 10 19 cm −3 . When using the method of Mahan [83], however, the predicted reduction was 1 meV or less due to the large static dielectric constant ( (0) ≈ 400) for PbTe. Following [51,79], we fit the experimental renormalization (for both Tauc and broadening methods) to an n 1/3 power law, including an additional constant term: RN = An 1/3 + RN,0 . The resulting renormalized fits to the optical and true band gap are plotted in figures 5(a) and (b) and are also shown in table 1. Figure 5(b) shows the true band gap, when considering renormalization, reduces rather significantly. For the three most heavily doped samples, the chemical potential, as estimated from Seebeck coefficient measurements, does not continue to increase at the same rate. As a result, the true gap approaches an approximately constant value of close to 0.15 eV. The n 1/3 model deviates significantly for carrier concentrations >∼ 6 × 10 19 cm −3 , resulting in a gap that rapidly approaches zero above this value. While the magnitude of the gap reduction is consistent with previously published results for heavily doped n-type PbTe [49], Si [51,53,79], Ge [81] and III-V semiconductors [52,73,75,76,84], the apparent discrepancy at high doping is not easily explained using the simple n 1/3 empirical model.
In Ravich's adaption of the Kane band model for lead chalcogenides, the band gap is a necessary parameter as it determines the non-parabolicity parameter-β = k B T E g . Because optical measurements have shown evidence that the band gap in PbTe decreases with doping as a result of band gap renormalization, it might be expected to affect the energy dependence of the band structure. The band edge effective mass in narrow gap semiconductors has been suggested to scale with the band gap itself [10,69]. Ravich suggests that the energy dependent effective mass for a Kane type band should scale with energy as where p is the kp matrix element coupling states between two bands: u 1 | p| u 2 . The first term will scale proportional to the band gap, while the second will scale with E, the electron energy. More often, the prefactor in equation (5) is treated as a constant-m * 0 , the band edge effective mass-which can be fit from the Seebeck coefficient versus carrier concentration Pisarenko plot. Optical measurements of the band gap renormalization in this work suggest that the gap decreases with increasing carrier concentration. In order to probe the effect that this might have on the estimate of the chemical potential and on the transport properties, a selfconsistent approach should be taken. For all of the previous calculations, the chemical potential has been estimated from Seebeck coefficient (equation (2)) assuming a constant Kane-band nonparabolicity parameter (β used in equations (1) and (2)) given by assuming a constant band  (2), m * 0 = 0.276 m 0 ) and points are calculated from Seebeck coefficient (equation (3)).
gap of 0.295 eV equal to that of the undoped PbTe sample. Using the fitted n-dependent gap: E g (n) = E g0 − RN (n), we can self-consistently calculate n as a function of chemical potential with an n-dependent nonparabolicity parameter, β = k B T E g (n) . The new value of chemical potential was used to recalculate the gap renormalization using the measured optical gaps-which was again fit to an n 1/3 model. After several iterations, the true gap appeared to converge to a selfconsistent value that was slightly higher than the previous result which assumed constant gap. The self-consistent solution yielded renormalization fitting parameters of 5.6 × 10 -8 eV cm and −0.068 eV for A and RN,0 respectively. The band gap, both fit (solid lines) and values from measurements (circles), for the model assuming constant β and the self-consistent form which includes an n-dependent band gap is shown in figure 6(a). Renormalized gap estimates were obtained using measured optical gaps and estimates of the electrochemical potential, ζ (shown in figure 6(b)). Here, we can see the effect of the band-nonparabolicity as an increasing band mass as the chemical potential raises. This is based on the assumption of a constant band edge effective mass: , which there is reasonable evidence for based on the Seebeck versus n H Pisarenko plot. At higher carrier concentrations (>6 × 10 19 cm −3 ), the renormalized band gap value becomes quite small, and errors become larger. Therefore, the fact that E g (n) levels out at about 0.15 eV may not be entirely accurate since the renormalization effect is probably overestimated at these doping levels. Further, PbTe is known to deviate from the Kane model for carrier concentrations greater than this value [16]. As in the constant β case, the empirical n 1/3 model may be an oversimplification in the case of increasingly narrow gap and/or the simplified Kane model of Ravich may also be an oversimplification which cannot account for these narrowing phenomena.
In the discussion of electronic band structure it is typically a good first approximation to assume the energy bands remain unaltered with doping-the rigid band approximation. Lee, Mahanti et al give some examples where this may not be a good assumption in Na doped PbTe [85], although the picture here is complicated by the variability in supercell calculations [86]. Even experimental measurements of effective mass can show small changes in band structure when using different dopants (I or La) in PbTe which may be related to a shift in band gap [87].With the goal of band engineering in mind it is important to be able to determine how the band structure is altered as the materials are doped or alloyed. While the rigid band approximation works reasonably well for many purposes, this work demonstrates that the differences between true, optical and thermal band gaps can be different of the order 0.1 eV. Most thermoelectric materials are heavily doped semiconductors, where E g,thermal > k B T . In this case, the electronic transport properties (electrical conductivity, Seebeck coefficient) are determined by a single band. In materials that contain multiple bands, the effect of a changing band gap with doping level will be important in developing a self-consistent transport model. The most obvious effect of a narrower band gap is the increased concentration of minority carriers. In the case of most thermoelectric materials, though, the chemical potential (and thermal band gap) is deep enough into the band to where the minority carrier population is still very small compared to majority carriers. The effects become more apparent at high temperatures when k B T ∼ E g,th .
While band gap renormalization is a well-known effect that has been studied in many materials, it is generally ignored in thermoelectric materials in favor of the rigid band approximation. This work suggests that in PbTe, both approximations may play a role. While it seems unlikely that the band gap in these materials might become very small (<0.1 eV), it is possible that if the material is sufficiently doped that bipolar effects can be suppressed and a reduced band gap would not be observed by measuring transport properties alone. Ultimately, the apparent discrepancy between the rigid band approximation and band gap renormalization may need to be resolved with either better estimates of the n-dependent chemical potential (using a six band model, rather than the Kane model), or by developing models that more accurately represent renormalization in narrow gap semiconductors. While the constant β single band Kane model does show a much lower chemical potential increase with doping, there is still a significant renormalization leading towards zero gap at very high doping levels, even when the self-consistent solution is considered. This probably is an artifact of the renormalization fit which is best for doping levels less than 6 × 10 19 cm −3 . While the n 1/3 scaling law works well in a free electron gas [80], there may be some corrections for narrow gap semiconductors-particularly as the gap becomes small. Even though band gap narrowing is thought to be well understood, the effects in narrow gap semiconductors-as most thermoelectric materials are-are difficult to reconcile with other measurements. Perhaps more experimental and theoretical work should be done to investigate what is different in narrow gap, heavily doped semiconductors from other wider gap materials.

Conclusions
While the DRIFTS method of measuring the optical absorption has been thoroughly explored in catalysis research and has shown some promise for quantifying chemical reactions [88,89], as a technique for precisely determining band gaps in semiconductors diffuse reflectance has only proven semi-quantitative (±0.1 eV) [90]. In this work, we detect shifts in band gap with doping of smaller than 0.01 eV in PbTe which can be analyzed and understood by methods consistent with current optical literature.
While electronic transport measurements are essential to determine the thermoelectric properties for a materials, optical properties can provide more direct knowledge about the band structure. Although diffuse reflectance measurements have been used in the thermoelectric literature, it is important to account for the effects of electron population on the absorption edge that alters the observed optical band gap. We have shown a series of iodine doped PbTe samples which show a Burstein-Moss shift in the absorption edge to higher frequencies. Using estimates of the chemical potential from transport measurements, it is possible to estimate that a gap shrinkage of up to 0.15 eV occurs at the highest doping levels appropriate to thermoelectrics. This is observed both by fitting the optical band gap using the Tauc method and by fitting the band gap directly from the spectrally dependent absorption coefficient with a Fermi function. An attempt is made to fit the true gap self-consistently using a single band Kane model which results in a slight reduction in the renormalization. While renormalization has been shown to fit well-behaved empirical models for many semiconductors, PbTe (and possibly other narrow gap materials) exhibits a Burstein-Moss shift and renormalization that are not readily explained with existing theory.