• C. Kittel, Introduction to Solid State Physics (7th ed.), John Wiley & Sons, New York, NY, 1996.
• R. A. Smith, Semiconductors, Cambridge University Press, 1961.
• T. P. McLean, J. E. Quarrington, and V. Roberts, Phys. Rev. 108, 1371 (1957) and Phys Rev. 111, 1245 (1958).
• F. Wooten, Optical Properties of Solids, Academic Press, 1972.
• J. I. Pankove, Optical Processes in Semiconductors, Dover Publications, N. Y., 1972.

The book by Jacques I. Pankove is particularly good (and is available as an inexpensive Dover paperback). It introduces you to the band structure of semiconductors, the fundamental absorption processes in direct and indirect band gap semiconductors, and the relationships between optical constants.

The goals of this lab are as follows:

• to become familiar with spectrophotometry, the study of the absorption, reflection and transmission properties of materials;
• to measure the energy gap, refractive index and (energy-dependent) attenuation coefficient of Si, Ge and GaAs; and
• to observe the different absorption behavior of direct band gap versus indirect band gap semiconductors.

A single atom has discrete electron energy levels, but when many such atoms are brought together to form a solid, the electron orbitals overlap and interact to form wider energy bands. Such bands represent a continuum of allowed energies and adjacent bands may either overlap or be separated by a band gap (where no electron energy states exist) depending on the structure and atoms that make up the material.

{ ${/download/attachments/143560140/fig_1.png?version=1&modificationDate=1455551596000&api=v2}$ Figure 1: An example of energy levels broadening into energy bands and energy gaps. Here, N atoms (each with 4 electrons) move from distinct s and p energy states at large separation (far right of the plot) to two nearly continuous bands at smaller separation. Notice that it is possible for electrons which previously were in the upper energy state when the atoms were separated to fall into the lower band when the atoms are brought together to form the crystal lattice. (After Pankove,  Fig. 1-1, page 2.)

Electrons organize themselves within bands of a solid in much the same was as in the discrete energy levels of a single atom. The innermost (core) electrons are bound tightly to the nuclei and fill the lowest energy bands. The outermost (valence) electrons contribute to the chemical bonds holding the solids together and reside in what is called the valence band. The next higher energy band is called the conduction band. Electrons in the conduction band are no longer bound to the nucleus and can thus move freely in the material. Electrons can be excited from the valence band into the conduction band by absorbing energy from incident photons or from phonons, the quantized lattice vibrations associated with thermal energy.

When an electron is excited from the valence to the conduction band, it leaves behind a hole in the valence band, of equal, but opposite charge. Such an excited electron can decay back to the valence band by recombination with a hole.

Solids fall into three categories based on their band structure.

(1)  In insulators, the valence and conduction bands are separated by a wide band gap, typically several eV or more. At room temperature, thermal energies are insufficient to excite electrons from the valence band to the conduction band. Thus, the electrons remain tightly bound to the atoms.

(2)  In metals, the valence and conduction bands overlap, meaning that the highest energy electrons in the valence band also lie in the conduction band. Therefore, thermal excitation is sufficient to excite electrons into the conduction band, even at temperatures down to T = 0.

(3)  In semiconductors, the valence and conduction bands are separated by a small band gap, typically 0.1-1 eV. At room temperature, some electrons are thermally excited to the conduction band, but most remain in the valence band. The smallness of the band gap, however, means electrons can be promoted from the valence to conduction bands by absorption of photons with energies near the visible range.

{ ${/download/attachments/143560140/fig_2.png?version=1&modificationDate=1455551816000&api=v2}$\\

Figure 2: A comparison of the band structure of insulators, metals and semiconductors. (After Kittel Fig. 7-1, page 158.)

There is no clear distinction between semiconductors and insulators, but roughly speaking, materials having energy gaps smaller than ~2.5 eV are considered semiconductors. Table 1 of Chapter 8 in Kittel's Introduction to Solid State Physics lists the energy gaps of a number of crystals.

One notices two trends which cause the energy gap to increase. First, the gap widens as atoms connected by covalent bonds get smaller and hence can get closer together. This is the case looking, for example, at the progression

Sn (0 eV) → Ge (0 .67 eV) → Si (1.11 eV) → Diamond (5.5 eV).

Second, the gap widens as the bonds go from being more covalent to more ionic as it becomes easier to strip off electrons rather than share them. This is seen, for example, in the progression

Ge (0.67 eV) → GaAs (1.43 eV) → ZnSe (2.7 eV) → KBr (7.6 eV).

The manipulation of this band gap energy and the number of available excited states above it provide the principal means of controlling electron flow used in transistors and other devices.

In addition to the energy discussed above, electrons in the solid also carry momentum and obey the normal kinetic energy relation

E = p_2/2_m*

where m* is the electron’s effective mass. This effective mass typically is different from the mass of the free electron, _m_e, and will also vary depending on the specifics of the solid’s band structure and whether we are talking about an electron in the conduction band or a hole in the valence band.

The momentum of the electron is further constrained by the de Broglie relation

{ ${/download/attachments/143560140/de%20Broglie.png?version=1&modificationDate=1455551416000&api=v2}$ where k is the electron’s wavevector, whose magnitude – called the wavenumber – is k = 2_π/λ_. In order to satisfy the appropriate boundary conditions on the electron wavefunction, the magnitude k takes only certain discrete values and these values depend on the physical properties of the lattice such as the lattice separation, a. Note, however, that just as the energy within a band can be considered continuous because there are N ~ 1023 closely spaced levels, momentum and wavenumber can likewise be considered continuous. The electron energy distribution is therefore a nearly continuous quadratic function of the wavenumber in one direction. (See Fig. 3.)

{ ${/download/attachments/143560140/fig_3.png?version=1&modificationDate=1455551880000&api=v2}$\\

Figure 3: E vs. k for an electron in a one-dimensional lattice. (After Pankove Fig. 1-2, page 4.)

Now imagine a three dimensional crystal where . If the lattice separation is not the same in all three orthogonal directions, then the wavevectors will have different dependencies in different directions and the energy may not have its global minimum at _k_x = _k_y = _k_z = 0. Instead, there may be multiple local peaks and valleys corresponding to the different planes of the crystal. (See Fig. 4.)

{ ${/download/attachments/143560140/fig_4.png?version=1&modificationDate=1455551900000&api=v2}$\\

Figure 4: E vs. k for an electron in a generic, non-cubic lattice. (After Pankove Fig. 1-4, page 5.)

In solid state physics, the band structure of a material is often graphically represented with such E vs. k pictures. Using the top energy of the valence band as the baseline, the energy “shape” of the two bands can be drawn. In the valence band, the effective mass of a hole is actually negative and therefore the curvature is downward. (See Fig. 5.)

{ ${/download/attachments/143560140/fig_5.png?version=1&modificationDate=1455551925000&api=v2}$\\

Figure 5: E vs. k showing a valence band and conduction band along with the band gap. (After Pankove Fig. 1-4, page 5.)

In the visible region of the spectrum, semiconductors can have similar optical properties to metals. Both metals and semiconductors are highly reflective – exhibiting a luster when polished – and highly absorptive – typically having coefficients of absorption of the order of α ≈ 105 cm-1.

The absorption coefficient is defined according to

where _I_i is the incident intensity of the light and _I_t is the transmitted intensity after traveling a distance d through a material with absorption coefficient α. The coefficient α depends on the photon frequency f or, equivalently, its energy E = hf where h is Planck’s constant. In semiconductors the absorption coefficient can vary radically with photon energy because of fundamentally different responses to photons with energies larger or smaller than the band gap energy.

As the incident photon energy decreases from visible toward the infrared, one observes a sudden increase in transmission followed by a spectral region in which semiconductors are more transparent. This fundamental absorption edge lies at a photon energy which is just sufficient to excite electrons across the energy gap _E_g which separates the valence band from the conduction band. The magnitude of the gap is an important factor determining the optical, electrical, and photoelectrical properties.

In this experiment you will determine the band gap energy _E_g for GaAs, Si, and Ge semiconductors from measurements of α(E) through the optical absorption edge. One might naively assume that this transition is sharp, but the change from opaque to transparent is gradual and encodes considerable information.

First, the rate at which a photon of a given energy is converted into an electronic excitation is given by Fermi’s Golden Rule. This rule states that the rate is proportional to both the density of electrons available to absorb the photon and the density of available unoccupied final states, and that the rate is proportional to the sum of probabilities for transitions between these states. The details of these terms depend on electron densities of states, but for our purposes we need to know only the following two things: (a) if the energy of the photon is less than the gap energy, no electron-hole pairs can be created and the absorption rate is zero, and (b) as the energy rises above the band gap energy, the number of final states increases in a calculable way. Appropriate formulas are derived in Pankove’s text and summarized in the next section below.

Second, one must also conserve momentum (or equivalently wavevector k) in any allowed transition. The photon's wavenumber – _k = 2__π/_λ where λ is the photon’s wavelength – is negligible compared to that of an electron, so we need only account for transitions where the sum of the wavevectors of the excited electron and hole add up to zero.

Finally, thermal excitations modify the above picture. Because of these, the initial state is not the ground state of the semiconductor. In particular, at room temperature vibrational modes are excited with energies of order kBT, about 1/40 eV at room temperature. To account for such excitations, one must average over the thermally excited initial states. Final states may also have vibrational modes in addition to the electron and hole. The discussion below explains how these vibrations affect the absorption of light.

Semiconductors fall into two categories based on the shape of their band structure.

A direct band gap semiconductor is one where initial and final states of the electron have the same wavevector and no intermediate processes are required to conserve momentum when a photon is emitted or absorbed. (See Fig. 6.) The transition rate is governed only by the photon energy.

{ ${/download/attachments/143560140/fig_6.png?version=1&modificationDate=1455552449000&api=v2}$\\

Figure 6: Bands near a direct gap. (After Pankove Fig. 3-1, page 35.)

According to Pankove (p. 36), the absorption coefficient is zero (that is, no absorption occurs) when E < _E_g, but then grows when E > _E_g as

where A is a constant.

In an indirect band gap semiconductor the highest occupied electronic state not only has a different energy than the lowest empty energy state, but a different wavevector as well. (See Fig. 7.) Optical transitions from the valence to the conduction band can therefore occur only when both a photon and a phonon are involved. The phonon must have a wavevector equal to the difference in wavevector of the two bands.

{ ${/download/attachments/143560140/fig_7.png?version=1&modificationDate=1455552367000&api=v2}$ Figure 7: Bands near an indirect gap. (After Pankove Fig. 3-2, page 37.)

If the phonon has energy _E_p, then the absorption coefficient for a transition with phonon absorption at photon energy E > _E_g - _E_p is

{ ${/download/attachments/143560140/eqn_3.png?version=2&modificationDate=1458159387000&api=v2}$

(3)

(where B is a constant), and the absorption coefficient for a transition with phonon emission at photon energy E > _E_g + _E_p is

{ ${/download/attachments/143560140/eqn_4.png?version=2&modificationDate=1458159425000&api=v2}$

(4)

(where C ≠ B again is a constant). (Pankove p. 38.) Note that when the photon has energy E > _E_p + _E_g, both absorption and emission are possible, and α(E) = _α_a(E) + _α_e(E).

SUMMARY:

A direct band gap semiconductor can be differentiated from an indirect band gap by the very different behaviors of their absorption coefficients.

• For a direct band gap semiconductor, α = 0 for E < _E_g, but increases as the square root of E for E > _E_g.
• For an indirect band gap semiconductor, α = 0 for E < _E_g - _E_p, but increases quadratically with E above this energy showing a “kink” (i.e. a change in the second derivative) at E = _E_g + _E_p where a second pathway for photon absorption opens up. 

The origins of the square root and quadratic dependences of α on E are due to differences in the density of states of electrons and phonons. Further details (including the explicit forms of the constants A, B, and C above) are beyond the scope of this course.

You will use a Beckman spectrophotometer to measure the fraction of light intensity as a function of wavelength transmitted through different semiconductor samples. This instrument uses a chopping technique to make alternating measurements of the intensity of the light with and without a sample in the optical path. The ratio of the two intensities is used to determine the fractional transmission, or transmittance, T of the sample,

{ ${/download/attachments/143560140/eqn_5.png?version=1&modificationDate=1455565526000&api=v2}$

(5)

where Ii is the intensity of the light in the absence of the sample and It is the intensity after passing through the sample.

{ ${/download/attachments/143560140/fig_8.png?version=2&modificationDate=1455565715000&api=v2}$ Figure 8: Optical path for Beckman Spectrophotometer.

A schematic of the instrument is shown in Fig. 8. Light from the source (A) is focused on the slit (E) by the condensing mirror (B). The directed beam from the condensing mirror (B) is deflected through the chopper (D) and upon the entrance slit (E) by the slit entrance mirror (C). Light focused on the slit (E) falls on the collimating mirror (F) and is rendered parallel and reflected to the quartz prism (G). The back surface of the prism is aluminized so that light is reflected back through the prism. The desired wavelength of light is selected by rotating the wavelength selector which adjusts the angular position of the prism. The spectrum is directed back to the collimating mirror (F) which focuses the selected wavelength in the slit (E). Light leaving the monochronometer is focused by the lens (H) into the sample compartment, where the beam is alternately switched between the reference path (J) and the sample path (M) by rotating mirrors (I and N) and stationary mirrors (L and K). The beam entering the photocell compartment is focused by the spherical condensing mirror (O) on either the lead sulfide detector (P) or the photomultliplier detector (Q).

In order to compensate for changes in measured intensity of the signal in the Reference channel, due to the spectral shape of the blackbody spectrum for example, the spectrophotometer uses a feedback circuit to control the width of the slit (E) so as to maintain constant intensity of the Reference signal. The ratio of the Sample signal to the Reference signal gives the fractional transmission of the sample.

This spectrophotometer was designed to scan automatically over a user-selectable wavelength range while plotting the data as transmittance versus wavelength using the built-in X-Y recorder. However, the speed of the scan is faster than the electronic and mechanical response time of the automatic slit control and you cannot obtain reliable measurements at the absorption edge where the transmission changes rapidly. To get around this limitation we have modified the apparatus so that the transmission output can be read as a voltage on a meter. You will take data by manually setting the wavelengths to be measured and recording the transmission data from the voltage output.

Before you begin to make measurements, calibrate the spectrophotometer so that you are making use of its full dynamic range of intensity. Make sure that the following controls are set as indicated:

• Power Switch → On.
• Motor → Set to Out position.
• Scan-Stop switch → Set to Stop.
• A-%T-E dial → Set to %T.
• Time Constant → Set to 0.1.
• Range → Set to 0-100%. (Full clockwise.)
• Sensitivity → Set to ~1.0.
• On the sample chamber make sure both dials are set to the lead sulfide detector.

Let the spectrophotometer warm up for about 10 minutes and then use the following procedure to calibrate the intensity scale.

1. With no samples in the sample chamber, place the lid on the chamber. - Set the wavelength to ~1000 nm using the Wavelength selector dial located on the left side, top of the instrument. - Open both the Sample and Reference shutters. The pen holder on the X-Y plotter should move up to near the 100% mark. - Adjust the 100% dial so that the pen holder points to the 100% mark. - Now close the Sample shutter. The pen holder should fall to near the 0% mark. - Adjust the Zero dial so that the pen holder points to 0%.

Note that since you will be recording the voltage output of the detector it is not crucial that the pen holder be set precisely to the 100% and 0% marks. The purpose of this calibration is to ensure that you are using most of the dynamic range of the detector. When properly calibrated, the voltage output should be approximately 25 mV at 100% and less than 1 mV at 0%. Once set, do not adjust the Zero or 100% dials.

The voltage output at these two extremes now set the values corresponding to 100% and 0% transmission. In principle, the response of the detector could vary as a function of wavelength. Although the automatic slit control feature of the Beckman spectrophotometer is designed to maintain a uniform detector response, you should quantify the detector response yourself as it is easy to do so. After measuring the 100% and 0% voltages for many wavelengths, you can look for any variation and assign final average values for 100% and 0% voltages (with uncertainties).

NOTEBOOK: Record measurements of the 100% and 0% voltages (with uncertainties) across a wide wavelength range, say 650 nm to 2500 nm.

There are three samples of Si with different thicknesses and one sample each of Ge and GaAs provided. You will make a detailed measurement of the transmittance versus wavelength for each in the vicinity of its absorption edge.

The samples are thin slices of semiconductor mounted in aluminum sample holders. The samples are polished to reduce light scattering. They are thin and delicate. Don't touch these surfaces. The GaAs sample is so thin that a free-standing film is not possible. Accordingly, it is mounted on a thicker film of another semiconductor, gallium phosphide (GaP). The light thus passes through both materials. The GaP has a wide band gap and creates negligible absorption at the light frequencies of interest. However, it does create reflection, thus complicating the multiple reflection analysis below. The index of refraction of GaP is very similar to that of GaAs, so reflections at the GaAs-GaP interface may be neglected.

Note that the samples are not in the center of their holders. They are positioned so that the light beam will pass through the samples when the holder is oriented properly in the sample chamber. Place the sample against the lower left corner of the sample chamber against the exit opening of the spectrophotometer, with the sample label visible.

For each sample, first manually scan though wavelengths to determine the approximate location of the absorption edge by observing the action of the plotter pen holder. Once you have a rough estimate, make careful measurements of the transmittance vs. wavelength through the region where the sample goes from opaque to relatively transparent. Take as many points as the resolution in the wavelength dial will allow through the transition region. You will not be analyzing transmittance, but instead the computed quantity, absorption; while you can show the features of the former with only a few points, a dense sampling will be necessary to see all the eventual features of the latter. Also, be sure to go sufficiently far in either direction to get a clear plateau at both extremes.

NOTEBOOK: For each sample, take as many points as the resolution in the wavelength dial will allow through the transition region. Make sure to collect sufficient points at lower and higher wavelengths as well to establish the baseline values. It will be useful to plot the data as you go. (Voltage as a function of wavelength is fine.)

NOTE: The mechanical linkage between the wavelength control knob and the optical elements is (intentionally) loose; the wavelength reading is accurate only when the knob is turned so that wavelength is decreasing. If you overshoot a desired wavelength, turn the knob up to a value larger than you want, and slowly dial back down to the proper point.

NOTEBOOK: To make an estimate of the uncertainties, go back and repeat your measurements for a few data points for each sample. (Remember to approach those points only while descending along the wavelength scale.) While such measurements ought to give the same value each time, the spread in your collected data will allow you to estimate the magnitude of random uncertainty.

Though the data collection was straightforward, the analysis for this experiment can be time consuming as you will need to convert your measured voltages to several intermediate quantities before arriving at the final absorption coefficients, α. In particular, you will need to be mindful of how you estimate uncertainties at each juncture since these will be propagated along from step to step.

Note that in the analysis which follows, we will fold both statistical and systematic uncertainties into our consideration. This will lead to larger than normal error bars (and a smaller than normal reduced chi-squared value on the one fit) since more than just random fluctuations are being included. Keep this in mind when interpreting your results.

REPORT: The analysis methods which follow will help you translate your voltage data (and uncertainties) into absorption coefficient data (and uncertainties). Since the report will rely heavily on plots of propagated quantities, it will be difficult to evaluate from the final values and error bars alone how uncertainties were estimated at each step and how these assumptions contributed to the final result. Therefore, chose a single data point – pick a wavelength and voltage pair in the middle of a transition region rather than at one extreme – and show explicitly all the calculations and uncertainty estimates leading to α for that one point. In this way – using a typical data point as a stand in for all data points – your TA will be able to better understand how you arrived at your results.

In order to convert from raw voltage to transmittance, we must use the data collected corresponding to 0% and 100% transmission.

REPORT: Present your 0% and 100% data (with uncertainties) in tabular or plot form. Determine single values for _V_0 and _V_100 to use for the calibration from voltage to transmittance and determine uncertainties in each based on the spread of your data and the range of values you measure in the experiment.

Once we have established our baselines, we can now convert from voltage to transmittance as

{ ${/download/attachments/143560140/eqn_5b.png?version=1&modificationDate=1457555563000&api=v2}$

(6)

where _V_0 and _V_100 are the voltages corresponding to 0% and 100% transmission, respectively. Likewise, we need to convert wavelength to energy as

{ ${/download/attachments/143560140/eqn_6b.png?version=1&modificationDate=1460039108000&api=v2}$

(7)

REPORT: For each sample, convert from voltage to transmittance and, if significant, propagate your calibration uncertainties. Plot transmittance (in percent) as a function of energy (in eV) with uncertainties. There should be separate plots for GaAs and Ge, but show all three samples of Si on one figure to directly highlight how transmission depends on thickness.

The plots should show a clear transition from high to low transmittance (i.e. from low to high absorption) over a range of energies. In coarse terms, the transition is expected to begin when the energy of the incident photon increases beyond the energy of the semiconductor energy gap.

REPORT: Make a coarse estimate of the energy gap by determining at what energy the transmittance begins to fall from its high plateau level and estimate an associated uncertainty. Note, however, that this will result in an underestimate of the true energy gap_._ In indirect gap semiconductors (like Si and Ge), the transmittance will begin to fall at the energy _E_g - _E_p due to phonon-assisted processes (where _E_p « _E_g is the phonon energy), and in both direct and indirect gap semiconductors absorption can occur at energies just below the energy gap due to impurities in the material. Your uncertainty should be large enough to reflect not just the difficulty of determining where the transition begins, but also the fact that the beginning of the transition is not exactly the energy gap energy. Use your judgement and remember that this is a _coars_e estimate. 

We need to extract the absorption coefficients from measurements of the fractional transmission of light through the samples. This analysis is complicated by the fact that not all of the observed light loss is due to absorption in the semiconductor. There is also loss of light due to surface reflections as the light enters and leaves the semiconductor. This reflection occurs whenever light is incident on the boundary between two materials with different indices of refraction.

Let us add up these light losses due to the effects of reflection and absorption as a beam of light passes through the semiconductor as shown in Fig. 9.

{ ${/download/attachments/143560140/fig_9.png?version=1&modificationDate=1455566585000&api=v2}$\\

Figure 9: Passage of light through a semiconductor sample.

When the incident beam Ii enters the semiconductor at interface (1), a fraction R of the light is reflected. The remaining light Ii (1-R) passes through the semiconductor suffering more loss due to absorption so that the intensity of the light at interface (2) is Ii (1-R)e- αd. At interface (2), some of the remaining light is reflected and the rest leaves the semiconductor. The light that was reflected at interface (2) propagates back to interface (1), with consequent loss due to absorption, where it is again reflected back to interface (2). Adding up all the losses due to multiple reflections and absorption leads to an infinite series for the fraction of light transmitted through the semiconductor,

{ ${/download/attachments/143560140/eqn_6.png?version=1&modificationDate=1455566573000&api=v2}$

(8)

Since _R2e-__αd < _1 we can evaluate the sum and express the transmittance as the ratio of the transmitted intensity It to the incident intensity Ii,

{ ${/download/attachments/143560140/eqn_7.png?version=1&modificationDate=1455567043000&api=v2}$

(9)

This complicated function can be inverted to find α as a function of T and R, leading to

{ ${/download/attachments/143560140/eqn_8.png?version=1&modificationDate=1455567049000&api=v2}$

(10)

The experiment measures T, but what about R? For the case of normal incidence when one of the materials is air (n ≈ 1), the fraction of light reflected, or reflectance, R is given approximately by

{ ${/download/attachments/143560140/eqn_10.png?version=1&modificationDate=1455567169000&api=v2}$

(11)

Neither R nor n is constant as a function of energy, but they are nearly constant near the absorption edge. A small bump at exactly the band gap energy is predicted, but we will neglect it.

NOTE: The index of refraction is actually a complex number n = n + iκ, where κ_is the__extinction coefficient,_ related to the absorption coefficient by_ α_ = 4π_κ/λ._Though this form does not suggest it, the index of refraction and extinction coefficient are not independent, but are uniquely related by the Kramers-Kronig relations, and therefore n also depends on α. The details of these relations are not necessary here; it is sufficient to note that in the vicinity of the absorption edge, n is nearly constant and several orders of magnitude larger than κ such that the complex contribution to n can be neglected.

To determine R from your data, consider the transmittance at photon energies _E « E_g where the semiconductor is transparent and therefore  α = 0. In this case, Eq. (9) reduces to

{ ${/download/attachments/143560140/eqn_11.png?version=1&modificationDate=1455567210000&api=v2}$

(12)

which can be rearranged to give

{ ${/download/attachments/143560140/eqn_12.png?version=1&modificationDate=1455567216000&api=v2}$

(13)

REPORT: Compute R from Eq. (13) for each sample, and use it with Eq. (11) to calculate the index of refraction. Discuss and propagate the appropriate uncertainties. You will likely find a refractive index n somewhat larger than the literature value. This is caused (i) by imperfectly polished surfaces, which scatter light and thus reduce T in the α = 0 region, yielding an erroneously large R value, and (ii) by the fact alluded to above that n increases somewhat as the energy approaches _E_g; the literature value of n is measured at smaller energy.

NOTE: Because GaAs sits on a substrate of GaP, its transmittance in the α = 0 region shows an interesting feature; you should observe a distinct oscillation with wavelength. The reason for this is that the thickness of the substrate is comparable to the wavelength of incident light and the substrate acts as a thin film. The oscillation is an interference pattern showing destructive interference when the the light transmitted and reflected off the surfaces of the GaP become out of phase. Use the highest value of the plateau (where there the interference is constructive) as your estimate for Tα = 0.

Once we have determined R, the absorption coefficient α is now a function of the T data only, given by Eq. (10). As this conversion is a complicated function of several calculated variables, we have determined the equation propagating the uncertainties for you as

{ ${/download/attachments/143560140/eqn_13.png?version=1&modificationDate=1457543021000&api=v2}$

(14)

As mentioned above, we have included both statistical uncertainties (e.g. Δ_T_ includes uncertainties from the raw voltage measurements) and systematic uncertainties (e.g. uncertainties from your 100% and 0% voltages may lead to a biased calibration and likewise with the determination of R). Note that this leads to an overestimate of the statistical variation in α, so keep this in mind while interpreting your results. Note also that depending on the magnitude of your uncertainties, it may not be necessary to keep all three terms in the square root; use your judgement to decide if one or more terms can be neglected.

REPORT: Plot α (in units of 1/mm or 1/μm as appropriate) as a function of energy (in eV) with uncertainties for all five samples. There should be separate plots for GaAs and Ge, but show all three samples of Si on one figure.

As direct and indirect band gap semiconductors absorb light via different mechanisms, the absorption coefficients should show quantitatively different behaviors as a function of energy for the two types. 

REPORT: Based on Eqs. (2)-(4), which of the three materials have a direct band gap and which have an indirect band gap?

As absorption coefficient is a property of the material (not of any specific property of a given sample), we expect the three curves for the three thicknesses of silicon to give the same results.

REPORT: For silicon, do the three thicknesses all collapse to the same absorption coefficient as expected?

For direct band gap semiconductors, the form of the absorption coefficient given by Eq. (2) suggests that the transition will be linearized if we plot α2 vs. energy.

REPORT: For your direct band gap material(s), plot plot α2 vs. energy. Look for a linear region in the data and fit to a straight line of the form _α_2 = A'(E - _E_g). The x-intercept of such a fit should be _E_g. Compare this result to the coarser estimate made directly from the transmittance plot.

WARNING: Do not fit to the generic function _α_2 = aE + b, but instead use the form given above, _α_2 = A'(_E - E_g). The first form is written in terms of the y-intercept, (which is located very far from where the data is actually clustered), and will therefore likely lead to a large uncertainty on b (since finding it involves a large extrapolation). It would be possible to calculate _E_g from a and b, but correctly calculating the uncertainty becomes tricky because the uncertainties on a and b are not independent and we cannot use the usual uncertainty propagation formulas. We avoid this issue by writing the fit function directly in terms of the x-intercept, (which is quite close to where the data is clustered), so that the correct uncertainty is automatically calculated by the fitting algorithm.

Fitting the indirect band gap absorption coefficients to their predicted forms is highly non-trivial and will not be required for the report.

When writing your report, consult the rubric and notes below for the appropriate quarter.