The Moss-Burstein effect results from the Pauli Exclusion Principle and is seen in semiconductors as a shift with increasing doping of the band-gap as defined as the separation in energy between the top of the valence band and the unoccupied energy states in the conduction band. The shift arises because the Fermi energy (EF) lies in the conduction band for heavy n-type doping (or in the valence band for p-type doping). The filled states therefore block thermal or optical excitation. Consequently the measured band gap determined from the onset of interband absorption moves to higher energy (i.e. suffers "a blue shift").
Provided that the effective masses of the valence and conduction bands are known reasonably accurately and it is assumed that the curvature and position of the bands are independent of the doping, the shift in energy can be used as an accurate and contactless method of determining the carrier concentration in the sample.
And Eg is the band gap found with undoped samples.
Note that the equations overleaf assume that the optical transition of an electron from the valence band to the conduction band is "vertical"; i.e. that the photon wavevector (k = 2π/λ) is small compared with that of the electrons at the Fermi energy (kF = (3π2)1/3). Check that you agree with this assumption taking λ (in air) as 3 microns, the refractive index of InAs as 3.5 and n = 1024 m-3.
By performing the following experiment, and assuming that the conduction and valence band effective masses are mc = 0.03m0 and mv = 0.4m0 respectively, calculate the carrier concentration (n) for the doped samples provided (the effective mass at the conduction band edge is 0.024m0. At the Fermi energy concerned the appropriate effective mass has increased by 50%).
You are provided with a Spex monochromator, a globar source with a germanium lens, a light chopper, lead selenide and lead sulphide photodetectors and a "Phase Sensitive Detector" which enhances and amplifies the signal and suppresses noise. A PC is also available for equipment control and data processing. See Figure 1 for a schematic of the experimental set-up.
Check the zero setting of the monochromator. If the grating were perfectly positioned it would act as a mirror for all wavelengths at a dial setting of 0000. You will almost certainly find that the maximum reflection occurs at a slightly different dial reading. Add or subtract this small "zero error" when you calculate the true wavelength. Use the "mirror" setting to align your source and detector. Provided that you are using the germanium lens you will not get any diffracted radiation through the spectrometer until a dial reading of about 3000 is reached and you should obtain a maximum signal on your detector at a dial reading of about 4000. Remember that the grating equation is m λ = a sin θ so that you may get second order (m = 2) radiation through the monochromator at dial settings greater that 6000 (assuming that the Ge lens cuts in at 3000). Assume that the grating scale is linear in wavelength and that the wavelength in microns is 6x10-4 of the dial reading; i.e. that a dial reading of 4000 corresponds to a wavelength of 2.4μm.
Take several 'background' recordings with no sample in place in order to familiarise yourself with the equipment and in order that background features due to atmospheric absorption may be at least partially removed by ratioing the transmission with the sample in place with the “background” values.
The 'background' spectrum is limited by the band gap absorption of the Ge lens at short wavelengths and by the cut-off for intrinsic excitation of the photodetectors at long wavelengths. Estimate the band gaps of Ge, PbS and PbSe from the respective background spectra.
Observe the effect of changing the slit-widths on the monochromator on the atmospheric features at wavelengths at 4.2μm and 2.7μm.
You are given four epitaxial films of InAs of thickness of a few microns grown on GaAs substrates. One sample is undoped and the others have been fairly heavily doped with silicon donors. Taking great care, place each metal sample mount in turn over the detector iris in order to take a transmission spectrum. On no account touch the surface of the semiconductor samples as they are very fragile.
The GaAs substrates do not absorb significantly in the wavelength range concerned. However both the InAs epitaxial film and the GaAs substrate have high refractive indices and reflect quite strongly. You should be able to observe Fabry-Perot interference fringes from the epitaxial film at wavelengths which are too long for the band gap absorption to occur. Because of the high refractive index the infrared radiation travels almost perpendicular to the film surfaces so the period of the Fabry Perot fringes is given by
δ(1/ λ) = l/2nrd where nr is the refractive index of the InAs
Use the above relationship to determine the thicknesses of the samples (assume that nr = 3.5 for InAs).
The GaAs substrate is much thicker and similar fringes occur in the very far infrared region of the spectrum. Because of the high refractive indices involved you may expect an insertion loss of the order of 70%.
The object of the experiment is to measure the onset of absorption for each sample. Having obtained the 'raw' transmission spectrum, ratio the results against a background spectrum taken with the same slit widths using the data analysis system or point-by-point.
In which d is the thickness of the sample and α is the (wavelength-dependent) absorption coefficient. In order to analyse your data assume that the onset of band-edge absorption occurs at the wavelength corresponding to α = 2000 cm-1. 1
1 Strictly speaking the variation of the real and imaginary parts for the dielectric constant should be analysed in detail in the region of a strong absorption feature such as band edge (Kramers-Kronig analysis). However such an analysis is time-consuming and would not be valid for the present samples
Assume that the interference fringes are only significant at wavelengths longer than that corresponding to α = 2000 cm-1. Draw a smooth curve through the centres of the interference fringes and assume that the intensity transmitted just before the increasing absorption becomes apparent corresponds to "I0". In fact the assumption that kBT « E is not very good at room temperature and thermal blurring of the Fermi-Dirac function is quite significant in broadening the absorption edge.
Finally calculate the carrier concentrations for each of the doped sample from the blue shift in the absorption "edge".
The maximum current which should be passed through the globar source is 10A. The detectors require a bias voltage of 50V. DO NOT EXCEED THESE SETTINGS unless instructed to by a demonstrator. The lead sulphide detector is more sensitive than the lead selenide but the PbS band gap is greater than that for InAs so you will not be able to observe the onset of band-gap absorption for the undoped sample with this detector (which occurs when hf = Eg). Take care not to reverse-bias the detectors.
The chopper frequency is not critical but should not be close to a harmonic of the mains frequency. A good choice would be about 120 Hz
Optimise the phase on the Phase Sensitive Detector (PSD for short - sometimes called a "lock-in" amplifier) to produce maximum output voltage. The recommended time constants are 1s and you should expect to use a sensitivity in the range 10mV to 100 μV depending on slit-widths, whether a sample is in the beam, choice of detector etc. Be careful that the instrument does not overload when the maximum signal is present.
The PSD reduces the amount of noise by combining the optical chopper frequency with that of the received signal. The PSD ignores any frequencies outside the chopper frequency as these would be produced by noise only.
The stepping intervals for the grating are set using the computer. You will need to choose the grating setting at which the run starts and the range of points to be recorded. Think about the sensitivity and scales you should be using, to record the relevant features.
At very low temperatures, the phonon density is very small [large denominator in Equation (3-11)]; therefore, αa is also small. The temperature dependences of αa and αeare illustrated in Fig. 3-3, where the square root of a is plotted to yield a linear dependence on hv. Such a plot, by extrapolation to α = 0, gives the values of Eg – Epand Eg+ Ep. Note that Eghas been shifted with temperature to reflect the temperature dependence of the energy gap.
As mentioned earlier, there are several types of phonons, one longitudinal-acoustic and two transverse-acoustic, which can participate in the transition process. In fact, they all participate, but with different probabilities.3
If the semiconductor is heavily doped, the Fermi level is inside the band (the conduction band in an n-type material) by a quantity ξn (Fig. 3-4).
Fig. 3-4 Energy-momentum diagram for degenerate w-type germanium in the  direction. Two phonon-assisted transitions are shown to illustrate the usual photon absorption mechanism.
Since the states below ξnare already filled, fundamental transitions to states below Eg+ ξn are forbidden; hence the absorption edge should shift to higher energies by about ξn. The shift of the absorption edge due to band filling is sometimes called the Burstein-Moss shift.4, 5 A calculation of the absorption coefficient was made for heavily doped n-type germanium6; the results are reproduced in Fig. 3-5. At 0°K, only the phonon-emission process is possible; √αe for pure germanium intercepts the abscissa at Eg+ Ef. The calculated intercept shifts by ξn, as expected. The drop of absorption at a given hv > Eg+ Ep+ ξn with increasing doping is due to the decrease in the number of available final states.
3 G. G. MacFarlane, T. P. McLean, J. E. Quarrington, and V. Roberts, Phys. Rev. 108, 1137 (1957) and 111, 1245 (1958).
4 E. Burstein, Phys, Rev. 93, 632 (1954).
5 T. S. Moss, Proc. Phys. Soc. (London) B76, 775 (1954).
6 J. I. Pankove and P. Aigrain, Phys. Rev. 126, 956 (1962).
From: J I Pankove, “Optical Processes in Semiconductors”, Dover (1971)