2B30: PRACTICAL ASTROPHYSICS FORMAL REPORT: SPECTROSCOPIC DETERMINATION OF THE BAND GAP IN SILICON

Transcription

1 2B30: PRACTICAL ASTROPHYSICS FORMAL REPORT: SPECTROSCOPIC DETERMINATION OF THE BAND GAP IN SILICON Adam Hill Tutor: Dr. Peter Storey 1

2 Abstract The purpose of this experiment was to determine the energy level difference between the valence and conduction bands - the band gap of a pure sample of silicon. This was achieved by measuring simultaneously the optical and electrical properties of the silicon sample when exposed to differing wavelengths of light. The photoelectric effect means that when a critical frequency was obtained electrons in the silicon would have a large enough energy to traverse the band gap, hence altering the electrical and optical properties of the silicon. The wavelength at which there was a sudden change in the electrical resistance and the optical transmission of the sample was found and from this the band gap energy was deduced. The results yielded an approximate value for the band gap energy in Silicon of / ev. 2

3 Introduction Materials can generally be broken into three separate categories: metals; insulators; and semiconductors. The properties of these three categories can in part be explained by the configuration of their internal energy band structure. The Pauli Exclusion Principle applies to states within energy bands i.e. there is a limit to the total number of electrons that can be contained within an energy band. In the case of metals, one band is partly filled and so it takes very little energy for the electrons to change state. This means that they can easily carry current when an electric field is applied. See Fig 1.1 Fig. 1.1 Metal band structure Empty Full Band Insulators have bands that are either completely full or completely empty. Electrons are normally unable to traverse the energy gap and so cannot respond to an electric current. See Fig Fig. 1.2 Insulator band structure Empty Band Band gap Full Band Semiconductors are arranged in the same way as insulators however the band gap is much smaller. Sufficiently smaller in fact to allow electrons to leave the full valence band and jump the energy gap to the higher empty band with a small supply of energy such as thermal excitation. Silicon is an example of an intrinsic semiconductor. The band gap energy is the energy difference between the highest energy quantum state in the valence band and lowest energy state in the conduction band. There are no electron states in between the two bands. At room temperature there are very few electrons in the conduction band as the thermal energy available to the electrons is much less than the energy gap. 3

4 Through the photoelectric effect energy can be imparted to the electrons by hitting the sample with photons of energy greater than or equal to that of the energy gap. If the incident photons have energy lower than the energy gap then the optical transmission of the sample will be high and the electrical resistance will be high. When the photons reach the critical energy the optical transmission will drop sharply and the electrical resistance also falls. By gradually increasing the energy of the incident photons these changes in the properties of silicon were observed. The critical photon energy was measured and an estimate of the band gap obtained. The photon energy (E) and the wavelength of the light (λ) are associated by the equation, E=hc/λ (1) where c is the speed of light and h is Planck s constant. Experimental Method The apparatus used in the experiment was set up as shown below, Fig. 2. Fig. 2: Experiment apparatus 1 4

5 The experiment light was broken into its constituent wavelengths using a reflection type diffraction grating with 600 lines mm -1. The dispersion law of the diffraction grating states, λ ( R) = 2( d / m) cos( ψ ) sin( R) (2) where λ is the wavelength; R is the angle from the zero position of the turntable; d is the grating line spacing; m is the spectrum order, 1 in this case; ψ is 15 in the case of the experimental set-up. In order to measure the optical transmission and electrical resistance of the silicon sample as it was exposed to different wavelengths of light the equipment below was set-up, fig.3. Fig. 3 Measurement equipment 2 5

6 The expected region for the critical photon energy was within the near infrared. However the first order infrared coincides with the second order visible when reflected onto the sample. This would have confused measurements so an optical filter was incorporated into the apparatus to block all wavelengths of less than 750nm. Through rotating the turntable upon which the diffraction grating was mounted it was possible to target different wavelengths of light onto the Silicon sample. The grating was blazed reflecting more light in the direction of anti-clockwise rotation and so the grating was rotated in that direction. For each measurement the AC. gain of the amplifiers was adjusted to give the most sensitive measurement. Starting from R = 8 the grating was rotated thought 1 steps up to 35. At each step the following measurements were made: grating table angle; the AC. gain on the resistance and transmission amplifiers; and the transmission and resistance voltmeter readings. As the grating was rotated the intensity of light falling on the sample varies. To compensate for this the readings were normalised. With the Silicon sample removed another set of readings measuring the light transmission were taken. These measurements allowed the experimental data to be corrected for fluctuations in the light intensity. Experimental Data Table 1: Initial data Undeviated light angle, θ = Measurements Grating table angle AC gain Transmission AC gain Resistance ( ) transmission amp. Voltmeter (V) resistance amp. Voltmeter (V)

7 Grating table angle AC gain Transmission AC gain Resistance ( ) transmission amp. Voltmeter (V) resistance amp. Voltmeter (V) Table 2: Normalising Readings Normalising readings Grating table angle AC gain Transmission ( ) transmission amp. Voltmeter (V)

8 Calculation of Results To analyse the results the true voltages were found by using the data in Tables 1 and 2. The measured voltages from both tables were divided by their associated gain readings. These results are displayed in Table 3. Table 3: Calculated true voltage values R 0 (undeviated beam) ( ) R= R 0 - angle of True sample True sample True normalising grating turntable ( ) transmission voltage (V) resistance voltage (V) Voltage (V) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-01 The results were then normalised by dividing the true resistance and transmission voltages by the true normalising voltage measured at the corresponding angle (data from Table 3). Using equations (2) and (1) the wavelength and photon energy for the corresponding R angle were calculated respectively. In order to plot both sets of data onto the same graph the data was scaled. A scaling factor was determined and the lower set of data multiplied up by this scaling factor. This data is found in Table 4. 8

9 Table 4: Final calculated data Normalised True sample Normalised True sample Wavelength Photon Energy Scaled normalised Scale transmission voltage resistance voltage (V) (V) (µm) (ev) resistance voltage Factor 7.168E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-03 9

10 Analysis of graphs Graph 1 shows the main results of the experiment. It is plotted from the data calculated in table 4. From graph 1, 2 and 3 the discontinuities in resistance change and transmission are obvious, however they are not as sharply defined as expected due to the simplification of the theory behind the process. As this region is not well defined in order to estimate the band gap energy the rising edge of the resistance and transmission change curves, of graph 1, were linearly interpolated. Horizontal lines approximating the maximum point of the transmission change curve and minimum point of the resistance change curve were also plotted. The intersection of the maxima/minima line and the rising edge interpolated lines provided two estimates of the band gap energy. The values obtained were: R R from the transmission change curve from the resistance change curve Using formula (2) λ ( R) = 2( d / m) cos( ψ ) sin( R) ψ = 15, d = 1/ lines m -1, m = 1 λ(22.50 ) = [2 x cos(15 ) x sin(22.5 )]/ = 1.23 x 10-6 m (3 s.f.) λ(23.75 ) = [2 x cos(15 ) x sin(23.75 )]/ = 1.30 x 10-6 m (3 s.f.) Using formula (1) E=hc/λ h = x Js, c = x 10 8 ms -1 E(22.50 ) = 1.01 ev E(23.75 ) = ev (3 s.f.) (3 s.f.) 10

11 Error Analysis The primary random error that was associated with this experiment was the human error in the reading of the rotation angle of the diffraction grating turntable. The error was made twice, when taking the initial readings and then again when taking the normalisation readings. The error in the reading of the turntable is +/ The overall error in R is given by, ( R/R) 2 = ( A/A) 2 + ( B/B) 2 A = original reading, B = normalising reading However A=B=R and so the error becomes, ( R) 2 = ( A) 2 + ( B) 2 = (2.5 ) 2 + (2.5 ) 2 R = (3 s.f.) The error in R will lead to an error in λ. Equation (2) is λ ( R) = 2( d / m) cos( ψ ) sin( R) Differentiating this gives, λ = 2( d / m) cos( ψ ) cos( R) R (3) (3) / (2) λ / λ = + / cot( R ) R / R (4) When R=22.5 λ = 1.23 x 10-6 x cot(22.5 ) x / 22.5 = 7.79 x 10-9 m -1 When R=23.75 λ = 1.30 x 10-6 x cot(23.75 ) x / = 7.34 x 10-9 m -1 These error will result in an error in the calculated threshold energy. ( E/E) 2 = ( λ/λ) 2 E = (E x λ )/ λ 11

12 When R = 22.5, E=1.01 ev E = 1.01 x 7.79 x 10-9 / 1.23 x 10-6 = 6.39 x 10-3 ev When R = 23.75, E=0.956 ev E = x 7.34 x 10-9 / 1.30 x 10-6 = 5.40 x 10-3 ev Summary of results The analysis of the experimental data and their associated uncertainties yielded the following results for the band gap energy of Silicon When R = 22.5 : Band gap energy is / ev. Corresponding wavelength is 1.23 x / x 10-6 m. When R = : Band gap energy is / ev. Corresponding wavelength is 1.30 x / x 10-6 m. Mean value of the band gap energy is / ev. Mean threshold wavelength is 1.26 x / x 10-6 m. 12

13 Conclusion The results of the experiment give an approximate value for the band gap energy of Silicon as / ev. The actual value for the band gap energy is ev ( 3 ). This is close to the mean value obtained by the experiment, however it falls outside of the range of the experiment uncertainties. This may be due to larger random errors than predicted or there may be a systematic error in the experiment as both of the experimental approximations were less than the true value. Also contributing to uncalculated errors would be the way in which the discontinuities were linearly interpolated in order to obtain the experimental results. The true value of the band gap energy can be seen, in graph 3, to fall midway in the transmission change discontinuity and at the peak of the resistance change discontinuity. At lower photon energies, < 0.9 ev, two anomalous readings were observed. These readings were not repeated as they did not fall within the critical region of the graph and so would not effect any calculations. The region of the graph describing wavelengths λ < 0.8 µm, shows a curve that according to theory should not exist as the filter should have blocked any signal in this region. As both the resistance and transmission change curves both exhibit the same characteristics in this region it suggests that this region is generated by a source of noise within the system which happens to correspond to the correct phase modulation allowing it to contaminate the signal. Accuracy of this experiment would be improved by using higher resolution equipment that could obtain more sensitive and accurate results within the wavelength region of interest. Elimination of more noise would also lead to better results. 13

14 Appendix Lock-in amplifiers In order to enable the experiment to be conducted outside of a dark room a lock-in amplifier or chopper amplifier is required to block out background light that would have polluted the light source used. The light source is modulated to a particular frequency by the interruption of the beam by the rotating chopper. The drive for the chopper feeds a reference signal at the same frequency for the amplifier. As a result the lock-ins are only sensitive to sources with this phase modulation and hence blocking out all other signals. A schematic circuit of a Lock-in amplifier is shown below (Fig. 4). Fig. 4 A schematic circuit of a Lock-in amplifier 4 14

15 References Figure 2 from 2B30 script for experiment CM4. Figure 3 from 2B30 script for experiment CM4. Solid State Physics, Gerald Burns, Academic Press Inc. Figure 6 from 2B30 script for experiment CM4. 15