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Department of Physics University of Jaffna Experimental Report of Band Gap D.D.A.D.J. Prabodha S 9493 Band Gap PHY301M4 1.0 Abstract This experiment was designed and conducted on behalf of finding the band gap of a given semiconductor. Here it has been given an n-Ge test piece. Practically it was obtained as 0.67698eV at room temperature (303K) and this value was deviated 4.15% from the theoretical value 0.65eV. Department of Physic 1 Janith Prabodha Band Gap PHY301M4 2.0 INTRODUCTION 2.1 Energy band and Fermi level The Bohr model of the atom tells us that electrons can only have certain discrete amounts of energy corresponding to certain electron orbits around the atomic nucleus. The electrons have a minimum energy state and only certain discrete higher energy states are allowed. A fixed amount of energy is needed to jump the electron up into a higher energy state and when the electron falls back into a lower energy state that energy is given up as electromagnetic radiation. The highest filled band is called the valence band. The next higher band is the conduction band, which is separated from the valence band by an energy gap, also called a band gap. This band gap represents the energy required to knock electrons out of atoms into the conduction band. In any material, for conduction to occur, there must be electrons available in the highest energy band. The energy required for an electron to jump the band gap can be provided by heat or some form of radiation or from an electric field and is usually expressed in electron Volts (eV) where 1 eV is equivalent to 1.6 X 10-19 Joules (J). In a metal the minimum energy needed to liberate an electron from the surface of the metal is called the "work function". CB Conduction Band (C B) CB Forbidden gap Over lap Valence Band (V B) VB Fermi level VB Conductor Fermi Level Fermi level Insulator 17 Figure2.1:- Energy band picture of materials Department of Physic Gap (1eV) (Several eV) 2 Semiconductor 21 Janith Prabodha 14 Band Gap PHY301M4 2.2 Conductor Energy Bands Already have free electrons in the conduction band which are available to carry the current and a small band gap which makes it easy to jump more electrons into the conduction band. In the conductors valence band & conduction band are overlap. Metals have a positive coefficient of resistance since the thermal agitation of the electrons increases with temperature and impedes electron flow. 2.3 Insulator Energy Bands Have a wide band gap typically higher than 5 eV (electron Volts) with no electrons in the conduction band, at ordinary temperatures no electron can reach the conduction band. Conduction can occur if a high enough field is used to force electrons into the conduction band but this usually results in the breakdown of the insulating material. 2.4 Semiconductor Energy Band Have few electrons in the conduction band making them poor conductors but their relatively low band gap (0.67 eV for Germanium and 1.12 eV for Silicon) permits their conductivity to be increased by using an external stimulus to raise the energy level of the electrons. When this occurs the higher energy electron breaks free from the covalent bond between the semiconductor atoms creating an electron-hole pair of new charge carriers thus increasing the conductivity of the semiconductor material. Adding small numbers of new current carriers by doping allows dramatic changes to the conductivity of the semiconductor. According to that there are two types of semiconductors such as intrinsic semiconductor and extrinsic semiconductor. Technically, there are no intrinsic or extrinsic conductors. Intrinsic and extrinsic conduction are terms that are applied to semiconductors. These terms refer to methods of enhancing the conductivity of semiconductors. The conductivity of semiconductors arising from thermal excitation is called intrinsic conductivity. Another way to increase the conductivity of a semiconductor is by doping (the addition of impurities). After doping, the resulting conductivity is called extrinsic conductivity. In doped semiconductors, extra energy levels are added. The increase in conductivity with temperature can be modeled in terms of the Fermi function, which allows one to calculate the population of the conduction band Department of Physic 3 Janith Prabodha Band Gap PHY301M4 At finite temperatures, the number of electrons which reach the conduction band and contribute to current can be modeled by the Fermi function. That current is small compared to that in doped semiconductors under the same conditions. More free electrons Free electrons Donor level Acceptor level (Impurity level) (Impurity level) Holes Less holes (a) Intrinsic semiconductor (b) n-type semiconductor More Lessholes free electrons (c) p-type semiconductor Figure 2.2:- Energy band diagram of semiconductors. 2.5 Germanium Energy Bands Germanium was the material on which early transistors were based. It has a low melting point and is easy to work with and a low resistivity which helps in achieving high frequency response. Unfortunately it also has a low maximum working temperature of 175 °C and suffers from inherent high leakage currents due to its low band gap of only 0.67 eV. For intrinsic semiconductors like silicon and germanium, the Fermi level is essentially halfway between the valence and conduction bands. Although no conduction occurs at 0 K, at higher temperatures a finite number of electrons can reach the conduction band and provide some current. Department of Physic 4 Janith Prabodha Band Gap PHY301M4 0K (No electrons in conduction band) Fermi level 300K 0.74eV Figure 2.3:- Energy band diagram of Germanium semiconductor Department of Physic 5 Janith Prabodha Band Gap PHY301M4 3.0 THEORY The conductivity of semiconductors is characteristically a function of temperature. There ranges can be distinguished at low temperatures we have extrinsic conduction. As the temperature rises charge carriers are activated from the impurities. At moderate temperatures We talk of impurity depletion (saturation range). Since a further temperature rises. At high temperatures it is intrinsic conduction. Figure 3.1:- Conductivity of semi-conductor as a function of the reciprocal of the temperature. Figure 3.2:- Conductivity of intrinsic range At high temperatures it is intrinsic conduction which finally predominates. In this instance charge carriers are additionally transferred by thermal excitation from the valence band to the conduction band. The temperature dependence is in this case essentially described by an exponential function. Eg 2 KT 0 exp ………………………... (1) Where, 𝐸𝑔 - Energy gap. K - Boltzmann’s constant. T - Absolute temperature. Here, Department of Physic 6 Janith Prabodha Band Gap PHY301M4 T Where, UT T0 UT - Voltage across the thermocouple 40V K T0 – Room temperature But, Resistivity Conductivity RA l 1 l RA ……………………... (2) From Ohm’s law V IR ……………………………… (3) By using equations (2) and (3), From equation (1) Eg 1 ln( 0 ) ln( ) 2K T y = -mx + c From the above linear equation y = - m x+ c gives, gradient m Eg m 2K So, Energy gap Where, E g 2Km K = 8.625x10-5 eV/K Department of Physic 7 Janith Prabodha Band Gap PHY301M4 4.0 PROCEDURE Figure 4.1:- Experimental Set Up As shown in the figure 4.1, the apparatus was arranged according to the setup diagram that has been given in the procedure card. A millivolt meter was connected to the output terminals which come out from the thermocouple that is placed behind the n-Ge test piece. An ac voltage source was applied to the heating coil at the back of the test piece and the applied voltage has a capability of varying. Constant DC voltage value was given n-Ge test piece and the given potential must not be exceeded the maximum allowable current between the test piece all over the processing time. If it does, it has to be decreased the DC value immediately. A miliampere meter and a voltage meter were connected accordingly on behalf of measuring the current and voltage through the test piece. 100 Ω resistor was connected as series, right next to the test piece for protecting from over current. First, the alternating voltage was set to 2V. Checked the motionless temperature. Again the alternating voltage was increased to 4V and notice the motionless temperature. Same method was done for 6V by keeping mind that the maximum temperature of test piece would not be exceeded. Then the test piece was allowed to cool and the current, voltage variations were recorded as a function of temperature in the steps of 5K. Steps of temperature were calculated from the millivolt meter readings that of 200µV was to 5K as the equation given for thermocouple, 40µV per Kelvin. Department of Physic 8 Janith Prabodha Band Gap PHY301M4 5.0 RESULTS Graph 5.1:- Logarithm of conductivity of Germanium test piece versus reciprocal of absolute temperature for graph m = 3924.50623 K Practical value of Band gap of Ge = 2 × 3924.50623 K × 8.625 × 10−5 eVK −1 = 0.67698eV Theoretical value of band gap of Ge = 0.65eV Deviation percentage of band gap = 4.15% Dimension of Ge sample = 20mm×10mm×1mm Room temperature = 303 K Department of Physic 9 Janith Prabodha Band Gap PHY301M4 6.0 CONCLUSION This practical was arranged to determine the band gap of n-Ge test piece. Logarithmic of conductivity versus reciprocal of absolute temperature graph was plotted. Gradient of the graph gave the opportunity to derive the band gap or 𝐸𝑔 of test piece and it was determined as 0.67698eV at room temperature (303K). The obtained value was deviated 4.15% from the theoretical value 0.65eV. Department of Physic 10 Janith Prabodha Band Gap PHY301M4 7.0 DISCUSSION Here we are dealing with a test piece which has a small energy gap and hence the values must be so accurate for a better answer. But the instruments which we used here were not ideal and even the connectors and connecting wires have a considerable resistance. This facts directly affect to the final result. As the readings; voltage across the test piece and the current flow through the test piece must be taken simultaneously for different voltages of thermocouple. It was too difficult take values simultaneously as they had been changing rapidly all over the practical. Values taking from a detector that is changing rapidly can be given an error values even at the noticing down. It will be affected to the calculations. Department of Physic 11 Janith Prabodha Band Gap PHY301M4 8.0 REFERENCES 1. https://www.gopracticals.com/physics/physics-determine-band-gap-semiconductor 2. http://vlabs.iitb.ac.in/vlabs-dev/labs/physics-basics/labs/energy-band-gap-iitk/simulation 3. Novak, G.S., Conversion of units of measurement. IEEE Transactions on Software Engineering, 1995. 21(8): p. 651-661. 4. Kittel, C., Introduction to solid state physics. 2005: Wiley. 5. https://www.hiram.edu/wp-content/uploads/2016/12/exp-band-gap.pdf. Department of Physic 12 Janith Prabodha Band Gap PHY301M4 9.0 Appendix Table 9.1(Readings of heating coil voltage 2v &4v & 6v) Temperature Temperature (T) (C) 130 125 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 35 30 (K) 403 398 393 388 383 378 373 368 363 358 353 348 343 338 333 328 323 318 313 308 303 4000 3800 3600 3400 3200 3000 2800 2600 2400 2200 2000 1800 1600 1400 1200 1000 800 600 400 200 0 Current Voltage 1 T (mA) 29.5 28.7 28 27.5 26.6 25.5 24.5 23.4 22.1 20.5 19.4 18.1 16.4 15 13.8 12 10.5 9.1 7.4 6.3 5.2 (V) 0.72 0.78 0.89 1 1.04 1.12 1.24 1.34 1.45 1.65 1.76 1.9 2.07 2.25 2.41 2.55 2.72 2.86 3.03 3.11 3.21 −1 0.0024814 0.0025126 0.0025445 0.0025773 0.002611 0.0026455 0.002681 0.0027174 0.0027548 0.0027933 0.0028329 0.0028736 0.0029155 0.0029586 0.003003 0.0030488 0.003096 0.0031447 0.0031949 0.0032468 0.0033003 σ −1 Ln( σ) −1 20.48611111 18.3974359 15.73033708 13.75 12.78846154 11.38392857 9.879032258 8.731343284 7.620689655 6.212121212 5.511363636 4.763157895 3.961352657 3.333333333 2.863070539 2.352941176 1.930147059 1.590909091 1.221122112 1.012861736 0.809968847 3.01974715 2.912211302 2.755591146 2.621038824 2.548543322 2.432202586 2.290414557 2.166919228 2.030866872 1.826502418 1.706812076 1.560910872 1.376585547 1.203972804 1.051894664 0.85566611 0.657596196 0.464305608 0.1997702 0.012779727 -0.210759492 Table 9.1:- Readings of heating coil voltage 2v &4v & 6v. Department of Physic 13 Janith Prabodha