HALL EFFECT In a Ga!ium Arsenide Semiconductor Jason Robin Fall 07 Phy Adv Lab Ha! Effect 1
HALL EFFECT In a Ga!ium Arsenide Semiconductor Jason Robin University of Rochester Fall 07 PHY ADV LAB Introduction In this lab, the hall effect is due to a current being applied over a semiconductor in a magnetic field causing a displacement of the charge carriers to wall 4 in the figure 1 creating a voltage between walls 3 and 4. Figure 1 The build of charge on walls 3 and 4 have a potential difference which will balance the Lorentz force which acts on the moving charge carriers as they are in the magnetic field. This effect occurs general in any conducting material but is especially interesting to study in semiconductors because of the interesting behavior of the charge carriers which is a direct Ha! Effect 2
effect of quantum mechanics and Fermi-Dirac statistics and will be more carefully discussed in the theoretical explanation section of the report. Theoretical Background Different types of materials have different levels of conductivities. Mathematically, if a material is a good conductor, such as most metals, the electrical conductivity has a high value (S m^-1). A material like glass, which is a poor conductor has a very low value of conductivity. What makes a material a conductor, or insulator? Every material has two regions where charge carriers can occupy, a valence band and conduction band. In the valence band of energy, the electrons (negative charge carriers) are bound to the atom and cannot leave. If an electron has enough energy to occupy the conduction band, then it is not bound to any specific atom and is free to roam around the lattice of the crystalline structure. Electrons will be able to occupy the conduction band if the Fermi level is at a higher energy the the bottom threshold of the conduction band guaranteeing that at zero temperature all states below the Fermi level are occupied. Figure 2. A comparison of band structures for conductors ad insulators In figure 2, on the left is Sodium which is a good conductor, the Fermi level reaches into the conduction band so there are some electrons which are not bound to any specific atoms and are free to act as charge carriers. On the right is the band spectrum from a Diamond which is an insulator or a poor conductor, the Fermi level is below the conduction band so there are no unbounded electrons that are free to move and act as charge carriers so this material Ha! Effect 3
is called an insulator. The fermi level is specific of materials and is determined by the density of free electrons and the number of states available for a material. The number of available states, or density of states, is a function of an energy interval and for free electrons is given by. Figure 3. the density of states function for a $ee electron While the density of states gives information regarding the number of available states, the Fermi-Dirac distribution is defined as the probability that a fermion, or electron in this case, occupies a specific energy level and is given by. Figure 4. Fermi-Dirac probability distribution. Here if the temperature is zero and the energy is below the fermi energy E_f then F(E) is equal to 1 and if the energy is above the fermi level, then F(E) is equal to 0. For temperatures greater than zero, then the Fermi-Dirac distribution is then plotted as normal which is illustrated in figure 5. Ha! Effect 4
Figure 5. Notice that for non-zero temperature the probability that particle energies are above the fermi energy is the same as the probability that it is not below the fermi level. This is due to the unity of probability and that for zero temperature the normalized probability to be below the fermi energy is 1, and now requiring that the probability sum to one gives that chance of a particle being above the fermi energy is equal to the chance that its not below it. For semiconductors, in the band structure of the material, the conduction band and valence band are very close to one another and hence have a small gap between them. Small changes in the internal energy, which is directly proportional to temperature can easily excite an electron from the valence band to the conduction band. It is for this reason that semiconductors have their name, for certain temperatures they are strictly insulators, but for low enough temperature they become conductors. For a crystalline semiconductor, electrons are not allowed to occupy energy regions which are not in the conduction or valence bands forcing the density of states to take a value of zero outside of those bands, which is illustrated in figure 7. Ha! Effect 5
Figure 6. the density of states function for a semiconductor The density of states from figure 3. describes free electrons, to properly describe the density of states in the conduction band a new density of states must be determined to require that the probability that a particle is found outside the conduction or valence bands be zero. Figure 7. density of states for an electron in the conduction band of a semiconductor To determine the number of electrons that are in the conduction band of a semiconductor one integrates the product of the density of states and the Fermi-Dirac distribution in the region of lowest allowed energy in the conduction band to infinity. Ha! Effect 6
Figure 8. Calculations of the number of electrons in the conduction band. The second calculation in Figure 8 gives the number of charge carriers which includes negatively charged electrons and the lack of electrons which are called holes and act like particles that have positive charge. When a current flows through a semiconductor, either the negative electrons in the conduction band are free to flow with the current or the positively charged hole move in the reverse direction of the conventional current which can both contribute to the overall current of the system. Figure 9. I!ustration of the flow of holes and conduction electrons in a Silicon semiconductor Ha! Effect 7
In the hall experiment there is a rectangular semiconductor as in figure 1. which is attached to an external voltage supply and then placed in a uniform magnetic field which runs perpendicular to the current density and hence velocity of the charge carriers in the semiconductor. The magnetic field will interact with the charge carriers via the Lorentz force which will force the negatively charged electrons to wall 4 and the holes onto wall 3 because of the cross product nature of the Lorentz force. Figure 10. The Lorentz force To balance the lorentz force, the negative charges that build up on wall 4 and positively charged holes on wall 3 create a scalar potential so in equilibrium we have. then define the hall coefficient to be Figure 10. Equilibrium condition of the ha! effect and the hall angle Figure 11. definition of the ha! coefficient Figure 12. definition of the ha! anle Ha! Effect 8
Experimental Set-Up Figure 13. Circuit Diagram of Set up Figure 13 is a circuit diagram of the set used in measuring the hall voltage across the gallium arsenide semiconductor. The computer has three ports, one for the the thermocoupler which is a way of measuring the temperature inside the dewar by comparing the current in the system with that of one with a known temperature of thirty two degrees F. Vhall is the port that measures directly the hall voltage from the gallium arsenide semiconductor, and Vr measures the resistance across the gallium arsenide semiconductor. A power supply drives the current through the semiconductor and and independent power supply feeds the heater which will warm the dewar throughout the experiment. Ha! Effect 9
Figure 12. The diagram of the dewar which houses the semiconductor Ha! Effect 10
In figure 12 the dewar is filled with liquid nitrogen and the heat diffuses out of the brass rod into the reservoir of liquid nitrogen. The hall generator is approximately where the semiconductor rests and the heat cooling supplies the heat to warm the semiconductor through a range of temperatures. Procedure First make sure that all of the leads from the computer to the dewar and thermocoupler are properly connected. Next fill the dewars reservoir with liquid nitrogen about 3/4 full. The author recommends filling the dewar with liquid nitrogen at least 1 hour prior to conducting to the experiment to let the system reach a thermal equilibrium due to the inertia of the system. Next power on the computers and make sure the voltmeters are properly working. Run the program for the Hall_Exp. Then, follow the on screen instructions. It is advisable to use a starting temperature of 190K and take at least 500 data points. Results Hall Voltage(T) vs Temp(K) 0-0.0125-0.0250-0.0375-0.0500 Hall Voltage vs Temp 187.500 203.750 220.000 Figure 13. Ha! Voltage(T) vs Temp(K) Ha! Effect 11
Hall Mobility(m/V-s) vs Temperature(K) 0-1000 -2000-3000 -4000 Hall Mobility vs Temperature 187.500 195.625 203.750 211.875 220.000 Figure 14. Ha! Mobility vs Temperature Hall Angle vs Temperature(K) 0-0.0025-0.0050-0.0075 Hall Angle vs Temperature -0.0100 187.500 195.625 203.750 211.875 220.000 Conclusions Figure 15. Ha! Angle vs Temperature After hours upon hours of trying to make sense of the data, it just doesn t work. It s the authors opinion that one of the main problems was that the lab manual written by the university of rochester recommended testing a temperature range of 130-340K, which is what was done by this lab group, and then when later looking through the specifications of the machine, the minimum operating temperature was 200K. Also, many hours were spent trying to find a theoretical explanation for why the data seemed so against the expected. Ha! Effect 12
Testing many different types of theoretical predications for the evolution of resistivity for intrinsic semiconductors with temperature yielded undesirable results which was discouraging. I have reasons to believe that the code in the computer has been altered because in the output file for the thermocoupler device the reading was always zero and it seems some uncorrelated value was used. Ha! Effect 13