EE 346: Semiconductor Devices

Transcription

1 EE 346: Semiconductor Devices Lecture /01/2017 Tewodros A. Zewde 1

2 The One-Electron Atom The potential function is due to the coulomb attraction between the proton and electron and is given by where e is the magnitude of the electronic charge. In spherical coordinates, Schrodinger's wave equation may be written as We will assume that the solution to the time-independent wave equation can be written in the form, and hence

3 The second term in the above equation is a function of one parameter, and we may then write that The solution to Equation is of the form. Since the wave function must be single-valued, we impose the condition that m is an integer, or m = The separation-of-variables constants l, n, and m are known as quantum numbers and are related by

4 Each set of quantum number corresponds to a quantum state which the electron may occupy. The electron energy may be written in the form where n is the principal quantum number. For the lowest energy state, n = l. l = 0, and m = 0, and the wave function is given by The probability of finding the electron at a particular distance from the nucleus, is proportional to the product *. The wave function is spherically symmetric, and the parameter a o is given by

5 The probability density function for the lowest energy state is plotted below. The most probable distance from nucleus is at r =a 0, and this leads to the concept of an electron cloud*. For n = 2, l = 0. and m = 0, the second energy shell is at a greater radius from the nucleus than the first energy shell, though there is still a small probability that the electron will exist at the smaller radius. *The electron cloud model says that we can not know exactly where an electron is at any given time, but the electrons are more likely to be in specific areas. It is theoretically possible, for an electron to be a nearly infinite distance away from the atomic nucleus it is orbiting, although the probability of an electron decreases dramatically the further away from the nucleus you search.

6 ALLOWED AND FORBIDDEN ENERGY BANDS Hydrogen The first two figures show the radial probability density function for the lowest electron energy state of the single, noninteracting hydrogen atom and for two atoms that are in close proximity to each other, respectively. This interaction or perturbation results in the discrete quantized energy level splitting into two discrete energy levels, schematically shown in the last figure.

7 Now, if we somehow start with a regular periodic arrangement of hydrogen type atoms that are initially very far apart, and begin pushing the atoms together, the initial quantized energy level will split into a band of discrete energy levels At the equilibrium interatomic distance, there is a band of allowed energies, but within the allowed band, the energies are at discrete levels.

8 Suppose the atom in this imaginary crystal contains electrons up through the n = 3 energy levels. If the equilibrium interatomic distance is ro. then we have bands of allowed energies that the electrons may occupy separated by bands of forbidden energies. This energy-band splitting and the formation of allowed and forbidden bands is the energy-band theory of single-crystal materials.

9 The actual hand splitting in a crystal is much more complicated than indicated the previous figure. Consider an isolated silicon atom, and its schematic representation is shown below. Since the first two energy shells are completely full and are tightly bound to the nucleus, we need only consider the valence electrons (i.e., n = 3 level). 4 empty states n=2: Complete Shell 2 2s electrons 6 2p electrons Silicon 4 Valence Shell Electrons n=1: Complete Shell 2 s electrons n=3: 2 3s electrons Only 2 of 6 3p electrons

10 4 electrons available for sharing (covalent bonding) in outer shell of atoms

11 SYRACUSE UNIVERSITY ENGINEERING & COMPUTER SCIENCE

12 SYRACUSE UNIVERSITY ENGINEERING & COMPUTER SCIENCE

13 SYRACUSE UNIVERSITY ENGINEERING & COMPUTER SCIENCE

14 SYRACUSE UNIVERSITY ENGINEERING &COMPUTER SCIENCE

15 SYRACUSE UNIVERSITY ENGINEERING &COMPUTER SCIENCE

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17 T=0K E C or conduction band Band Gap where no states exist E V or valence band

18 For (E thermal =kt)=0 No electrons in conduction band means no electron conduction is possible Ec Ev

19 For (E thermal =kt)>0 Electron free to move in conduction band Ec + Ev As the negatively charged electron breaks away from its covalent bonding position, a positively charged "empty state" is created in the original covalent bonding position in the valence band. As the temperature further increases, more covalent bonds are broken, more electrons jump to the conduction hand, and more positive "empty states" are created in the valence band.

20 Carrier Movement Under Bias For (E thermal =kt)>0 Electron free to move in conduction band Ec + Ev Direction of Current Flow Hole movement in valence band Direction of Current Flow The movement of a valence electron into the empty state is equivalent to the movement of the positively charged empty state itself.

21 Electron free to move in conduction band Ec + Ev Direction of Current Flow Hole movement in valence band Direction of Current Flow The crystal now has a second equally important charge carrier that can give rise to a current. This charge carrier is called a hole.

22 Electron free to move in conduction band Ec + Ev Direction of Current Flow Hole movement in valence band Direction of Current Flow

23 Electron free to move in conduction band Ec + Ev Direction of Current Flow Hole movement in valence band Direction of Current Flow

24 Current is due to the net flow of charge. If we had a collection of positively charged ions with a volume density N (cm 3 ) and an average drift velocity v d (cm/s), then the drift current density would be J = q Nv d A/cm 2 If, instead of considering the average drift velocity, we considered the individual ion velocities, then we could write the drift current density as where u i is the velocity of the i th ion. The summation is taken over a unit volume so that the current density J is still in units of A/cm 2. Similarly, drift current density due to the motion of electrons as

25 Assuming at this point that no external forces are applied so the electron and "empty state" distributions are symmetrical with k. If an external force is applied to the electrons in the conduction band, there are empty energy states into which the electrons can move: therefore, because of the external force, electrons can gain energy and a net momentum.

26 Only these particles carry electricity. Thus, we call these carriers Clarification of confusing issues: Holes and Electrons Terminology Electrons: Sometimes referred to as conduction electrons: The electrons in the conduction band that are free to move throughout the crystal. Holes: Missing electrons normally found in the valence band (or empty states in the valence band that would normally be filled). If we talk about empty states in the conduction band, we DO NOT call them holes! This would be confusing. The conduction band has mostly empty states and a few electrons. If we talk about filled states in the valence band, we DO NOT call them electrons! This would be confusing. We can call them Valence Electrons to indicate they are bond to atoms (in the valence shells of atoms). The valence band has mostly filled states and a few holes. For the vast majority of this class we only talk about electrons (conduction band electrons) and holes (empty states in the valence band)!

27 Material Classification based on Size of Bandgap Ease of achieving thermal population of conduction band determines whether a material is an insulator, semiconductor, or metal. ~0 Electrons in Conduction Band ~ cm -3 Electrons in Conduction Band without help ~10 22 cm -3 Electrons in Conduction Band

28 DENSITY OF STATES FUNCTION Since current is due to the flow of charge. an important step in the process is to determine the number of electrons and holes in the semiconductor that will be available for conduction. The number of carriers that can contribute to the conduction process is a function of the number of available energy or quantum states since, by the Pauli exclusion principle, only one electron can occupy a given quantum state. Hence, we must determine the density of discrete energy levels, or allowed energy states as a function of energy in order to calculate the electron and hole concentrations. As the energy of this free electron becomes small, the number of available quantum states decreases.

29 We derived a general expression for the density of allowed electron quantum states using the model of a free electron with mass rn bounded in a three-dimensional infinite potential well We can extend this same general model to a semiconductor to determine the density of quantum states in the conduction band and the density of quantum states in the valence band. The density of allowed electronic energy states in the conduction and valance hand are given as follow: As the energy of the electron in the conduction band decreases, the number of available quantum states also decreases.

30 The plot of the density of quantum states as a function of energy. Quantum states do not exist within the forbidden energy band, so g(e) = 0 for E v <E<E c. If the electron and hole effective masses were equal, then the functions g c (E) and g v (E) would be symmetrical about the energy midway between E c and E v.