Microelectronic: The Thomson Model, The Thomson Model, and Semiconductors - Assignment Example

Logic and Programming Microelectronic Assignment Question Thomson’s Model The Thomson Model, otherwise known asthe plum pudding model or the blueberry muffin model is a discovery from the experiments carried out by a British physicist, John Joseph Thomson in 1904. He suggested a model atom, which integrated the newly discovered electron in 1897. The abandonment of the plum pudding model followed the discovery of atomic nucleus. In the plum pudding model, the atom consists of electrons, which are particles surrounded by a collection of positively charged particles that balances the negativity of the electrons. Thomson epitomizes the balance using raisins (with negative charges) surrounded by pudding (positive charges). The concept behind this model was that the electrons were positioned in the entire body of the atom, with the support of various structures for the positioning of the large number of electrons. Joseph Thomson uses this concept to explain the rotation of the rings of electrons as shown in the figure 1 below. Figure 1: The Plum and Pudding Model Instead of using the reference of positive charges as soup, Thomson argued that the atom was surrounded by a cloud of positively charged particles. After discovering this model, Thomson decided to abandon his previous discovery of the hypothesis of nebular atom. This hypothesis claimed that the atom was made up of inconsequential whirlpools. The 1909, Ernest Marsden and Hans Geiger was able to disapprove Thomson’s model by his new discovery known as the gold foil experiment. Ernest Rutherford further interpreted this experiment in 1911, arguing that the gold foil was an extremely small nucleus of the atom, which contained a large amount of highly positive charges that could balance approximately 100 electrons. This led to the discovery and development of the Rutherford Model. After the appearance of Rutherfords paper in 1911, there was a sudden suggestion by Antonius Van den Broek, which claimed that atomic number of any element is the nuclear charge. This suggestion prompted the dire need for further experiment to make a substantial decision. In 1913, Henry Moseley did an experiment to prove that the atomic number was not the nuclear charge but that the atomic number is just one unit away from the effective nuclear charge. This experimental work ended up in the design and development of the Bohr model, a quantum-limited atomic model in 1913. In The Bohr’s Model, a nucleus with the atomic number being the positive charge is in the center of an the same number of electrons, moving around it to form a set of orbital shells. The Thomsons model received its name “Plum Pudding” from a comparison made between it and a dessert in Britain with the same name “Plum Pudding”, though not originating from Thomson Himself. The atom of any element is made up of a specific number of negatively charged particles covered in a spherical container of the same number of positive charges. In the Joseph Thomson’s model, the electrons are freely hanging in the atom and are free to rotate inside the cloud of positively charged substance. The orbits in which the electrons move are made stable inside the model from the concept that when electrons flow away from the middle of the positively charged cloud, it feels a greater positive pull towards the inner region of the cloud. This is because the inner orbit was filled with more material holding the opposite charges. In the operation of the Thomsons model, the negative charges freely rotate in form of rings. These rings are further stabilized through the interactions between the various electrons. The spectra are explained by the difference in energy between the different orbits (energy levels). Joseph Thomson attempted to use his model to give explanations to most of the major spectral lines that had been identified in some elements. However, this was not with a success. Still, Thomsons model was the source of later discoveries of successful models including the Bohr model of atoms (a solar-system-like model). A very important mathematical application of the connection to the plum pudding model was the optimal distribution of a similar point charges on a sphere of a diameter of one unit known as the Thomson problem. This problem comes as a natural result of the Thomson’s plum pudding model when there is no uniformly positive charge. The conventional electrostatic management of negatively charged particles confined to a sphere quantum of dots. This is also the same as the way they are treated inside the plum pudding model. Among all the physicists who worked in partnership with the determination of the structure of an atom, Joseph Thomson remained most steadfast to the contemporary chemical community. He developed non-mathematical theory known as the atomic theory. This theory accounts for the chemical bonding in relation to the molecular structure. In 1913, joseph Thomson made publication of a significant profile that urged chemists’ to make use of the spectrograph in their analyses. Question 2: Experiment: Experiment of the Wave nature of Electrons The experiment for confirmation of the wave nature of electrons is the Davisson- Germer experiment, which confirms the Louis DeBroglie’s Hypothesis. Louis DeBroglie, a student in the University of Paris carried out an experiment after being impacted by the relativity and after studying the photoelectric effect. The two subjects had been initiated during his life time. The photoelectric effect was a pointer to the fact that light had particle properties, which were thought of to be in a wave form. He wanted to investigate whether the electrons among other "particles" would be able to demonstrate the perceived wave-like properties. As a student, he applied the two newly found ideas to light that suggested an intriguing possibility. Figure 2: Confirming the DeBroglie Hypothesis 2.1. Wavelike Properties All forms of quanta such as photons and electrons a common property referred to as the wave function. This is a cloudlike movement having a probability that the quantum resides inside that cloud. The quantum is most likely to be found in the region with the greatest density of the cloud. However, it is very difficult to locate the exact position of the quantum inside the cloud (the wave function) at any moment. 2.2. Electron Wave Examples There are two examples of models that support the wave property of electrons which DeBroglie suggests in his hypothesis. The two models are: Discrete atomic energy levels Diffraction of electrons From the Bohr model of atom energy levels, the electrons are considered as waves moving along the circumference of the orbits of electrons to experience a profitable interference. Figure 3: Hydrogen Energy Levels En = 13.6 eV / n^2 Figure 4: Electron Diffraction The wave nature of electrons must be used in the explanation of electron behavior and their confinement to the size and the order of an atom. This wave property is applied in the quantum mechanical concept of particles inside a box. The outcome of the calculation is used in the description of the density of energy states in regard to electrons of solid substances. 2.3. DeBroglie Hypothesis De Broglie developed this hypothesis around 1923. In this hypothesis, the path moving to the expression of wavelength for particles is comparable to the analogy of the photon momentum. Starting with the Einstein formula E = Mc ^ 2 = KE + M0C^2. This is also expressed as: E = MC^2 = √ (P^2C^2 + M0^2C^4) For this reason, any particle with rest mass = 0 P = E / c For a Photon, E = hv = hc / λ Since De Broglie wavelength λ = h / p P = h / λ Therefore p = hc / cλ The relationship between the wavelength and the momentum for a photon is derived from the formula. The DeBroglie wavelength connections are also applicable in the calculation of properties of other particles. 2.4. Calculation of DeBroglie Wavelengths The evidence in the Davisson-Germer experiment demonstrates the fact that electrons give the DeBroglie wavelength by: Wavelength λ = h / p The calculation below used a pitched baseball to check if the relationship between wavelength and the momentum is the same for all particles. V = 40 m / sec M = 0.15kg λ = h / mv λ = (6.6260 * 10^-34 J.s) / [(0.15kg) (40m / sec)] = 1.10 * 10^-34m In this calculation, the atomic diameter = 10^-10 and nuclear diameter = 10^-34. If an electron accelerates through 100 volts, v = 5.9 * 10^6 m / sec λ = (6.6260 * 10^-34 J.s) / [(9.11 * 10^-31kg) (5.9 * 10^6 m / sec)] = 1.20 * 10^-10m This is shorter than the smallest wavelength of visible light, which is approximately 390nm. Upon the practical exploration of the wavelengths of normal macroscopic items such as baseballs, it is clear that the various DeBroglie wavelengths for each of them are extremely small. By comparing the wavelength exponential functions, the wavelengths of normal objects are smaller than the nucleus by a very large margin. This implies that ordinary objects have no evidence of the wave nature. Additionally, they can be viewed as particles for every practical reason. In the exploration of the wavelengths for normal macroscopic items like baseballs, they have very small DeBroglie wavelengths. Question 3 There are two categories of semiconductors; intrinsic and extrinsic semiconductors. 3.1. Intrinsic Semiconductor Intrinsic semiconductors are pure chemical having extremely poor conductivity. These materials have a balance between the number of electrons (negative charges) and protons (positive). A good example is the silicon crystal. A slight change in the temperature on top of the absolute zero temperature causes the knock off of an electron at the lattice from the normal position. This leaves an unfilled electron position referred to as a hole. Upon the application of a voltage, both the hole and the electron support a slight flow of electric current. It is possible to model the conductivity of semiconductors using the band theory of solid substances. The semiconductor band model argues that in normal temperatures, it is possible for electrons to move, reaching the conduction band. This contributes to the semiconductor conducting electricity. 3.2. Extrinsic Semiconductor Extrinsic semiconductors are formed by introducing impurities into pure element semiconductor materials in a doping process to improve their conductivity. Doping creates two semiconductors categories; the n - type (negative charge) and the p – type (positive charge). This means that every atom and its four closest neighbors in the edge form a normal tetrahedron. The atom itself remains at the center of the tetrahedron. Apart from the semiconductors of pure elements, some semiconductors are made up of alloys and chemical compounds. The compound semiconductors are better because they have a wide spectrum of mobility and energy gaps. This ensures that the materials are able to satisfy specific requirements. 3.3. Summary of the Difference between Intrinsic and extrinsic Semi-conductors INTRINSIC SEMICONDUCTORS EXTRINSIC SEMICONDUCTORS They are pure semi-conductor materials without any additional impurity atoms. They are produced by the doping process in which a small amount of impurity atoms are added to the pure semi-conductor material Examples of intrinsic semiconductor materials are crystals of pure germanium and silicon. Examples of extrinsic semiconductor materials are silicon and germanium crystals having impurity atoms from As, Sb etc. They have equal number of freely moving electrons inside the conducting band and the number of empty holes in the valence band. The number of empty holes and that of free electrons are not equal. Electrons are in excess in the n-type materials while there are excess holes in the p-type materials. They have very low electrical conductivity. They have very high electrical conductivity Their electrical conductivity is only affected by the temperature Their electrical conductivity is affected by the temperature and the amount of impurity atoms used in the doped process Table 1: Difference between Intrinsic and extrinsic Semi-Conductors 3.4. N-type Semi-Conductor These are extrinsic semiconductor materials, formed by the doping of impurity atoms of the fifth group in the periodic table up to pure Si and Ge semi-conductor. The introductions of impurity atoms generate extra electrons inside the structure called the donor atoms. The electrons are considered the major carriers (majority) while the holes are the minor carriers (minority). The density of the electrons (ne) is higher that the density of holes (nh). The energy level of the donor is nearer to its conduction band than in the p type semiconductors. The energy donor level is far from the atom’s valence band. Between the conduction band and the donor energy level, there is the Fermi-energy. 3.5. P-type Semi-Conductor This is an extrinsic semi-conductor. It is formed by the doping of impurity atoms of elements in the third group in the periodic table up to the pure Si and Ge semi-conductor. These impurity atoms form holes called acceptor electrons in the structure. The holes are majority carriers and electrons are minority carriers. The density of holes is higher than that of electrons. The energy level of the acceptor is nearer to the valence band and far from the conduction band. The Fermi-energy level is found between the valence band and the acceptor energy level. N - Type Semiconductor P - Type Semiconductor This is an extrinsic semiconductor. It is obtained by the doping of the impurity atoms of elements such as phosphorus, antimony, arsenic, etc. to pure semiconductor atoms such as Si and Ge. This is an extrinsic semiconductor formed by the doping of the group three impurity atoms including indium, gallium etc to pure elements such as Si and Ge. The impurity atoms introduced into it generates extra electrons inside its structure. These are referred to as donor atoms. The impurity atoms introduced, make holes inside the structure. Because of the holes, the atoms are referred to as the acceptor atoms. The electrons are considered the majority carriers of the charge while holes are the minority carriers. The holes are considered the majority careers of charge and the electrons are the minority charge carriers. Table 2: Difference between N - Type and P - Type Semi-Conductors Question 4: Effect on the silicon electrical properties of doping with Doping 4.1. Donor 4.1.1. Doping using Group 5 Impurity atoms In this study, we use Phosphorus atoms (P), which has five available electrons to form Bonds with neighbouring atoms. Figure 5: Doping using Phosphorus on Silicon Four atoms move to create bonds with the neighboring Silicon atoms Si. The 5th electron remains unpaired and with a weak bounding to P. The thermal excitation is normally sufficient to break down the bond between the unpaired electron and the P atom. This causes the ionization of the donor atom P which bonds with the Si network, as well as a free electron, the donor. The donor electron carries the charge. Figure 6 below represent the ionization process using the diagram of energy levels. Figure 6: Energy Level Diagram Conduction Band c d Valence v Band Figure 7: Conduction band and Valence Band When T = 0, all the free electrons have very weak bonding to the P atoms, which is at the level marked d inside the figure 7 (energy level diagram). Naturally, the energy spent is calculated as c – d =  0.01 – 0.1eV. This is less or equal to thermal energies at the room temperature (kT = 0.0250eV). Therefore, at room temperature, many donor atoms released their extra electron (excited from the Ed level to move to the conduction band. The concentration of electrons in the conduction band increases greatly. Similarly the conductivity also increases greatly. Semiconductors which are doped using donors are the n-type. 4.2. Acceptor 4.2.1. Doping using Group 3 Impurity Atoms e.g. B Atom B contains 3 electrons available for the formation of bonds with neighboring atoms. B makes a tetrahedral bond with a broken bond. The broken bonding shared among all the four bonds with the neighboring Si’s. This considers the tetrahedral bond formed by the B to bind a vacancy. Figure 8 below shows the bonding. Figure 8: Doping using Group 3 Impurity B is able to receive or accept one electron from a bond between a Si and a Si atom from external element. This creates a negative ionization of the B atom (referred to as the acceptor), and a separate broken bond between a Si and a Si atom, referred to as a free or a mobile hole. This can be visualized as B that donates a free (mobile) hole. Figure 9 below shows the process of ionization by the acceptor. Figure 9: Doping of Si Using B This process takes place very easily at room temperature where there is thermal energy. Energy Level drawing when T = 0 Ec Ea Ev Ec Ea Ev Figure 10: Energy Level Diagram with The Acceptor Majority of the holes are excited to move down to the valence band. Again, many electrons are excited to move upwards from the valence band to the levels at Ea. This creates a very large number of holes (vacancies) inside the valence band. The doped semiconductor material in this case is refered to as the acceptor, which is the p – type semiconductor. n-Type The n type is dopped with the impurity atoms of elements in group 5 of the periodic table, such as P, As, etc. Upon the ionization, the impurities give out (donate) exess electron o the conduction band. Almost all the positively ionized atoms (donors) donate electrons at Room Temperature. Therefore, the number of nc increase greatly at Room Temperature. There is also a great increase in the  at room temperature. p-Type P Type semiconductors are doped with Group 3 acceptor impurities suct as B. Upon ionization, The impurities receive excess electron from the valence band. This creates an extra hole inside the valence band. Almost all the ionized (acceptor) create the holes at Room Temperature. Therefore, there is a great increase in pv at normal Room Temperature, and a great increase in  at normal Room Temp. It is possible to estimate the bonding energy for the donor electrons inside the n - type semiconductor using the a model called quasi-Bohr model. In the Bohr model, electron moving round the positively charged center(with protons). The quasi model applies to the n - type semiconductor materials which are doped with P. Weak bonding of the unpaired causes the electron to orbit the P nucleus with one positive charge at the centre.  r e- Figure 11: Doping with P Electrons go round the P+ under the influence of the central Coulomb force. Question 5 There are two categories of free careers for semiconductors. These are electrons and empty holes. This creates a possibility of two different currents to move in various junctions. These are drift current and diffusion currents. 5.1. The p - n diode Currents inside a p - n diode are controlled by diffusing the minority carriers inside the neutral zone, where the net charges totals to zero. The current is derived on the basis of the following assumptions: The contacts are reasonable and cannot show any bending of the band The contacts are able to absorb and produce the required carriers to fix the bulk circumstance carrier concentrations There is no voltage dropped across the neutral zones The voltage used falls only across the depletion zone The quantity of the introduced carriers across the p - n junction because of the voltage used is minor as compared to the main carrier concentration controlled by the doping process. In the forward bias, the injection of electrons takes place from the n - zone in the p - zone and the empty holes are brought in from the p – zone to the n – zone because of the external application of the potential that reduces the energy obstruction between the p and n type zone. The injection of carriers makes the minor carrier gradient of the electrons inside the p – type zone and a gradient of a hole inside the n - type zone. The gradients of these carriers trigger the diffusion current. Gradients of the same nature take place in reverse bias. However, they are very small, being the minority carriers but not the majority carriers brought in across the barrier. There is a small amount of leakage current in the reverse bias depending on the number of empty holes present and produced inside the n - type zone. The second factor is the amount of present and produced inside the p - type zone. The reverse leakage current under normal conditions is less than the forward leakage current because there are a small number of the minority carriers present. Figure 11: PN Junction Energy Band Diagram Figure 12: Depletion with no external bias The separation of the ionized charges inside the depletion zone is triggered by the inner electric field. The inner electric field is because of the voltage built inside because of the potential difference of the contact. Figure 13 below shows the p - n junction in the forward bias. Figure 13: Forward bias condition in the p - n diode Forward bias practically reduces the potential blockage across the P - n junction with a value controlled by the voltage used. In this regard, more carriers carry more energy than the blockage cross the P – n junction. The electric current flows and always increases in an exponential order. The reverse bias increases total value of the voltage used on the built-in voltage and increases the potential blockage. In that regard, there are no carriers with enough energy to move through the barrier. The minority carriers drift across the P - n junction. Because of the low number of the minority carriers, the electric current is very low and under the control of the minority carrier concentration as opposed to the electric field. This makes the off-current inside a diode low and steady as function of the reverse bias voltage while there is no mechanism for break down. Figure 14 below shows the Reverse bias condition and the energy band drawing of the p - n diode in reverse bias. Figure 14: Reverse bias condition and the energy band drawing of the p - n diode in reverse bias Question 6: Calculation of the resistances along the length of the silicon i. The silicon has a donor concentration of 5x1022m-3 Resistance R = Rs * (W / L), where Rs is the Sheet resistance, W is the silicon width and L is the Length R = Rs * (W / L), Rs = 1 / (μqN(x) dx) =1/ (0.048*5*1022*10^16) Rs = 4.0769 * 10^-19 R = 4.0769 * 10^-19 * (2 / 1000) / (1/100) R = 4.0769 * 10^-19 * (2 / 10) R = 8.15 * 10^-20Ω ii. The silicon has an acceptor concentration of 5x1022m-3 R = Rs * (W / L), Rs = 1 / (μqN(x) dx) =1 * 1.2 * 10^16/ (0.048 * 5 * 1022 * 10^16) Rs = 0.004892 R = 0.004892 * (2 / 1000) / (1/100) R = 0.004892 * (2 / 1000) / (1/100) * (2 / 10) R = 0.000196Ω R = 1.96 * 10^-4 Ω iii. The silicon is intrinsic It will behave like the donor. Resistance R = Rs * (W / L), where Rs is the Sheet resistance, W is the silicon width and L is the Length R = Rs * (W / L), Rs = 1 / (μqN(x) dx) =1/ (0.048*5*1022*10^16) Rs = 4.0769 * 10^-19 R = 4.0769 * 10^-19 * (2 / 1000) / (1/100) R = 4.0769 * 10^-19 * (2 / 10) R = 8.15 * 10^-20Ω Read More