Microsoft PowerPoint - 半導體元件與物理1

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1 半導體元件與物理 Semiconductor Devices and physics 許正興國立聯合大學電機工程學系

2 1. Crystal Structure of Solids 2. Quantum Theory of Solids 3. Semiconductor in Equilibrium and Carrier Transport phenomena 4. PN Junction and Metal-Semiconductor 5. BJT 6. MOSFET 7. Optical Device

3 Part I 1. Extensions of the Atoms 2. Basic viewpoints of quantum Mechanics descriptive methods: Classical mechanics -> plants and satellites, Quantum mechanics -> electrons and high-frequency electromagnetic waves 3. Schrodinger s wave equation and application Quantum Mechanics -> Wave mechanics, the wave equation will be applied to the oneelectron, or hydrogen atom

4 Atoms Atoms are the basic structural unit of all engineering materials => consist primarily of three basic subatomic particle: protons, neutrons, and electrons. => the simple model of an atom envisions a very small nucleus of about m in diameter surrounded by a relatively thinly dispersed electron cloud of varying density so that the diameter of the atom is of the order of m. ( nucleus accounts for almost all the mass of the atom => contains the protons and neutrons. A proton mass : g A unit charge : C Neutron is slightly heavier than the proton but no charge : g The electron has a relatively small mass of g (1/1836 that of the proton and a unit charge of C) => The electron charge cloud thus constitutes almost all the volume of the atom but accounts for only a very small part of its mass.

5 The electrons, particularly the outer ones, determine most of the electrical, mechanical, chemical, and thermal properties of the atoms.

6 Atomic Numbers and Atomic Masses The atomic number is an atom indicates the number of protons (positively charged particle) => neutral atom the atomic number is also equal to the number of electrons in its charge cloud. (each element has its own characteristic atomic number)

7 Atomic Masses Relative atomic mass of an element is the mass in grams of atoms (Avogadro s number N A ) of the element. -> carbon atom with 6 protons and 6 neutrons is the carbon-12 atom and is the reference mass for atomic masses. => atomic mass unit (u) is defined as exactly one-twelfth of the mass of a carbon atom. (a mass of 12 u) ; one molar relative atomic mass of carbon 12 has a mass of 12 g on this scale. Ex. : a. what is the mass in grams of one atom of copper? b. How many copper atoms are in 1 g of copper? a. The atomic mass of copper is g/mol ; in g of copper there are atom => 63.54(g/mol)/ (atoms/mol)=x (g Cu)/1 (atom) => x = g b. The number of copper atoms in 1 g of copper is => (atom/mol)/63.54 (g/mol) = x (atom of Cu)/1 g => x = atoms

8 The electronic structure of atoms Hydrogen atom is the simplest atom and consist of one electron surrounding a nucleus of one proton. => certain definite energy levels (orbitals) are allowed when consider the motion if the hydrogen electron its nucleus. => The term atomic orbital does not represent the orbit of an electron around the nucleus represent the energy state of the electron The reason for the restricted energy values is that electrons obey the laws of quantum mechanics, => allow certain energy values and not any arbitrary ones. If the hydrogen electron is excited to a higher energy level ->energy is absorbed by a discrete amount ; if the electron drops to a lower energy level -> energy is emitted by a discrete amount (which the form of electromagnetic radiation called a photon) the change energy E associated with the transition of the electron from one level to another is related to the frequency ν (nu) of the photon => E =hν (h : Planck s cont. = joule-second) ; c = λν => E =hc/λ

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10 Ex. Calculate the energy in joules (J) and electron volts (ev) of the photon whose wavelength λ is nanometers (nm). (1.00 ev = J ; h = Js ; 1nm = 10-9 m) Sol: E = hc/λ => E = [( Js)( m/s)]/(121.6 nm)(10-9 m/nm) = J = J (1 ev/ J) = 10.2 ev

11 Using hydrogen spectral data, Niels Bohr in 1913 developed a model for the hydrogen atom that consisted of a single electron orbiting a proton at a fixed radius. The Bohr equation=>

12 Ex: A hydrogen atom exists with its electron in the n = 3 state. The electron undergoes a transition to the n = 2 state. Calculate (a) the energy of the photon emitted, (b) its frequency, and (c) its wavelength. Sol: (a) (b) Energy of the photon emitted is E = ev/n 2 => E = E 3 -E 2 = (-13.6)/3 2 (-13.6)/2 2 = 1.89 ev = 1.89 ev ( J)/eV = J The frequency of the photon is E = hν (c) => ν= E/h = J/ JS = s -1 = Hz The wavelength of the photon is E = hc/λ =>λ = hc/ E = ( Js)( m/s)/ J = m = 659nm

13 In modern atomic theory, the n in Bohr s equation is designed the principal quantum number and represents the principal energy levels for electrons in atoms. The energy level of the hydrogen electron in its ground state ( 電子基態 ) is ev (where n = 1) When the hydrogen electron is excited to higher energy level, its energy is raised but its numerical value is less. If the hydrogen electron is excited to the free state where n =, the electron has a zero value of energy If the energy required to remove the electron completely from the hydrogen atom is 13.6 ev -> called ionization energy of the hydrogen electron

14 Heisenberg s uncertainly principle: Electrons can have noncircular orbits around the uncleus, the position and momentum of a small particle such as an electron cannot be determined simultaneously. => the exact position of the electron at any instant cannot be determined because is so small a particle. electron charge cloud density distribution is sometimes used to represent the position of the hydrogen electron in its motion about its nucleus.

15 Quantum Numbers of Electrons of Atoms Modern atomic theory states that the motion of an electron about its nucleus and its energy is characterized by not just one quantum number but four quantum numbers : (i) principal quantum number: n, (ii) the subsidiary quantum number: l, (iii) the magnetic quantum number, m l, (iv) the electron spin quantum number, m s Principal Quantum Number : corresponds to the n in the Bohr equation, and represents the main energy levels for the electron or shells => the higher the value of n ->the higher the energy of the electron and the higher the possibility of the electron being farther away from the nucleus. ( positive integers and range from 1 to 7) Subsidiary Quantum Number : the second quantum number is the subsidiary quantum and specifies subenergy levels within the main energy levels (subshell) where the probability of finding an electron is high if that energy level is occupied. (l = 0, 1, 2,., n-1 ; the letters s, p, d, f ) => Number designation l = => The s, p, d, and f subenergy levels of an electron are terms orbitals Letter designation l = s p d f

16 Magnetic Quantum Number : The third quantum number ml, specifies the spatial orientation of a single atomic orbital and has little effect on the energy of an electron. => the m l quantum number has permissible values of l to l, including zero. => in general, there are 2l+1 allowed values for m l. (maximum s : one orbital ; p : three orbitals ; d : five orbitals ; f : seven orbitals) Electron Spin Quantum Number : the fourth quantum number, specifies two allowed spin directions (clockwise and counterclockwise) for an electron spinning on its own axis => +1/2 and -1/2 (very minor effect on the energy of an electron) According to the Pauli exclusion principle of atomic theory, on two electrons can have the same set of four quantum numbers.

17 Electronic Structure of Multielectron Atoms Maximum Number of Electrons for Each Principal Shell : Atoms consist of the principal shells of high densities as dictated by the laws of quantum mechanics. The maximum number of electrons that can be contained in each shell in an atom is defined by different sets of the four quantum numbers (Pauli principle) and is 2n 2 (n is principal quantum number) => Atomic Size : Each atom can be considered to a first approximation as a sphere with a definite radius. => The radius of the atomic sphere is not a constant but depends to some extent on its environment.

18 Some trends in atomic size are evident. -> principal quantum number increases => the size of atom increases ; however, a few exceptions where the atomic size actually gets smaller. Electron Configurations of the Elements : electron configuration of an atom describes the way in which electrons are arranged in orbitals in an atom.=> are written in a conventional that lists the principal quantum number list, which is then followed by an orbital letter s, p, d, f.

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20 Principles of Quantum Mechanics Basic concept : delve into mathematics of quantum mechanics, there are three principles => principle of energy quanta, the wave-particle duality principle, and uncertainty principle. One experiment that demonstrates an inconsistency between experimental results and the classical theory of light is called the photoelectric effect. according to classical physics, if the intensity of the light us large enough, the work function of the material will be overcome and an electron will be emitted from the surface independent of the incident frequency. ( Maybe incorrect ) This result is not observed <real case: the maximum kinetic energy of the photoelectron varies linearly with frequency with a limiting frequency (ν=ν 0 ) at a constant incident intensity => below which no photoelectron is produced> If the incident intensity varies at a constant frequency, the rate of photoelectron emission changes, but the maximum kinetic energy remains the same. Planck postulated in 1900 that thermal radiation is emitted from the a heated surface in discrete packets of energy called quanta (E = hν). Einstein interpreted the photoelectric results by suggesting that the energy in a light wave is also contained in discrete packets or bundles. particle-like packet of energy : photon (E=hν) in 1905 The photoelectric effect: (a) electrons are ejected from the surface of a metal when exposed to light of frequency n in a vacuum; (b) plot of the maximum kinetic energy of ejected electrons vs. frequency of the incoming light.

21 A photon with sufficient energy can knock an electron from the surface of the material. The minimum energy required to remove an electron is called the work function of the materials and any excess photon energy goes into the kinetic energy of the photoelectron. The photoelectric effect : the discrete nature of the photon and demonstrates the particle-like behavior of the photon. The maximum kinetic energy of the photoelectron can be written as T max = mv 2 /2 = hν-hν 0 (ν ν 0 ) : hν is the incident photon energy and hν 0 is the minimum energy (work function), required to remove an electron from the surface.

22 Wave-particle duality The particle-like behavior of electromagnetic waves was also instrumental in the explanation of the Compton effect. An x-ray beam was incident on a solid, a portion of the x-ray beam was deflected and the frequency of the deflected wave had shifted compared to the incident wave. the observed change in frequency and the deflected angle corresponded exactly to the expected results of a billiard ball collision between an x-ray quanta or photon and an electron in which both energy and momentum are conserved. In 1924, de Broglie postulated the existence of matter wave. => suggested that since waves exhibit particle-like behavior, then particles should be expected to show wave-like properties. The hypothesis of de-broglie was the existence of a wave-particle duality principle. wave exhibit particle-like behavior ( the momentum of a photon : p=h/λ) particle exhibit wave-like behavior ( the wavelength of a particle : λ=h/p) Wave nature of electrons has been tested in several ways. -> in 1927, Davisson and Germer

23 Electrons from a heated filament were accelerated at normal incidence onto a single crystal of nickel. A detector measured the scattered electrons as a function of angle. the existence of a peak in the density of scattered electrons can be explained as a constructive interference of waves scattered by the periodic atoms in the planes of the nickel crystal. In order to gain some appreciation of the frequencies and wavelengths involved in the wave-particle duality principle a wavelength of 72.7Å obtained in the next example is in the ultraviolet range. Typically, we will be considering wavelengths in the ultraviolet and visible range.

24 Uncertainty principle The Heisenberg uncertainty principle in 1927, also applied primarily to very small particles, and states that we cannot describe with absolute accuracy the behavior of these subatomic particles. the uncertainty principle describes a fundamental relationship between conjugate variables, including position and momentum and also energy and time. The first statement of the uncertainty principle is that it is impossible to simultaneously describe with absolute accuracy the position and momentum of a particle. -> If the uncertainty in the momentum is p and the uncertainty in the position is x, then the uncertainty principle is stated as p x ħ, where ħ is h/2π = J-s (Planck s constant) include angular position and angular momentum The second statement of the uncertainty principle is that it is impossible to simultaneously describe with absolute accuracy the energy of a particle and the instant of time the particle has this energy. -> uncertainty in the energy is E and the uncertainty in the time is t, then the uncertainty principle is stated as E t ħ the uncertainty principle is only significant for subatomic particles one consequence of the uncertainty principle is that we cannot, for ex., determine the exact position of an electron -> we will, instead, determine the probability of finding an electron at a particular position. next develop a probability density function : determine the probability that an electron has a particular energy.

25 Schrodinger equation Schrdinger equation : K.E.+P.E. = total E The various experimental results involving electromagnetic waves and particles, which could not be explained by classical laws of physical, showed that a revised formulation of mechanics was required. Schrodinger, in 1926, provided a formulation called wave mechanic, which incorporated the principles of quanta introduced by Planck, and wave-particle duality principle introduced by de-broglie. <based on the wave-particle duality principle, we will describe the motion of electrons in a crystal by wave theory> this wave theory is described by Schrodinger s wave equation.

26 The wave equation The one-dimensional, nonrelativistic Schrodinger s wave equation is give by where Ψ(x,t) is the wave function, V(x) is the potential function assumed to be independent of time, m is the mass of the particle. There are theoretical arguments that justify the form of schrodinger s wave equation, but the equation is a basic postulate of quantum mechanics. The wave equation Ψ(x,t) will be used to describe the behavior of the system and, mathematically, Ψ(x,t) can be complex quantity. We may determine the time-dependent portion of the wave function and the positiondependent, or time-independent, portion of the wave function by using the technique of separation of variables. Assume the wave function can be written as where ψ(x) is a function of the position x only and φ(t) is a function of time t only. Substituting this form of the solution into Schrodinger s wave equation If we divide by the total wave function, them

27 From above equation, we will denote this separation of variables constant by η. The timedependent portion of above equation is then written as where again the parameter η is called a separation constant. The solution of above equation can be written as The form of this solution is the classical exponential form of a sinusoidal wave where η/ħ is the radian frequency ω. We have that E=hν or E=hω/2π. Then ω=η/ħ=e/ħ so that the separation constant is equation to the total energy E of the particle. The time-independent portion of Schrodinger s wave equation can now be written as where the separation constant is the total energy E of the particle. Then the equation may be written as This time-independent Schrodinger s wave equation can also be justified on the basis of the classical wave equation as

28 Physical meaning of the wave function We are ultimately trying to use the wave function Ψ(x,t) to describe the behavior of an electron in a crystal. The function Ψ(x,t) is a wave function, so it is reasonable to ask what the relation is between the function and the electron. The total wave function is the product of the positiondependent, or time-independent, function and the time-dependent function. Since the total wave function Ψ(x,t) is a complex function, it cannot by itself represent a real physical quantity. 2 Max Born postulated in 1926 that the function Ψ( x, t) dx is the probability of finding the particle between x and x+dx at a given time, or that Ψ( x,t) 2 is a probability density function where Ψ * (x,t) is the complex conjugate function. Therefore Then the product of the total wave function and its complex conjugate is given by Therefore, we have that is the probability density function and is independent of time. One major difference between classical and quantum mechanics is the in classical mechanics, the position of a particle or body can be determined precisely, whereas in quantum mechanics, the position of a particle is found in terms of a probability.

29 Boundary conditions Since the function Ψ( x,t) 2 represent the probability density function, then for a single particle, we have that The probability of finding the particle somewhere is certain. The above equation allows us to normalize the wave function and is one boundary condition that is used to determine some wave function coefficients.

30 Applications of Schrodinger s wave equation Schrodinger s wave equation in several example using various potential functions. These examples will demonstrate the techniques used in the solution of Schrodinger s differential equation provide an indication of the electron behavior under these various potentials. Electron in free space Consider the motion of an electron in free space. If there is on force acting on the particle, then the potential function V(x) will be constant and we must have E > V(x). Assume, for simplicity, that the potential function V(x)=0 for all x. then the timeindependent wave equation can be written from above equation The solution to this differential equation can be written in the form Recall that the time-dependent portion of the solution is Then the total solution for the wave function is given by

31 This wave function solution is a traveling wave, which means that a particle moving in free space is represented by a traveling wave. the coefficient A is a wave traveling in the +x direction ; the coefficient B is a wave traveling in the x direction ; the value of these coefficients will be determined from boundary conditions => see the traveling-wave solution for an electron in a crystal or semiconductor material. Assume we have a particle traveling in the +x direction for a moment, which will be describes by the +x traveling wave => B = 0 the traveling-wave solution in the form where k is a wave number and is k=2π/λ The parameter λ is the wavelength and is From de Broglie s wave-particle duality principle, the wavelength is also given by A free particle with a well-defined energy will also have a well-defined wavelength and momentum. The probability density function is Ψ(x,t)Ψ*(x,t) = AA*, which is a constant independent of position. A free particle with a well-defined momentum can be found anywhere with equal probability. this result is in agreement with the Heisenberg uncertainty principle in that a precise momentum implies an undefined position.

32 Infinite Potential Well The problem of a particle in the infinite potential well is a classic example of a bound particle. The potential V(x) as a function of position for this problem is shown in fig. The particle is assume to exist in region II so the particle is contained within a finite region of space. The time-independent Schrodinger s wave equation is given as where E is the total energy of the particle If E is finite, the wave function must be zero, or ψ(x)=0, in both regions I and III. A particle cannot penetrate these infinite potential barrier, so the probability of finding the particle in regions I and III is zero. -> The time-independent Schrodinger s wave equation in region II, where V=0, becomes A particular form of solution to this equation is given by where

33 One boundary condition is that the wave function ψ(x) must be continuous so that Applying the boundary condition at x=0, we must have that A 1 = 0. At x=a, we have This equation is valid if Ka=nπ, where the parameter n is a positive integer, or n=1, 2, 3. The parameter n is referred to as a quantum number. We can write => -n are not considered The coefficient A 2 can be found from the normalization boundary condition that was given as * ψ ( x) ψ ( x) dx =1. If we assume that the wave function solution ψ(x)=ψ*(x). Substituting the wave function into probability density function, we have Evaluation this integral gives Finally, the time-independent wave solution is given by This solution represents the electron in the infinite potential well and is a standing wave solution. The free electron was represented by a traveling wave, and now the bound particle is represented by a standing wave.

34 The parameter K in the wave solution was defined by above equation. Equating these two expressions for K, we obtain The total energy can be written as For the particle in the infinite potential well, the wave function is now given by Where the constant K must have discrete values, implying that the total energy of the particle can only have discrete values. This result means that the energy of the particle is quantized. That is, the energy of the particle can only have particular discrete value. We may note that as the energy increase, the probability of finding the particle at any given value of x becomes more uniform. Figure a shows the first four allowed energies for the particle in the infinite potential well, and fig. b and c show the corresponding wave functions and probability functions.

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36 Step Potential Consider a step potential function as shown in fig. we considered a particle being confined between two potential barriers, and we will assume that a flux of particles is incident on the potential barrier. Assume that the particles are traveling in the +x direction and that they originated at x=-. A particularly interesting result is obtained for the case when the total energy of the particle is less than the barrier height, or E<V 0. Consider the time-independent wave equation in each of the two regions. This general equation was given in above equation as The wave equation in region I (V=0) is The general solution to this equation can be written in the form Where the constant K 1 is The first term in above equation is a traveling wave in the +x direction that represents the incident wave, and the second term is a traveling wave in the x direction that represents a reflected wave. As in the case of a free particle, the incident and reflected particles are represented by traveling waves.

37 For the incident wave, A 1 A 1* is the probability density function of the incident particles. If we multiply this probability density function by the incident velocity, then v i A 1 A 1* is the flux of incident particles ; the v r B 1 B 1* is the flux of the reflected particles.< v r is the velocity of the reflected wave.> In region II, the potential is V=V 0. If we assume that E<V 0, then the differential equation describing the wave function in region II can be written as The general solution may then be written in the form where One boundary condition is that the wave function ψ 2 (x) must remain finite, which means that the coefficient B 2 =0. the wave function is now given by The wave function at x=0 must be continuous so that Then we obtain Since the potential function is everywhere finite, the first derivative of the wave function must also be continuous so that

38 Using above equation, we obtain From several above equation and solve them, to determine the coefficients B 1 and A 2 in terms of the incident wave coefficient A 1 => The reflected probability density function is given by and define a reflection coefficient, R, as the ratio of the reflected flux to the incident flux, which is written as where v i and v r are incident and reflected velocities, respectively, of the particles. In region I, V=0 so that E = T, where T is the kinetic energy of the particle. => so that the constant K 1 => The incident velocity : ; reflected velocity : The incident and reflected velocities (magnitudes) are equal => Substituting the expression =>

39 The result of R=1 implies that all of the particles incident on the potential barrier for E<V 0 are eventually reflected. => Particles are not absorbed or transmitted through the potential barrier. This result is entirely consistent with classical physics and one might ask why we should consider this problem in terms of quantum mechanics. The interesting results is in terms of what happens in region II. The wave solution in region II was given as ψ 2 (x) = A 2 e -K 2x. The coefficient A2 is A1+B1, which we derived from the boundary conditions. For the case of E<V 0, the coefficient A 2 is not zero. If A 2 is not zero, then the probability density function ψ 2 (x) ψ 2* (x) of the particle being found in region II is not equal to zero. This result implies that there is a finite probability that the incident particle will penetrate the potential barrier and exist in region II. The probability of a particle penetrating the potential barrier is another difference between classical and quantum mechanics: The quantum mechanical penetration is classically not allowed. Q A 2 = A 1 + B 1 0 Tunneling Phenomenon ϕ ( x) = A e 2 2 k x 2 0 Tunneling

40 The Potential Barrier Consider the potential barrier function, which is shown in figure. The more interesting problem is in the case when the total energy of an incident particle is E<V 0. Assume that we have a flux of incident particles originating on the negative x axis traveling in the +x direction. => solve Schrodinger s time-independent wave equation in each of the three regions. The solutions of the wave equation in regions I, II, III are given as where and The coefficient B 3 represents a negative traveling wave in region III. however, once a particle gets into region III, there are no potential changes to cause a reflection; therefore, the coefficient B3 must be zero. we have four boundary relations for the boundaries at x=0 and x=a corresponding to the wave function and its first derivative being continuous we can solve for the four coefficients B 1, A 2, B 2, and A 3 in terms of A 1

41 One particular parameter of interest is the transmission coefficient, in this case defined as the ratio of the transmitted flux in region III to the incident flux in region I. Then the transmission coefficient T is where v t and v i are the velocity of the transmitted and incident particles. Since the potential V=0 in both regions I and III, the incident and transmitted velocities are equal. The transmission coefficient may be determined by solving the boundary condition equations. For the special case when E<<V 0, we find that Above equation implies that there is a finite probability that a particle impinging a potential barrier will penetrate the barrier and will appear in region III. This phenomenon is called tunneling and it, too, contradicts classical mechanics. The wave functions through the potential barrier