ace XIlI (HEMT), the heterojunction bipolar transistor (HBT), and the laser diode are analyzed Chapter 10 is dedicated to the most recent semiconductor devices After introducing the tunnel effect and the tunnel diode, the physics of low-dimensional devices(two-dimensional electron gas, quantum wire and quantum dot) is analyzed. The characteristics of the single-electron transistor are derived. Matlab problems are used to visualize tunneling through a potential barrier and to plot the density of states in lov dimensional devices Chapter ll introduces silicon processing techniques such as oxidation, ion implantation, lithography, etching and silicide formation. CMOS and BJT fabrication processes are also described step by step. Matlab problems analyze the influence of ion implantation and diffusion parameters on MOS capacitors, MOSFETS, and BJTs The solutions to the end-of-chapter problems available to Instructors to download a solution manual and Matlab files corresponding to the end-of-chapter problems, please go to the following Url:http://www.wkap.nl/prod/b/1-4020-7018-7 This book is dedicated to Gunner. David. Colin-Pierre. Peter. Eliott and michael. The late Professor f. van de wiele is acknowledged for his help reviewing this book and his mentorship in Semiconductor Device hysics Cynthia A. Colinge Jean-Pierre Colinge California State University University of californio
Preface xiii (HEMT), the heterojunction bipolar transistor (HBT), and the laser diode, are analyzed. Chapter 10 is dedicated to the most recent semiconductor devices. After introducing the tunnel effect and the tunnel diode, the physics of low-dimensional devices (two-dimensional electron gas, quantum wire and quantum dot) is analyzed. The characteristics of the single-electron transistor are derived. Matlab problems are used to visualize tunneling through a potential barrier and to plot the density of states in lowdimensional devices. Chapter 11 introduces silicon processing techniques such as oxidation, ion implantation, lithography, etching and silicide formation. CMOS and BJT fabrication processes are also described step by step. Matlab problems analyze the influence of ion implantation and diffusion parameters on MOS capacitors, MOSFETs, and BJTs. The solutions to the end-of-chapter problems are available to Instructors. To download a solution manual and the Matlab files corresponding to the end-of-chapter problems, please go to the following URL: http://www.wkap.nl/prod/b/1-4020-7018-7 This Book is dedicated to Gunner, David, Colin-Pierre, Peter, Eliott and Michael. The late Professor F. Van de Wiele is acknowledged for his help reviewing this book and his mentorship in Semiconductor Device Physics. Cynthia A. Colinge California State University Jean-Pierre Colinge University of California
Chapter 1 ENERGY BAND THEORY l.1. Electron in a crystal This Section describes the behavior of an electron in a crystal. It will be demonstrated that the electron can have only discrete values of energy, and the concept of "energy bands"will be introduced. This concept is a key element for the understanding of the electrical properties of semiconductors l.1.1. Two examples of electron behavior An electron behaves differently whether it is in a vacuum, in an atom, or in a crystal. In order to comprehend the dynamics of the electron in a semiconductor crystal, it is worthwhile to first understand how an electron behaves in a simpler environment. We will, therefore, study the classical"cases of the electron in a vacuum (free electron)and the electron confined in a box-like potential well (particle-in-a-box) 1.1.1.1. Free electron The free electron model can be applied to an electron which does not interact with its environment. In other words. the electron is not submitted to the attraction of the atoms in a crystal; it travels in a medium where the potential is constant. Such an electron is called a free electron. For a one-dimensional crystal, which is the simplest possible structure imaginable, the time-independent Schrodinger equation can be written for a constant potential V using Relationship A3. 12 from Annex 3. Since the reference for potential is arbitrary the potential can be set equal to zero(V=0)without losing generality. [ The time-independent Schrodinger equation can, therefore, be written as
Chapter 1 ENERGY BAND THEORY 1.1. Electron in a crystal This Section describes the behavior of an electron in a crystal. It will be demonstrated that the electron can have only discrete values of energy, and the concept of "energy bands" will be introduced. This concept is a key element for the understanding of the electrical properties of semiconductors. 1.1.1. Two examples of electron behavior An electron behaves differently whether it is in a vacuum, in an atom, or in a crystal. In order to comprehend the dynamics of the electron in a semiconductor crystal, it is worthwhile to first understand how an electron behaves in a simpler environment. We will, therefore, study the "classical" cases of the electron in a vacuum (free electron) and the electron confined in a box-like potential well (particle-in-a-box). 1.1.1.1. Free electron The free electron model can be applied to an electron which does not interact with its environment. In other words, the electron is not submitted to the attraction of the atoms in a crystal; it travels in a medium where the potential is constant. Such an electron is called a free electron. For a one-dimensional crystal, which is the simplest possible structure imaginable, the time-independent Schrödinger equation can be written for a constant potential V using Relationship A3.12 from Annex 3. Since the reference for potential is arbitrary the potential can be set equal to zero (V = 0) without losing The time-independent Schrödinger equation can, therefore, be written as:
Chapter I 2m故24x)=Exy (1.1.1) where E is the electron energy, and m is its mass. The solution to Equation 1.1.1 is Y(x)=CI exp(jkx)+C 2 exp(ilx wh 2me 加2k2 E (1.1.3) quation 1.1.2 represents two waves traveling in opposite directions. Cl expljkx) represents the motion of the electron in the +x direction, while C2 exp(-jkx) represents the motion of the electron in the -x direction What is the meaning of the variable k at first it can be observed that th unit in which k is expressed is m-I or cm-l; k is thus a vector belonging to le reciprocal space. In a one-dimensional crystal, however, k can be onsidered as a scalar number for all practical purposes. The momentum operator, Px, of the electron, given by relationship A3.2,is Considering an electron moving along the +x direction in a one dimensional sample and applying the momentum operator to the wave function Y(x)=CI exp(jkx)we obtain 方d(x) Px y(x) CI Ak expl)=hk y(x) The eigenvalues of the operator Pr are thus given by Hence. we can conclude that the number k. called the wave number. is equal to the momentum of the electron, within a multiplication factor h In classical mechanics the speed of the electron is equal to v=phm, which yields v=nk/m. We can thus relate the expression of the electron energy, by Expression 1.1.3, to that derived from classical mechanics 2k2 hm→E The energy of the free electron is a parabolic function of its momentum k, as shown in Figure 1. 1. This result is identical to what is expected from classical mechanics considerations: the free"electron can take any value of energy in a continuous manner. It is worthwhile noting that electrons
2 Chapter 1 where E is the electron energy, and m is its mass. The solution to Equation 1.1.1 is : where: Equation 1.1.2 represents two waves traveling in opposite directions. represents the motion of the electron in the +x direction, while represents the motion of the electron in the -x direction. What is the meaning of the variable k? At first it can be observed that the unit in which k is expressed is or k is thus a vector belonging to the reciprocal space. In a one-dimensional crystal, however, k can be considered as a scalar number for all practical purposes. The momentum operator, of the electron, given by relationship A3.2, is: Considering an electron moving along the +x direction in a onedimensional sample and applying the momentum operator to the wave function we obtain: The eigenvalues of the operator px are thus given by: Hence, we can conclude that the number k, called the wave number, is equal to the momentum of the electron, within a multiplication factor In classical mechanics the speed of the electron is equal to v=p/m, which yields We can thus relate the expression of the electron energy, given by Expression 1.1.3, to that derived from classical mechanics: The energy of the free electron is a parabolic function of its momentum k, as shown in Figure 1.1.This result is identical to what is expected from classical mechanics considerations: the "free" electron can take any value of energy in a continuous manner. It is worthwhile noting that electrons
1. Energy Band Theor with momentum k or -k have the same energy. These electrons have the same momentum but travel in opposite directions Figure 1.1: Energy vs. k for a free electron Another interpretation can be given to k. If we now consider a three- dimensional crystal, k is a vector of the reciprocal space. It is the called the wave vector. Indeed, the expression exp(ikr), where r=(x, y, z) is the position of the electron, and represents a plane spatial wave moving in the direction of k. The spatial frequency of the wave is equal to k, and its spatial wavelength is equal to/2z 1.1.1. 2. The particle-in-a-box approach After studying the case of a free electron, it is worthwhile to consider a situation where the electron is confined within a small region of space The confinement can be realized by placing the electron in an infinitely deep potential well from which it cannot escape. In some way the electron can be considered as contained within a box or a well surrounded by infinitely high walls( Figure 1.2). To some limited extent, the particle in-a-box problem resembles that of electrons in an atom, where the attraction from the positively charge nucleus creates a potential well that traps the electrons
1. Energy Band Theory 3 with momentum k or -k have the same energy. These electrons have the same momentum but travel in opposite directions. Another interpretation can be given to k. If we now consider a threedimensional crystal, k is a vector of the reciprocal space. It is the called the wave vector. Indeed, the expression exp(jkr), where r=(x,y,z) is the position of the electron, and represents a plane spatial wave moving in the direction of k. The spatial frequency of the wave is equal to k, and its spatial wavelength is equal to 1.1.1.2. The particle-in-a-box approach After studying the case of a free electron, it is worthwhile to consider a situation where the electron is confined within a small region of space. The confinement can be realized by placing the electron in an infinitely deep potential well from which it cannot escape. In some way the electron can be considered as contained within a box or a well surrounded by infinitely high walls (Figure 1.2). To some limited extent, the particlein-a-box problem resembles that of electrons in an atom, where the attraction from the positively charge nucleus creates a potential well that "traps" the electrons
Chapter l n=3 Figure 1. 2: Particle in a box: a) Geometry of potential well; b) Energy levels; c)Wave functions; d) Probability density for n=1, 2 and 3 By definition the electron is confined inside the potential well and therefore, the wave function vanishes at the well edges: thus the boundary conditions to our problem are: Y(xs0)=y(x>a)=0. Within the potential well(0 sx sa), where V=0, the time-independent Schrodinger equation can be written as n2d2甲(x)=E甲(x) which can be rewritten in the following form d2 y(x)+k2Y(x) 2k2 0 with k=v2mE/h2 The solution to this homogenous, second-order differential equation is Y()=A sin(k)+ b cos(k) (1.1.8) Using the first boundary condition Y (x=0)=0 we obtain B= 0. Using the second boundary condition Y(a)=o we obtain A sin(ka)=0 and therefore
4 Chapter 1 By definition the electron is confined inside the potential well and therefore, the wave function vanishes at the well edges: thus the boundary conditions to our problem are: Within the potential well where V = 0, the time-independent Schrödinger equation can be written as: which can be rewritten in the following form: The solution to this homogenous, second-order differential equation is: Using the first boundary condition we obtain B = 0. Using the second boundary condition we obtain A sin(ka) = 0 and therefore: