Book 2 Basic Semiconductor Devices for Electrical Engineers Professor C.R. Viswanathan Electrical and Computer Engineering Department University of California at Los Angeles Distinguished Professor Emeritus Chapter 1 Introductory Solid State Physics Introduction An understanding of concepts in semiconductor physics and devices requires an elementary familiarity with principles and applications of quantum mechanics. Up to the end of nineteenth century all the investigations in physics were conducted using Newton’s Laws of motion and this branch of physics was called classical physics. The physicists at that time held the opinion that all physical phenomena can be explained using classical physics. However, as more and more sophisticated experimental techniques were developed and experiments on atomic size particles were studied, interesting and unexpected results which could not be interpreted using classical physics were observed. Physicists were looking for new physical theories to explain the observed experimental results. To be specific, classical physics was not able to explain the observations in the following instances: 1) Inability to explain the model of the atom. 2) Inability to explain why the light emitted by atoms in an electric discharge tube contains sharp spectral lines characteristic of each element. 3) Inability to provide a theory for the observed properties in thermal radiation i.e., heat energy radiated by a hot body. 4) Inability to explain the experimental results obtained in photoelectric emission of electrons from solids. Early physicists Planck (thermal radiation), Einstein (photoelectric emission), Bohr (model of the atom) and few others made some hypothetical and bold assumptions to make their models predict the experimental results. There was no theoretical basis on which all their assumptions could be justified and unified. In the 1920s, a revolutionary and amazing observation was made by De Broglie that particles also behave like waves. Einstein in 1905 had formulated that light energy behaves like particles called photons to explain the photoelectric results. De Broglie argued that energy and matter are two fundamental entities. The results of the photoelectric experiments demonstrated that light energy behaves also like particles and therefore matter also should exhibit wave properties. He predicted that a particle with a momentum of magnitude will behave like a plane wave of wavelength, equal to 풑 = 흀 (1.1) ℎ where is Planck’s constant and numerically equal to 휆 푝 풉 = 6.625 10 Joules - sec. −34 This led to the conclusionℎ that both particles푥 and energy (light) exhibited dual properties of behaving as particles and as waves. - 2 - Since Planck’s constant is extremely small, the De Broglie wavelength is correspondingly small for particles of large mass and momentum and therefore, the wave properties are not noticeable in our daily life. However, subatomicℎ particles such as electrons have such a small mass and their momentum is small to make the wavelength large enough to observe interference and diffraction effects in the laboratory. A particle moving with a momentum is represented by a plane wave of amplitude ( , ) given by: 풑��⃑ 훹 푟⃑ 푡 ( ) ( , ) = = 푝��⃑ ∙ 푟��⃑ (1.2) 푖�푘�⃑ ∙��푟⃑ − 휔 푡� 푖 ℏ − 휔 푡 where is the propagation vector of magnitude equal to , = , = 2 times the frequency 훹 푟⃑ 푡 퐴 푒 퐴 푒 2 휋 ℎ of the wave and , the radius vector is equal to 풌��⃑ 푘 휆 ℏ 2 휋 휔 휋 푟⃑ = + + and , and are the unit vectors푟⃑ along푎⃑ the푥푥 three푎⃑푦 coordinate푦 푎⃑푧 푧 axes. The radius vector denotes the position where the amplitude is evaluated. The wavelength of the plane wave according to equation 풂��⃑풙 풂��⃑풚 풂��⃑풛 (1.1), is given by = = (1.3) | | | | ℎ 2 휋 ( , ) is also called the wave-function or state휆 function푝 . The푘 expression for the plane wave given in equation (1.2) shows that the amplitude of the plane wave varies sinusoidally with time and position . For훹 illustrating푟⃑ 푡 the plane wave properties we will consider a plane wave travelling from negative infinity to positive infinity. Figure (1.1a) shows the amplitude of the plane wave at as a sinusoidal풕 function of 풙 time at some value = . Figure (1.1b) shows the amplitude varying as a sinusoidal function of position at some time = . 풕 풙 풙ퟏ 풙 풕 풕ퟏ - 3 - Plane wave 흍 흍 Amplitude, Amplitude, a) Amplitude as a function of time, t at x = x1 흍 흍 Amplitude, Amplitude, b) Amplitude as a function of position, x at t = t1 Figure (1.1): (a) Plane wave travelling from negative infinity to positive infinity, a) the amplitude of the plane wave as a function of time, t at some value = , (b) the amplitude of the plane wave as a function of x at some time = 푥 푥1 The dynamic behavior of a physical system such as a single푡 푡or1 a collection of particles can be deduced from , the wave-function of the system. In quantum mechanics, the wave-function is obtained by solving an equation called Schrödinger equation. Another consequence of applying quantum mechanical훹 principles to electrons is that the momentum and position of the particle say an electron, cannot be measured or known accurately at the same time to any precision that the measuring equipment is capable of. Either the momentum or the position alone can be measured to any accuracy the measuring equipment is capable of; but both of them cannot be measured at the same time to the accuracy or precision that each property can be measured separately. This is called the Heisenberg Uncertainly Principle. That is, the more precisely position or momentum is measured, the other can be measured only less precisely. The physics underlying the uncertainty principle is that the act of - 4 - measurement of one property disturbs the physical state of the system making measurement of the other property less accurate. In quantum mechanics, there are other examples of pairs of physical properties that obey the Heisenberg Uncertainty Principle. For example, energy and time is another pair that obeys the Uncertainty Principle. Mathematically the uncertainty principle says that the product of the uncertainty in simultaneous measurement of position and momentum of a particle has to be higher than a minimum value of the order of Planck’s constant , . The uncertainly principle is not observable in the size and mass of particles in our daily life where the uncertainties of the properties are too small to 푥 observe because of the small value of Planck’s풉 ∆푃 ∆constant.푥 ≥ ℎ When we deal with atomic particles such as electrons the uncertainty effects become significant. When the wave-function is determined by solving Schrödinger equation for an electron that is confined to a narrow region of space along the -axis, say between = 0 and = , the result shows that the - component of momentum can have only discrete values given by 푥 푥 푥 퐿푥 푥 = 푥 푥 ℎ 푃 푛 ( = ±1, ±2, ±3, . ) where is called a quantum number, and is an integer 퐿푥 . Thus the - component of momentum, , can have only discrete values and are said to be discretized or quantized 풙 and hence풏 is called a quantum number. If the electron 푛is constrained to move푒푡푒 in a restricted푥 region in 푷풙 the - plane, then 2 quantum numbers and will be needed to quantize the and components 풏풙 of the momentum. In the case when the electron is confined to a small three dimensional region, the 푥 푦 풏풙 풏풚 푥 푦 electron state will be specified by three quantum numbers and and , as discussed in the next section. 풏풙 풏풚 풏풛 Free Electron Model The electric current in a conducting solid such as a metal is explained in solid state physics using the Free Electron Model. According to this model, each atom of the solid contributes one (or two in some cases) electron to form a sea of electrons in the solid and these electrons are free to roam around in the solid instead of being attached to the parent atom. If an electric field is applied in the solid, the electrons will move due to the force of the electric field and each electron will carry a charge – coulomb thus giving rise to the flow of an electric current. Thus free electrons in the solid are treated quantum mechanically as equivalent to electrons in a box of the same dimensions as the solid. We푞 will now derive some properties of the free electrons in a conducting solid by using free electron model. Let us now consider an electron of mass that is confined to move within a three dimensional box. Let the box be rectangular with dimensions , and . As shown in Figure (1.2), we will choose 푚 the origin of our Cartesian coordinate system at one corner and the , and axes to be along the 퐿푥 퐿푦 퐿푧 three edges of the box. 푥 푦 푧 We assume that the potential energy to be constant and that it does not vary with position inside the box. Since force is equal to the gradient of the potential energy, there is no force acting on the - 5 - electron because the gradient is zero. The electron is said to be in a force free region. Let the potential energy be taken as zero. A particle moving in force-free region has a constant momentum and is therefore represented by a plane wave as discussed earlier. The wave-function for the particle is therefore given by ( ) ( , ) = = = ( ) (1.4) 푖�푘�⃑ ∙푟⃑− 휔 푡� 푖 푘�⃑ ∙푟⃑ −푖휔푡 −푖휔푡 where 훹 푟⃑ 푡 퐴 푒 퐴 푒 푒 휓 푟⃑ 푒 ( ) ( ) = �⃑ ( ) 푖 푘 ∙푟⃑ can be written as, in rectangular coordinates,휓 푟⃑ 퐴 푒 흍 풓� ⃑ ( ) = ( , , ) = 푖푘푥푥 푖푘푦푦 푖푘푧푧 where 휓 푟⃑ 휓 푥 푦 푧 퐴푥 푒 퐴푦 푒 퐴푧 푒 = 퐴 퐴푥퐴푦퐴푧 Figure (1.2): Electron in a box We can also write ( , , ) = ( ) ( ) ( ) (1.5) Where 휓 푥 푦 푧 휓푥 푥 휓푦 푦 휓푧 푧 ( ) = (1.6) 푖푘푥푥 휓푥 (푥) = 퐴푥 푒 (1.7) 푖푘푦푦 휓푦 (푦) = 퐴푦 푒 (1.8) 푖푘푧푧 휓푧 푧 퐴푧 푒 - 6 - The boundary condition on the wave-function is that the wave-function is the same in amplitude and phase when is incremented by , is incremented by and is incremented by i.e.