2 Solid-State Physics for Electronics 1. 2. Quantum mechanics: some basics 1. 2. 1. The wave equation in solids: from Maxwell,'s to Schrodinger's equation via In the theory of wave-particle duality, Louis de broglie associated the wavelength() with the mass(m) of a body, by makin 入 h For its part, the wave propagation equation for a vacuum(here the solid is thought of as electrons and ions swimming in a vacuum) is written as: i d2s If the wave is monochromatic. as in s=A(x,y, z)e o =A(x,y, =)e then As =AAe or and 33=-02Ae o(without modifying the result we can interchange a wave with form s =A(x, y, =k/ot =A(x, y, )e'2rv ). By introducing ?=2T-(length of a wave in a vacuum), wave propagation equation [1.2] can be written as △4+ A=0
2 Solid-State Physics for Electronics 1.2. Quantum mechanics: some basics 1.2.1. The wave equation in solids: from Maxwell’s to Schrödinger’s equation via the de Broglie hypothesis In the theory of wave-particle duality, Louis de Broglie associated the wavelength (O) with the mass (m) of a body, by making: . h mv O [1.1] For its part, the wave propagation equation for a vacuum (here the solid is thought of as electrons and ions swimming in a vacuum) is written as: 1 ² 0. ² ² s s c t w ' w [1.2] If the wave is monochromatic, as in: 2 (, ,) (, ,) i t i t s Ax yze Ax yze Z SQ then 's = 'A i t e Z and ² ² ² s t i t Ae w w Z Z (without modifying the result we can interchange a wave with form 2 ( , , ) ( , , ) ). i t i t s Ax yze Ax yze Z SQ By introducing 2 c Z O S (length of a wave in a vacuum), wave propagation equation [1.2] can be written as: ² 0 ² A A c Z ' [1.3] R 4 ² 0. ² A A S ' O [1.3’]
A particle(an electron for example) with mass denoted m, placed into a time- dependent potential energy V(x,y, =) has an energy my2+k (in common with a wide number of texts on quantum mechanics and solid-state physics, this book will inaccurately call potential the "potential energy"-to be The speed of the particle is thus given by 2(E 14 The de broglie wave for a frequency v= can be represented by the function p(which replaces the s in equation [1.2D -i 2TTVI e /(r Accepting with Schrodinger that the function y(amplitude of p) can be used in an analogous way to that shown in equation [1.3'l, we can use equations [1. 1] and [1. 4] with the wavelength written as 入= [.6] 2m(e-I so that W+=(E-)=0 [1.7 This is the Schrodinger equation that can be used wit ith crystals(where vis periodic) to give well defined solutions for the energy of electrons. As we shall see
Representations of Electron-Lattice Bonds 3 A particle (an electron for example) with mass denoted m, placed into a timeindependent potential energy V(x, y, z), has an energy: 1 v² 2 E mV (in common with a wide number of texts on quantum mechanics and solid-state physics, this book will inaccurately call potential the “potential energy” – to be denoted V ). The speed of the particle is thus given by 2 v . E V m [1.4] The de Broglie wave for a frequency E h Q can be represented by the function < (which replaces the s in equation [1.2]): 2 2 ȥ ȥ . E E it it h i t i t e e ee S SQ Z < \ \ = [1.5] Accepting with Schrödinger that the function \ (amplitude of < ) can be used in an analogous way to that shown in equation [1.3’], we can use equations [1.1] and [1.4] with the wavelength written as: , v 2 h h m mE V O [1.6] so that: 2 ȥ ( )ȥ 0. ² m ' E V = [1.7] This is the Schrödinger equation that can be used with crystals (where V is periodic) to give well defined solutions for the energy of electrons. As we shall see
4 Solid-State Physics for Electronics these solutions arise as permitted bands, otherwise termed valence and conduction bands, and forbidden bands(or"gaps" in semiconductors) by electronics specialists 1. 2. Form of progressive and stationary wave functions for an electron with known energy(E In general terms, the form(and a point defined by a vector r) of a wave unction for an electron of known energy(E)is given by y(, 1)=y()e /o=y(e where v(r) is the wave function at amplitudes which are in accordance with Schrodingers equation [1.71 if the resultant wave P(r, t) is a stationary wave, then v(r) is real; if the resultant wave P(r, t)is progressive, then y(r) takes on the form v()=f(B/k' where f()is a real function, and k=mi is the wave vector 1. 2.3. Important properties of linear operators 1. 1. f the hwo(linear) operators H and T are commutative, the proper functions of one can also be used as the proper functions of the other For the sake of simplicity, non-degenerate states are used. For a proper function u of H corresponding to the proper non-degenerate value(a), we find that = Multiplying the left-hand side of the equation by T'gives TH V=TaY =oTy
4 Solid-State Physics for Electronics these solutions arise as permitted bands, otherwise termed valence and conduction bands, and forbidden bands (or “gaps” in semiconductors) by electronics specialists. 1.2.2. Form of progressive and stationary wave functions for an electron with known energy (E) In general terms, the form (and a point defined by a vector r) G of a wave function for an electron of known energy (E) is given by: (,) () () , E j t j t rt re re Z < \ \ = GG G where \( ) r G is the wave function at amplitudes which are in accordance with Schrödinger’s equation [1.7]: – if the resultant wave <(,) r t is a stationary wave, then \( ) r G is real; – if the resultant wave <(,) r t G is progressive, then \( ) r G takes on the form . () () jk r \ r f re G G G G where f ( ) r G is a real function, and 2 k uS O G G is the wave vector. 1.2.3. Important properties of linear operators 1.2.3.1. If the two (linear) operators H and T are commutative, the proper functions of one can also be used as the proper functions of the other For the sake of simplicity, non-degenerate states are used. For a proper function \ of H corresponding to the proper non-degenerate value (D), we find that: H \ D\ Multiplying the left-hand side of the equation by T gives: TH T T \ D\ D \
As H T=0, we can write HTY =aTy This equation shows that Tw is a proper function of H with the proper value a. Hypothetically, this proper value is non-degenerate. Therefore, comparing the latter equation with the former(Hu=av, indicating that w is a proper function of H for the same proper value a), we now find that Ty and y are collinear. This is written as T This equation in fact signifies that y is a proper function of T with the proper value being the coefficient of collinearity(O)(QED) 1.2.3.2. f the operator H remains invariant when subject to a transformation using coordinates), then this operator H commutes with operator) associated with the transformation Here are the respective initial and final states(with initial on the left and final to he right) energy:E→E Hamiltonian: H-TH=H=H(invariance of H under effect ofT) wave function:y→Ty=y Similarly, the application of the operator T to the quantity H v, with H being invariant under Ts effect, gives: Hv→T(Hy)=H'y H HT
Representations of Electron-Lattice Bonds 5 As >H T, 0, @ we can write: HT T \ D \. This equation shows that T \ is a proper function of H with the proper value D. Hypothetically, this proper value is non-degenerate. Therefore, comparing the latter equation with the former ( , H \ D\ indicating that \ is a proper function of H for the same proper value D), we now find that T \ and \ are collinear. This is written as: T t \ \. This equation in fact signifies that \ is a proper function of T with the proper value being the coefficient of collinearity (t) (QED). 1.2.3.2. If the operator H remains invariant when subject to a transformation using coordinates (T), then this operator H commutes with operator (T) associated with the transformation Here are the respective initial and final states (with initial on the left and final to the right): energy: ' Hamiltonian: = ' = (invariance of under effect of ) wave function: = '. T T T E E H TH H H H T T o o \o \ \ Similarly, the application of the operator T to the quantity H \, with H being invariant under T’s effect, gives: ' = ' = ( ) = ' ' = ' = . T H H T H T H H H HT \ \ \o \ \ \ \
6 Solid-State Physics for Electronics We thus find TH Y=HTY from which. IH, T=0 QED 1. 2.3.3. The consequence e If the operator H is invariant to the effect of the operator T, then the proper tions of T can be used as the proper functions of H. 1.3. Bonds in solids: a free electron as the zero order approximation for a weak bond; and strong bonds 1.3.1. The free electron: approximation to the zero order The electric conduction properties of metals historically could have been derived from the most basic of theories that of free electrons This would assume that the conduction(or free)electrons move within a flat-bottomed potential well. In this model, the electrons are simply imprisoned in a potential well with walls that coincide with the limits of the solid. The potential is zero between the infinitely high walls. This problem is studied in detail in Chapter 2 with the introduction of the state density function that is commonly used in solid-state electronics. In three dimensions, the problem is treated as a potential box In order to take the electronic properties of semiconductors and tors into account (where the electrons are no longer free), and indeed understanding of metals, the use of more elaborate models is required. The finer interactions of electrons with nuclei situated at nodes throughout the solid are brought into play so that the wells flat bottom(where V=v=0) is perturbed or even strongly modified by the generated potentials. In a crystalline solid where the atoms are spread periodically in certain directions, the potential is also periodic and has a depth which depends on the nature of the solid. Two approaches can be considered, depending on the nature of the bonds. If the well depth is small (weak bond) then a treatment of the initial problem(free electron) using perturbation theory is possible. If the wells are quite deep,for example as in a covalent crystal with electrons tied to given atoms through strong
6 Solid-State Physics for Electronics We thus find: TH HT \ \, from which: >H T, 0 @ QED. 1.2.3.3. The consequence If the operator H is invariant to the effect of the operator T, then the proper functions of T can be used as the proper functions of H. 1.3. Bonds in solids: a free electron as the zero order approximation for a weak bond; and strong bonds 1.3.1. The free electron: approximation to the zero order The electric conduction properties of metals historically could have been derived from the most basic of theories, that of free electrons. This would assume that the conduction (or free) electrons move within a flat-bottomed potential well. In this model, the electrons are simply imprisoned in a potential well with walls that coincide with the limits of the solid. The potential is zero between the infinitely high walls. This problem is studied in detail in Chapter 2 with the introduction of the state density function that is commonly used in solid-state electronics. In three dimensions, the problem is treated as a potential box. In order to take the electronic properties of semiconductors and insulators into account (where the electrons are no longer free), and indeed improve the understanding of metals, the use of more elaborate models is required. The finer interactions of electrons with nuclei situated at nodes throughout the solid are brought into play so that the well’s flat bottom (where V = V0 = 0) is perturbed or even strongly modified by the generated potentials. In a crystalline solid where the atoms are spread periodically in certain directions, the potential is also periodic and has a depth which depends on the nature of the solid. Two approaches can be considered, depending on the nature of the bonds. If the well depth is small (weak bond) then a treatment of the initial problem (free electron) using perturbation theory is possible. If the wells are quite deep, for example as in a covalent crystal with electrons tied to given atoms through strong