Appendix A Fundamental Constants Table A.1 Fundamental constants Quantity & symbol CGS MKS (SI) Speed of light in vacuum c 2.997925 1010 cm sec−1 108 ms−1 −7 −6 −1 Magnetic constant μ0 = 4π × 10 1.25664 10 Hm (permeability of free space) 2 −12 −1 Electric constant 0 = 1/μ0c 8.8541853 10 Fm (permittivity of free space) Electron charge e 1.60219 10−19 C 4.80324 10−10 esu Electron volt eV 1.60219 10−12 erg eV−1 10−19 JeV−1 −28 −31 Electron rest mass m0 9.10956 10 gm 10 kg Planck’s constant h 6.626196 10−27 erg sec 10−34 Js Planck’s constant = h/2π 1.05459 10−27 erg sec 10−34 Js 2 2 −8 −10 Bohr radius a0 = 4π0 /m0e 0.529177 10 cm 10 m = 2 2 Rydberg constant Ry /2m0a0 13.6058 eV eV Fine structure constant 7.297351 10−3 10−3 2 α = e /20hc 23 −1 23 −1 Avogadro’s constant L,NA 6.022169 10 mol 10 mol 25 −3 Loschmidt’s constant NL 2.68719 10 m −16 −1 −23 −1 Boltzmann’s constant kB = R/L 1.380622 10 erg K 10 JK Faraday constant F = Le 9.64867 104 Cmol−1 7 −1 −1 −1 −1 Universal gas constant R = LkB 8.31435 10 erg K mol JK mol Mechanical equivalent of heat 4.184 107 erg cal−1 J cal−1 −21 −1 −24 −1 Bohr magneton μB = e/2m0c 9.2741 10 erg G 10 JT −24 −27 Proton rest mass mp 1.67251 10 gm 10 kg −24 −1 −27 −1 Nuclear magneton μN = e/2mpc 5.0508 10 erg G 10 JT −24 −27 Neutron rest mass mn 1.67492 10 gm 10 kg Stefan-Boltzmann constant 5.6697 10−8 Wm−2 K−4 = 5 4 3 2 σ 2π kB /15h c Y. Fu, Physical Models of Semiconductor Quantum Devices, 381 DOI 10.1007/978-94-007-7174-1, © Springer Science+Business Media Dordrecht 2014 382 A Fundamental Constants Table A.1 (Continued) Quantity & symbol CGS MKS (SI) Gravitational constant G 6.6732 10−11 Nm2 kg−2 −2 Acceleration of free fall gn 9.80665 m s 1.0 [eV]=2.41796 × 1014 [Hz]=8.0655 × 103 wavenumber [cm−1]= 1.1604 × 104 [K]=1.239855 wavelength [µm] W: watt, G: gauss, T: tesla, N: newton, C: coulomb Appendix B Quantum Physics B.1 Black Body Radiation Planck’s law for the energy density distribution for the radiation from a black body at temperature T is 8πhf3 1 w(f,T ) = (B.1) c3 ehf / k B T − 1 where h is the Planck’s constant, and f is the frequency. Note that ω = 2πf is the angular frequency, and = h/2π. The low-frequency Rayleigh-Jens law 8πf 2k T w(f,T ) = B c3 is obtained when hf kB T . Stefan-Boltzmann’s law for the total radiation energy per unit volume can be derived ˆ ∞ 8π 5k4 4σ W(T)= w(f,T ) f = B T 4 = T 4 d 3 3 (B.2) 0 15h c c = 5 4 3 2 where σ 2π kB /15h c is the Stefan-Boltzmann constant. Wien’s law for the wavelength λmax at which the energy density has its maximum value can be derived from Eq. (B.1) b λ = (B.3) max T where b = 0.2898 cm·K is a universal constant. Note λ = c/f . B.2 The Compton Effect A photon with initial wavelength λ incident upon an electron at rest with a rest mass m0. After the collision, the photon has a wavelength of λ scattered into a direction Y. Fu, Physical Models of Semiconductor Quantum Devices, 383 DOI 10.1007/978-94-007-7174-1, © Springer Science+Business Media Dordrecht 2014 384 B Quantum Physics at an angle θ with respect to its initial propagation direction, which is also referred to be the photon scattering angle. h λ = λ + (1 − cos θ)= λ + λC(1 − cos θ) (B.4) m0c where λC = h/m0c is called the Compton wavelength of the electron. B.3 Electron Diffraction Diffraction pattern of electron from a crystal can be explained by the dual particle- wave nature of matter. A particle having a momentum p is associated the so-called de Broglie wavelength h λ = (B.5) p B.4 Operators in Quantum Physics In quantum mechanics, classical physical quantities are translated into quantum op- erators. The measurement of a physical quantity A is described quantum mechan- ically by ˆ ˆ = ∗ = ∗ A ψ1 Aψ2dr (Aψ1) ψ2dr (B.6) where ψ is the wave function that describes the quantum mechanical state of the system under measurement. The only possible values that can be obtained when measuring a physical quantity A are the eigen values of the quantum operator A. Thus, the operator is hermitian and its eigen values are real, i.e., Aun = anun, where un is the eigen function, an is the eigen value which is real. Subscript n denotes the index of the eigen functions, i.e., operator A may have multiple eigen functions and eigen values. B.5 The Schrödinger Equation The momentum operator is given by p =−i∇, and the position operator is r = i∇p. The energy operator is ∂ E = i ∂t B.5 The Schrödinger Equation 385 The Hamiltonian of a particle with mass m and potential energy V is given by p2 H = + V 2m From the eigen equation Hψ = Eψ follows the time-dependent Schrödinger equa- tion 2 ∂ψ(r,t) − ∇2ψ(r,t)+ Vψ(r,t)= i (B.7) 2m ∂t where ψ(r,t), which is a function of space r and time t, is the wave function that describes the quantum mechanical state of the particle. The wave character of a par- ticle is described by ψ(r,t), and can be interpreted as a measure for the probability ρ(r,t)to find the particle at r and t ρ(r,t)= ψ(r,t)2 (B.8) The normalizing condition for the wave function is ˆ ∗ ψ|ψ= ψ ψdr = 1 At steady state, ∂ψ(r,t) i = Eψ(r,t) (B.9) ∂t so that − ψ(r,t)= ψ(r)e iEt/ (B.10) we have the time-independent Schrödinger equation 2 − ∇2ψ(r) + Vψ(r,t)= Eψ(r) (B.11) 2m for ψ(r). The probability flus s is given by ∗ ∗ s(r,t)= ψ (r,t)∇ψ(r,t)− ψ(r,t)∇ψ (r,t) (B.12) 2im which is related to the current density of the particle j = qs.Hereq is the electric charge of the particle (for electron q =−e). The following conservation law holds ∂ρ(r,t) =−∇·s(r,t) (B.13) ∂t 386 B Quantum Physics The time dependence of an operator A is given by (Heisenberg): dA ∂A [A,H] = + (B.14) dt ∂t i where H is the Hamiltonian and [A,B]≡AB − BA is the commutator of A and B. For hermitian operators the commutator is always complex. If [A,B]=0, the oper- ators A and B have a common set of eigen functions. By applying this to p and r it follows (Ehrenfest) d2r m t =−∇V (B.15) dt2 which is the classical Newton’s second law of motion. 1 + A classical product AB becomes 2 (AB BA) in quantum mechanics. B.6 The Uncertainty Principle The uncertainty ΔA in A is defined as (ΔA)2 = ψA −A2ψ = A2 −A2 it follows 1 ΔA · ΔB ≥ ψ[A,B]ψ (B.16) 2 from which it follows: ΔE · Δt ≥ ,Δpx · Δx ≥ (B.17) 2 2 B.7 Parity The parity operator in one dimension is given by Pψ(x) = ψ(−x).Ifthewave function is split into even and odd functions, it can be expanded into eigen functions of P: 1 1 ψ(x)= ψ(x)+ ψ(−x) + ψ(x)− ψ(−x) (B.18) 2 2 + − even: ψ odd: ψ with + 1 − 1 ψ = (1 + P)ψ(x), ψ = (1 − P)ψ(x) 2 2 both of which satisfy the Schrödinger equation. Hence, parity is a conserved quan- tity. Moreover, [P,H]=0. B.8 The Tunneling Effect 387 B.8 The Tunneling Effect The wave function of a particle in a one-dimensional infinitely high potential step from x = 0tox = a is given by 2 ψ (x) = sin(k x) (B.19) n a n where kn = nπ/a, n is an integer. The energy levels are given by n22π En = 2ma2 where m is the mass of the particle. If the wave function with energy E meets a potential barrier of W , while E<W, the wave function will, unlike the classical case, be non-zero within the potential barrier. Consider a one-dimensional system along the x axis. Denote the potential barrier as ⎧ ⎨0forx<0 = ≤ V(x) ⎩W for 0 x<a (B.20) 0forx ≥ a For an wave Aeikx running into the potential barrier from x =−∞, we express the total wave function as ⎧ ⎨Aeikx + Be−ikx for x<0 = iqx + −iqx ≤ ψ(x) ⎩Ce De for 0 x<a (B.21) Aeikx for x ≥ a with q2 = 2m(E − W)/2 and k2 = 2mE/2. Boundary conditions require that ψ(x) and ∂ψ(x)/∂x continuous at x = 0 and x = a, which give B, C and D and A in terms of A. The transmission coefficient T of the incident wave Aeikx through the potential barrier is defined by 2 2 2 2 |A | (1 + k/q) − (1 − k/q) T = = (B.22) |A|2 (1 + k/q)2e−iqa − (1 − k/q)2eiqa which is not zero even when E<W.