Transport Theory, Kinetic Theory - Notes | ESM 215, Study notes of Environmental Science

Download Transport Theory, Kinetic Theory - Notes | ESM 215 and more Study notes Environmental Science in PDF only on Docsity! ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 1 Transport Theory#1 Kinetic Theory Kinetic theory is a formalism for describing the motion of a population of “free” particles in a solid in response to an externally applied force, such as an electric field Key Assumptions: • All energy and momentum is transferred by the particles in motion. • In a statistical sense each particle is its own subsystem, i.e., the particles are fully distinguishable from each other, and are in quasi-equilibrium with temperature bath. • Collisions are instantaneous events that randomize particle motion. Features of Kinetic Theory: 1) Can define fluxes, each one being defined as the amount of some physical quantity crossing a unit area per unit time. • Particle flux: = ⋅ r r nJ n υ , where n is the particle density and v is the velocity. Both of these quantities are possibly functions of time and position. • Charge flux: = ⋅ ⋅ r r qJ n q υ (also called the electrical current) • Energy flux = ⋅ r r U 2J n [( 1 / 2 )mv ]υ , since all the energy is kinetic 2) Clearly, the particle velocity vector is very important in kinetic theory. In a first analysis, it is usually determined by classical mechanics. For example, in one dimension, we can solve for xυ from Newton’s law = = dvx x x dt F ma m where ax is the x component of acceleration. For an electron in a uniform electric field, the mechanical force is Fx = eEx. So, 0+= ⇒ = d eEx x x xdt m eE m υ υυ , where v0 is the initial velocity. 3) Clearly, a velocity increase with time cannot persist forever. It will be interrupted by collisions that can be modeled as a damping (scattering) term in Newton’s equation. + = md mx x eExdt υ υ τ (1) ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 2 This is a linear 1st order inhomogeneous differential equation with constant coefficients. The solution is the sum of a homogeneous solution (setting eEx = 0) and a particular solution. Taking the initial conditions once again as v(t=0) = v0, we get ( )0 ( ) 1t te Ee em τ ττυ υ − −= + − • The homogeneous solution is transient and vanishes for long time scales t >> τ. • The particular solution has a steady state term for long time scales of e E m τ υ → ≡ µE where µ is the mobility. • Using the kinetic flux for electric current, we get 2 = = ≡ ne τ E J nq σ Eq m υ (Ohm’s Law !) Where σ is called the electrical conductivity 2ne τ m ≡ . Transport conductivities usually connect a given transport flux to the spatial gradient of some macroscopic potential that arises from the non-equilibrium condition. The gradients vanish in equilibrium. For example: the charge flux is ( )= = − ∇ r rv J Eq σ σ φ , φ → electrostatic potential Similarly, the kinetic energy flux can be written = − ∇ r r J K TU where T is the temperature and K is the thermal conductivity. This makes sense because heat is just the macroscopic representation of kinetic energy at the microscopic level. Physical Interpretation of τ • From Newton’s equation, 1/τ is the rate at which the initial velocity changes to the steady- state value. This certainly makes sense for a continuum (“jellium”) model of the particles Homogeneous Particular ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 5 Mathematically, this can be written as PN(t+δt) = PN (1- δt/t) + PN-1(δt/t) where (1 –δt/t) is the probability that no collisions occur during δt, and δt /t is the probability for a single collision in this time. This equation can be re-arranged to yield 1( ) ( ) ( ) −+ − + =N N N NP t t P t P t P t δ δ τ τ which can be thought of as a recursion relation relating PN to PN-1. In the limit that δt → 0, the far-left time becomes a derivative and we get the inhomogeneous, first-order, ordinary differential equation 1( ) ( ) −+ =N N NdP t P t P dt τ τ (3) This is an important equation of probability theory and leads to the Poisson density function, as shown next. Derivation of Poisson statistics in kinetic collisions. Using the technique of an integrating factor and the initial condition PN(t = 0) = 0, one finds the following solution to the differential Eqn (3):ii 1 0 exp( / )( ) exp( '/ ) ( ')− − = ⋅∫ t N N tP t t P t dtτ τ τ (4) where t’ is a dummy variable of integration. Fortunately, we already know the solution to this for N = 0, i.e., from Eqn (2), P0(t) ≡ f(t) = exp(-t/τ). Substitution of this into Eqn (4) yields ii W.E. Boyce and R.C. DiPrima, “Elementary Differential Equations and Boundary Value Problems,” 2nd Ed., (Wiley, New York, 1969, Sec. 2.1. ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 6 1( ) exp( / )= − tP t t τ τ Substitution of this back into (4) and iterating yields 2 2 1( ) exp( / ) 2 ⎛ ⎞= −⎜ ⎟ ⎝ ⎠ tP t t τ τ By logical deduction, the solution for an arbitrary number N has the form ( / )( ) exp( / ) ! = − N N tP t t N τ τ This is the famous Poisson density function of probability theory. Like all bonafide pdfs, it approaches a Gaussian in the limit of large samples, in this case the limit of large N. This limiting behavior is well known in probability theory through the central limit theorem. It leads to the prediction that in a solid sample having a large number of charge carriers, one expects the collision rate to fluctuate about some mean value with Gaussian statistics. This can usually be associated with a Gaussian fluctuation in the electrical conductivity of the sample: a fundamental result of fluctuation theory known as the Johnson-Nyquist theorem. The resulting Gaussian fluctuations in the open-circuit voltage or short-circuit current through electrical contacts on the sample is a phenomena of paramount importance in solid-state electronics known as Johnson-Nyquist noise. AC Behavior of Carriers in Kinetic Theory To understand the response of charged particles in motion to time-varying external electric fields, we go back and re-write the Newton equation of motion in terms of the instantaneous position of each particle x. We do this because it allows us to develop expressions for the electrodynamic response in the same physical terms as for electrostatics, ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 7 namely in terms of a dipole moment or an electrical susceptibility. Since v ≡ dx/dt, we can re- write (1) as ( )/m x x eτ+ = Ε&& & (5) We solve this for the special case of sinusoidal time-varying field, for which we can apply the phasor form 0 −= j tE E e ω and write the position variable as 0 −= j tx x e ω . Substitution into (5) yields, 2( ) = − − j eE x m ω τω From the general (electrostatic) definition, the dipole moment is p = qx and 2 2( ) = − − j e E p m ω τω Again from electrostatics, the electric susceptibility is defined by χe ≡ P/(ε0E), where P is the macroscopic polarization given by P = np. Hence, we can write ( ) 2 2 0 e -ne χ m ω + = → ∞ jω τε (6) This has two interesting properties: First, it diverges to infinity as ω → 0 (electrostatic limit), which makes sense physically. Since the carriers are “free”, they undergo large displacement in the electric field, limited only by the size of the solid sample. In the present analysis, this size was not constrained, so the susceptibility should diverge ! Second, it is always negative, meaning that the free-electron response is truly “dia”-electric. When ωτ << 1, Eqn (6) can be re-written in a useful approximate form ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 10 ( )q Bυ × → rr Lorenzt force = e υ →⎛ ⎞− ×Β →⎜ ⎟ ⎝ ⎠ r for electron In general the Lorentz term complicates the solution greatly unless B r is uniform. Let’s assume 0 ˆ= r B B z , and write general form ˆ ˆ ˆ= + +r x y zx y zυ υ υ υ . So, 0 0ˆ ˆ → × = − + r x yB B y B xυ υ υ Matching cartesian components on both sides, we get 0 0 ˆ : ˆ : ˆ : + = + + = − + = x x x y y y y x z z z dv mx m qE q B dt dv m y m qE q B dt dv mz m qE dt υ υ τ υ υ τ υ τ Again, each equation is a 1st order linear, inhomogeneous differential equation having both a homogeneous and a particular solution: 0 0 0 0 0 ( )(1 ) ( )(1 ) (1 ) − − − − − − = + + − = + − − = + − t t x x x y t t y y y x t t z z z q ee E B e e m q ee E B e e m q ee E e e m τ τ τ τ ττ τυ υ υ τυ υ υ τυ υ The key result of the Lorentz force is that it couples components of velocity lying in the plane perpendicular to → B . In the steady state and for τ>>t . ( ) ( ) x x c y y y c x z z qE e qE e qE e υ µ ω τυ υ µ ω τυ ν µ → + → − ⎛ ⎞→ ⎜ ⎟ ⎝ ⎠ ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 11 where ωc ≡ eB0 /m is called the cyclotron frequency (a ubiquitous constant in magneto- transport problems, much like the Larmor frequency and Bohr magneton occur in magnetostatic problems). Suppose the lateral extent of a sample is finite along y and z axes, and an electric field xE is applied along x̂ . In steady state, υy and υz must be zero if charge carriers are confined to sample. This reasoning leads to: 0 0 0 = ⇒ = = ⇒ = z z y y c x E E υ υ µ ω τυ (6) and x x q E e υ µ⎛ ⎞= ⎜ ⎟ ⎝ ⎠ (as expected) Eqn (6) implies 0c y x c x x BqE E q E e m ω τ τυ ω τ µ = = = Linear in vx Linear in B The linear dependence on υx is consistent with experimental observation of linear dependence on xJ (electrical current density). By noting that x xJ nqυ= (kinetic theory), we get 0 0( )( ) x xy B J B Je mE q m q e nq nq τ τ ⎛ ⎞⎛ ⎞= =⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ; a famous result called the Hall effect. The ratio 0 1y x E B J nq = is called the Hall Coefficient for electrons ≡ RH RH → negative for electrons (or any negatively charged particles), and positive for holes (or any positively charged particles). Note: RH does not depend on mass (or effective mass). The Hall effect is very important historically and in modern technology as well. ECE215B/Materials206B Fundamentals of Solids for Electronics E.R. Brown/Spring 2008 12 Hall Effect Example (Hall Bar) We know for the geometry in Fig. 2 that if the external electric field is applied along the x axis and external magnetic field is applied along the z axis, then there will be a response electric field along y axis given by nq JBE x0y = where xJ is the current density and n is the mobile charge density. But if current is uniform cb I A IJ xxx ⋅ == . And if yE is uniform, b V E yy = , so we get nqbc IB b V xoy = or nqc IBV x0y = . b a c x y z Vy Ix + - Fig. 2. Hall bar geometry and interesting physical quantities.