ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First - PowerPoint PPT Presentation

ICTP/Psi-k/CECAM School on Electron-Phonon Physics from First Principles Trieste, 19-23 March 2018

Lecture Wed.2 Introduction to the Boltzmann transport equation Samuel Ponc´ e Department of Materials, University of Oxford Ponc´ e, Lecture Wed.2 02/33

Lecture Summary • Carrier transport • Quantum Boltzmann equation • Boltzmann transport equation • Self-energy relaxation time approximation • Lowest-order variational approximation • Ionized impurity scattering Ponc´ e, Lecture Wed.2 03/33

Carrier transport: experimental evidences • Lattice scattering • Impurity scattering • Ionized impurity scattering Ponc´ e, Lecture Wed.2 04/33

Carrier transport: experimental evidences Figure from S. M. Sze, Physics of Semiconductor Device , Wiley (2007) Ponc´ e, Lecture Wed.2 05/33

Carrier transport: experimental evidences ( Lecture Thu.2) Figure from S. M. Sze, Physics of Semiconductor Device , Wiley (2007) Ponc´ e, Lecture Wed.2 06/33

Carrier transport: experimental evidences 3 ) 10 18 Carrier concentration ( cm 10 16 10 14 Putley and Mitchell Ludwing and Watters 10 12 Morin and Maita DFT 1.0 1.5 2.0 2.5 1000/T ( K 1 ) Ponc´ e, Lecture Wed.2 07/33

Carrier transport Calculated evolution of the Fermi level of Si as a function of temperature and impurity concentration. 0.8 Conduction band 0.6 0.4 10 12 4 × 10 13 1 . 3 × 10 17 0.2 E F − E i (eV) n-type 0.0 p-type -0.2 10 12 10 14 2 × 10 17 -0.4 -0.6 Valence band -0.8 0 100 200 300 400 500 600 700 T (K) Ponc´ e, Lecture Wed.2 08/33

Carrier transport: experimental evidences Figure from S. M. Sze, Physics of Semiconductor Device , Wiley (2007) Ponc´ e, Lecture Wed.2 09/33

Carrier transport: experimental evidences Figure from S. M. Sze, Physics of Semiconductor Device , Wiley (2007) Ponc´ e, Lecture Wed.2 10/33

Quantum Boltzmann equation • Most general transport theory that describes the evolution of the particles distribution function f ( k , ω, r , t ) = − iG < ( k , ω, r , t ) , where G < is the FT of the lesser Green’s function G < ( r , t, R , T ) = i � ψ † ( R − 0 . 5 r , T − 0 . 5 t ) ψ ( R + 0 . 5 r , T + 0 . 5 t ) � with ( R , T ) for the center of mass. • Finding G < requires to solve a complex set of 2x2 matrix Green’s function [non-equilibrium Keldysh formalism] • Involves G ret that describes the dissipation of the system • Valid for out of equilibrium systems G. D. Mahan, Many-Particle Physics , Springer, 2000 Ponc´ e, Lecture Wed.2 11/33

Gradient expansion approximation Assumes • Homogeneous system ( ∇ r = 0 ) • In steady state ( ∇ t = 0 ) energy distribution el-ph self-energies A ( k , ω ) 2 ∂n F ∂ω E · { ( v k + ∇ k Re [Σ ret ])Γ + σ ∇ k Γ } = Σ > G < − Σ < G > 2Γ Γ = − Im Σ ret , σ = ω − ε k − Re Σ ret A = σ 2 + Γ 2 , L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics , Benjamin, 1962 Ponc´ e, Lecture Wed.2 12/33

Electric current • The steady-state electric current J is related to the driving electric field E via the mobility tensors µ as: J α = e ( n e µ e ,αβ + n h µ h ,αβ ) E β � = − e Ω − 1 � Ω − 1 d k f n k v n k ,α BZ n where v n k ,α = � − 1 ∂ε n k /∂k α is the band velocity. • We need to find the occupation function f n k which reduces to the Fermi-Dirac distribution f 0 n k in the absence of the electric field Ponc´ e, Lecture Wed.2 13/33

Mobility • Experimentalists prefers to measure mobility as it is independent of the carrier concentration n µ e ,αβ = σ αβ = 1 ∂J α n e n e ∂E β � � � � � d k f 0 = − d k v n k ,α ∂ E β f n k n k . n ∈ CB n ∈ CB (similar expression for hole mobility) • We need to evaluate the linear response of the distribution function f n k to the electric field E . Ponc´ e, Lecture Wed.2 14/33

Boltzmann transport equation (BTE) Electron can be treated as classical particle but electron scattering is the result of short-range forces and must be treated quantum mechanically. The BTE is a semi-classical treatment which • describes carrier dynamics using Newton’s law without treating explicitly the crystal potential. The influence of the crystal potential is treated indirectly through the electronic bandstructure (= effective masses). • carrier scattering is treated quantum mechanically. M. Lundstrom, Fundamentals of Carrier Transport , Cambridge (2000) Ponc´ e, Lecture Wed.2 15/33

Boltzmann transport equation Like in QBE, we start from the carrier distribution function f ( k , ω, r , t ) . At equilibrium d f/dt = 0 the change of the distribution function is given by the Boltzmann equation: � d dt = ∂f f ∂t + v · ∂f ∂ r + ∂ k ∂t · ∂f ∂ k + ∂T ∂t · ∂f ∂T + ∂f � = 0 � ∂t � scatt Approximations: G. D. Mahan, Many-Particle Physics , Springer, 2000 Ponc´ e, Lecture Wed.2 16/33

Boltzmann transport equation Like in QBE, we start from the carrier distribution function f ( k , ω, r , t ) . At equilibrium d f/dt = 0 the change of the distribution function is given by the Boltzmann equation: ✓ � d dt = ∂f f ∂f ∂ r + ∂ k ∂t · ∂f ∂ k + ∂T ∂t · ∂f ∂T + ∂f ✓ � ∂t + v · = 0 � ✓ ∂t � scatt Approximations: • Homogeneous field (independent of r ) G. D. Mahan, Many-Particle Physics , Springer, 2000 Ponc´ e, Lecture Wed.2 16/33

Boltzmann transport equation Like in QBE, we start from the carrier distribution function f ( k , ω, r , t ) . At equilibrium d f/dt = 0 the change of the distribution function is given by the Boltzmann equation: ✓ ✓ � d dt = ∂f f ∂f ∂ r + ∂ k ∂t · ∂f ∂ k + ∂T ∂T + ∂f ∂f ✓ ✓ � ∂t + v · ∂t · = 0 � ✓ ✓ ∂t � scatt Approximations: • Homogeneous field (independent of r ) • Constant temperature G. D. Mahan, Many-Particle Physics , Springer, 2000 Ponc´ e, Lecture Wed.2 16/33

Boltzmann transport equation Like in QBE, we start from the carrier distribution function f ( k , ω, r , t ) . At equilibrium d f/dt = 0 the change of the distribution function is given by the Boltzmann equation: ✓ ✓ ✓ � d f ∂f ∂f ∂ r + ∂ k ∂t · ∂f ∂ k + ∂T ∂f ∂T + ∂f ✓ ✓ ✓ � dt = ∂t + v · ∂t · = 0 � ✓ ✓ ✓ ∂t � scatt Approximations: • Homogeneous field (independent of r ) • Constant temperature • DC conductivity G. D. Mahan, Many-Particle Physics , Springer, 2000 Ponc´ e, Lecture Wed.2 16/33

Boltzmann transport equation Like in QBE, we start from the carrier distribution function f ( k , ω, r , t ) . At equilibrium d f/dt = 0 the change of the distribution function is given by the Boltzmann equation: ✓ ✓ ✓ � d f ∂f ∂f ∂ r + ∂ k ∂t · ∂f ∂ k + ∂T ∂T + ∂f ∂f ✓ ✓ ✓ � dt = ∂t + v · ∂t · = 0 � ✓ ✓ ✓ ∂t � scatt Approximations: • Homogeneous field (independent of r ) • Constant temperature • DC conductivity • No magnetic field ∂ k 1 ∂t = − ( − e ) E − 137 v × H G. D. Mahan, Many-Particle Physics , Springer, 2000 Ponc´ e, Lecture Wed.2 16/33

Boltzmann transport equation Like in QBE, we start from the carrier distribution function f ( k , ω, r , t ) . At equilibrium d f/dt = 0 the change of the distribution function is given by the Boltzmann equation: � ∂f n k ( T ) = ( − e ) E · ∂f n k ( T ) � Quantum → ← Semi-classical � ∂t ∂ k � scatt Approximations: • Homogeneous field (independent of r ) • Constant temperature • DC conductivity • No magnetic field ∂ k 1 ∂t = − ( − e ) E − 137 v × H G. D. Mahan, Many-Particle Physics , Springer, 2000 Ponc´ e, Lecture Wed.2 16/33

Linearized Boltzmann transport equation � ∂f n k ( T ) = ( − e ) E · ∂f n k ( T ) � Quantum → ← Semi-classical � ∂t ∂ k � scatt If E is small, f n k can be expanded into f n k = f 0 n k + O ( E ) . Keeping only the linear term in E becomes ∂f 0 ( − e ) E · ∂f n k ( T ) n k = ( − e ) E · v n k ∂ k ∂ε n k This is the collisionless term of Boltzmann’s equation for a uniform and constant electric field, in the absence of temperature gradients and magnetic fields Ponc´ e, Lecture Wed.2 17/33

Linearized Boltzmann transport equation ∂f 0 � ∂f n k ( T ) n k � = ( − e ) E · v n k � ∂t ∂ε n k � scatt This is the modification of the distribution function arising from electron-phonon scattering in and out of the state | n k � , via emission or absorption of phonons with frequency ω q ν Ponc´ e, Lecture Wed.2 18/33