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

1.0 1.5 2.0 2.5

1000/T (K

1)

1012 1014 1016 1018

Carrier concentration (cm

3) Putley and Mitchell Ludwing and Watters Morin and Maita DFT

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.

100 200 300 400 500 600 700 T (K)

• 0.8
• 0.6
• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 EF − Ei (eV)

Valence band Conduction band 1014 4 × 1013 2 × 1017 1.3 × 1017 1012 1012 n-type p-type 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) = iψ†(R − 0.5r, T − 0.5t)ψ(R + 0.5r, T + 0.5t) 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 Gret 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 ∂nF ∂ω E · {(vk + ∇kRe[Σret])Γ + σ∇kΓ} = Σ>G< − Σ<G> A = 2Γ σ2 + Γ2 , Γ = −ImΣret, σ = ω − εk − ReΣret

• 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: Jα = e (ne µe,αβ + nh µh,αβ)Eβ = −e Ω−1

n

Ω−1

BZ

• dk fnk vnk,α

where vnk,α = −1∂εnk/∂kα is the band velocity.

• We need to find the occupation function fnk which reduces to

the Fermi-Dirac distribution f0

nk 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 µe,αβ = σαβ ne = 1 ne ∂Jα ∂Eβ = −

• n∈CB
• dk vnk,α ∂Eβfnk

n∈CB

• dk f0

nk.

(similar expression for hole mobility)

• We need to evaluate the linear response of the distribution

function fnk 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 f dt = ∂f ∂t + v · ∂f ∂r + ∂k ∂t · ∂f ∂k + ∂T ∂t · ∂f ∂T + ∂f ∂t

• scatt

= 0 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 f dt = ∂f ∂t + v ·

✓ ✓ ✓

∂f ∂r + ∂k ∂t · ∂f ∂k + ∂T ∂t · ∂f ∂T + ∂f ∂t

• scatt

= 0 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 f dt = ∂f ∂t + v ·

✓ ✓ ✓

∂f ∂r + ∂k ∂t · ∂f ∂k + ∂T ∂t ·

✓ ✓ ✓

∂f ∂T + ∂f ∂t

• scatt

= 0 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 dt =

✓ ✓ ✓

∂f ∂t + v ·

✓ ✓ ✓

∂f ∂r + ∂k ∂t · ∂f ∂k + ∂T ∂t ·

✓ ✓ ✓

∂f ∂T + ∂f ∂t

• scatt

= 0 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 dt =

✓ ✓ ✓

∂f ∂t + v ·

✓ ✓ ✓

∂f ∂r + ∂k ∂t · ∂f ∂k + ∂T ∂t ·

✓ ✓ ✓

∂f ∂T + ∂f ∂t

• scatt

= 0 Approximations:

• Homogeneous field (independent of r)
• Constant temperature
• DC conductivity
• No magnetic field ∂k

∂t = −(−e)E − 1 137v × 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: ∂fnk(T) ∂t

• scatt

= (−e)E · ∂fnk(T) ∂k Quantum → ← Semi-classical Approximations:

• Homogeneous field (independent of r)
• Constant temperature
• DC conductivity
• No magnetic field ∂k

∂t = −(−e)E − 1 137v × H

• G. D. Mahan, Many-Particle Physics, Springer, 2000

Ponc´ e, Lecture Wed.2 16/33

Linearized Boltzmann transport equation

Quantum → ← Semi-classical ∂fnk(T) ∂t

• scatt

= (−e)E · ∂fnk(T) ∂k If E is small, fnk can be expanded into fnk = f0

nk + O(E). Keeping

• nly the linear term in E becomes

(−e)E · ∂fnk(T) ∂k = (−e)E · vnk ∂f0

nk

∂εnk 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

∂fnk(T) ∂t

• scatt

= (−e)E · vnk ∂f0

nk

∂εnk This is the modification of the distribution function arising from electron-phonon scattering in and out of the state |nk, via emission

• r absorption of phonons with frequency ωqν

Ponc´ e, Lecture Wed.2 18/33

Linearized Boltzmann transport equation

∂fnk(T) ∂t

• scatt

= (−e)E · vnk ∂f0

nk

∂εnk ∂f0

nk

∂εnk vnk · (−e)E = 2π

• mν

dq ΩBZ |gmnν(k, q)|2 ×

• (1 − fnk)fmk+qδ(εnk − εmk+q + ωqν)(1 + nqν)

+ (1 − fnk)fmk+qδ(εnk − εmk+q − ωqν)nqν − fnk(1 − fmk+q)δ(εnk − εmk+q − ωqν)(1 + nqν) − fnk(1 − fmk+q)δ(εnk − εmk+q + ωqν)nqν

• This is the modification of the distribution function arising from

electron-phonon scattering in and out of the state |nk, via emission

• r absorption of phonons with frequency ωqν
• G. Grimvall, The electron-phonon interaction in metals, North-Holland, 1981

Ponc´ e, Lecture Wed.2 19/33

The electron-phonon matrix element

gmnν(k, q) = umk+q|∆qνvSCF|unkuc ∆qνvSCF =

• καpe−iq·(r−Rp)
• 2Mκωqν

eκα,ν(q) ∂ VSCF(r) ∂τκαp

Lattice-periodic part of wavefunction Variation of the Kohn-Sham potential Zero-point amplitude Phonon polarization Displacement of a single ion Incommensurate modulation κ Atom in the unit cell α Cartesian direction p Unit cell in the equivalent supercell

(Lecture Tue.1)

Ponc´ e, Lecture Wed.2 20/33

Linearized Boltzmann transport equation

We take the derivatives of the Boltzmann equation with respect to E to obtain the iterative Botlzmann transport equation (IBTE): ∂Eβfnk =e∂f0

nk

∂εnk vnk,βτ 0

nk+ 2πτ 0 nk

• mν

dq ΩBZ |gmnν(k, q)|2 ×

• (1 + nqν − f0

nk)δ(εnk − εmk+q + ωqν)

+(nqν + f0

nk)δ(εnk − εmk+q − ωqν)

• ∂Eβfmk+q

having defined the relaxation time: 1 τ 0

nk

= 2ImΣFM

nk = 2π

• mν

dq ΩBZ |gmnν(k, q)|2 ×

• (1 − f0

mk+q + nqν)δ(εnk − εmk+q − ωqν)

+ (f0

mk+q + nqν)δ(εnk − εmk+q + ωqν)

• Ponc´

e, Lecture Wed.2 21/33

Self energy relaxation time approximation (SERTA)

We can approximate IBTE by neglecting ∂Eβfmk+q ∂Eβfnk =e∂f0

nk

∂εnk vnk,βτ 0

nk

The intrinsic electron mobility is therefore: µe,αβ = −

• n∈CB
• dk vnk,α ∂Eβfnk

n∈CB

• dk f0

nk

= −e ne Ω

• n∈CB
• dk

ΩBZ ∂f0

nk

∂εnk vnk,α vnk,β τ 0

nk

Ponc´ e, Lecture Wed.2 22/33

Intrinsic carrier mobility

Electron and hole mobility in silicon (EPW) 100 200 300 400 500 Temperature (K) 102 103 104 Mobility (cm2/Vs)

• S. Ponc´

e et al., Physical Review B, in press (2018) and can be found on arXiv:1803.05462

Ponc´ e, Lecture Wed.2 23/33

Intrinsic Si carrier mobility at 300K

250 500 750 1000 1250 1500 Mobility (cm2/Vs) electron hole PBE EXP SOC+FIT+SCR+IBTE 1305 PBE EXP SOC+GW+SCR+IBTE+RE 1366 PBE EXP SOC+GW+SCR+IBTE 1438 PBE EXP SOC+GW+SCR 1357 PBE EXP SOC+GW 1488 PBE EXP SOC 1419 PBE EXP No SOC 1457 LDA EXP No SOC 1463 PBE No SOC 1343 LDA No SOC 1555 PBE EXP SOC+FIT+SCR+IBTE 502 PBE EXP SOC+GW+SCR+IBTE+RE 658 PBE EXP SOC+GW+SCR+IBTE 693 PBE EXP SOC+GW+SCR 700 PBE EXP SOC+GW 799 PBE EXP SOC 820 PBE EXP No SOC 755 LDA EXP No SOC 739 PBE No SOC 722 LDA No SOC 743

• S. Ponc´

e et al., Physical Review B, in press (2018) and can be found on arXiv:1803.05462

Ponc´ e, Lecture Wed.2 24/33

Intrinsic carrier mobility

Electron mobility in GaAs using IBTE and SERTA (dashed)

T.-H. Liu et al., Phys. Rev. B 95, 075206 (2017)

Ponc´ e, Lecture Wed.2 25/33

Lowest-order variational approximation (LOVA)

From Eliashberg theory of phonon-driven superconductivity, Pinkski, Butler and Allen developed a framework based on this variational principle to compute electrical and thermal resistivities of metals. One can go from the BTE to the LOVA introducing energy integrals and using the following approximations:

• Isotropic relaxation time τ
• Assume the DOS at the Fermi level is slowly varying

δ(εnk − ε) ≈ δ(εnk − εF ) → valid for metals only !

• P. B. Allen, Phys. Rev. B 13, 1416 (1976)
• P. B. Allen, Phys. Rev. B 17, 3725 (1978)
• F. J. Pinski, P. B. Allen, and W. H. Butler, Phys. Rev. B 23, 5080 (1981)

Ponc´ e, Lecture Wed.2 26/33

Lowest-order variational approximation (LOVA)

Carrier resistivity: ρLOVA

αβ

= 2πΩBZkBT e2n(εF )vα(εF )vβ(εF ) ∞ dω ω (ω/2T)2α2

trF(ω)

sinh2(ω/2T) With the isotropic transport spectral function: α2

trF(ω) =

1 n(εF )v(εF )2

• nmν
• BZ

dkdq Ω2

BZ

|gmn,ν(k, q)|2

• vnk · vnk − vnk · vmk+q
• δ(εnk − εF )δ(εmk+q − εF )δ(ω − ωqν)
• P. B. Allen, Phys. Rev. B 13, 1416 (1976)
• P. B. Allen, Phys. Rev. B 17, 3725 (1978)
• F. J. Pinski, P. B. Allen, and W. H. Butler, Phys. Rev. B 23, 5080 (1981)

Ponc´ e, Lecture Wed.2 27/33

Lowest-order variational approximation (LOVA)

Resistivity of Pb with and without spin-orbit coupling

100 200 300 400 500 600

T (K)

10 20 30 40 50 ;(7+ cm)

Present work without SOC Present work with SOC Hellwege et al. Moore et al. Cook et al. Leadbetter et al.

10 20 0.5 1

• S. Ponc´

e et al., Comput. Phys. Commun. 209, 116 (2016)

Ponc´ e, Lecture Wed.2 28/33

Lowest-order variational approximation (LOVA)

Resistivity of Al with IBTE , SERTA (dashed line) and LOVA (dotted line)

• W. Li, Phys. Rev. B 92, 075405 (2015)

Ponc´ e, Lecture Wed.2 29/33

Brooks-Herring model for impurity scattering

Semi-empirical Brooks-Herring model for the hole of silicon: µi = 27/2ǫ2

s(kBT)3/2

π3/2e3m∗

d niG(b)

cm2 Vs

• ,

where G(b) = ln(b + 1) − b/(b + 1), b = 24πm∗

dǫs(kBT)2/e2h2n′,

and n′ = nh(2 − nh/ni). Here m∗

d = 0.55m0 is the silicon hole density-of-state effective mass.

• H. Brooks, Phys. Rev. 83, 879 (1951)
• S. S. Li and W. R. Thurber, Solid-State Electronics 20, 609 (1977)

Ponc´ e, Lecture Wed.2 30/33

Brooks-Herring model for impurity scattering

Because the electron mass is anisotropic in silicon, we used the Long-Norton model: µLN

i

= 7.3 · 1017T 3/2 niG(b) cm2 Vs

• ,

The mobility total phonon (µl) and impurity (µi) mobility is: µ = µl

• 1 + X2{ci(X) cos(X) + sin(X)(si(X) − π

2 )}

• X2 = 6µl/µi and ci(X) and si(X) are the cosine and sine integrals.
• P. Norton, T. Braggins, and H. Levinstein, Phys. Rev. B 8, 5632 (1973)

Ponc´ e, Lecture Wed.2 31/33

Ionized impurity scattering

Electron and hole mobility in silicon (EPW) 1015 1016 1017 1018 1019 Impurity concentration (cm−3) 400 800 1200 1600 Mobility (cm2/Vs)

• S. Ponc´

e et al., Physical Review B, in press (2018) and can be found on arXiv:1803.05462

Ponc´ e, Lecture Wed.2 32/33

References: insightful books

• J. M. Ziman, Electrons and Phonons, Oxford (1960)
• L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, Benjamin

(1962)

• G. Grimvall, The electron-phonon interaction in metals, North-Holland

(1981)

• G. D. Mahan, Many-Particle Physics, Springer (2000)
• M. Lundstrom, Fundamentals of Carrier Transport, Cambridge (2000)
• S. M. Sze, Physics of Semiconductor Device, Wiley (2007)

Ponc´ e, Lecture Wed.2 33/33

Supplemental Slides

Ponc´ e, Lecture Wed.2 01/01