Electron and Phonon Dispersion

Home , Phonon

ECE 656: Electronic Transport in Semiconductors Fall 2017

Mark Lundstrom

Electrical and Computer Engineering Purdue University, West Lafayette, IN USA

Lundstrom8/22/17 ECE-656 F17 Waves in general

! ! ! ! i(β⋅r−ωt) General wave: u(r,t) = A β e β = 2π λ ( ) ! ! ! ! ! i(β⋅r−ω(β )t) 3 U (r,t) = A β e d β ω ∫∫∫ ( )

Wave packet: β ≈ β0 υg = dω dβ U (x) group velocity β x

β0 Δβ 1 υ p = ω β Δβ Δx > Δx 2 2 phase velocity Lundstrom ECE-656 F17 outline

1) Electron dispersion (band structure)

2) Simple band structures

3) Phonon dispersion

4) Simple phonon dispersions

5) Electrons vs. phonons

3 Lundstrom ECE-656 F17 Electrons in atoms Hydrogen atom:

!2 " " " " ! q2 − ∇2ψ (r ) + U (r )ψ (r ) = Eψ (r ) U (r ) = − 2m 4πε r 0 0

U = 0 ! + U r ( ) ! 13.6 U (r ) En = − 2 eV n

4 Lundstrom ECE-656 F17 Electrons in free space

2        i t − ∇2ψ (r ) +U (r )ψ (r ) = Eψ (r ) Ψ(r,t) =ψ (r )e ω 2m 0 E = !ω ! U r = 0 free space, constant potential: ( )

2   2     − ∇ ψ (r ) = Eψ (r ) ψ (r ) = Aeikir k = 2m E  2m 0 0 ! ! β → k 2 2   ! " k momentum: p = k E k = ( ) 2m 0   2 position: ψ *(r )ψ (r ) = A

5 Lundstrom ECE-656 F17 Electrons in crystals

U = 0 U = 0 ! a U r C ( )

+ + + + + +

Crystals: 2 ! 2 " " " " ! ! ! − ∇ ψ (r ) + UC (r )ψ (r ) = Eψ (r ) UC (r + a) = UC (r ) 2m 0 “crystal potential” Lundstrom ECE-656 F17 6 Electrons in crystals

2  2        − ∇ ψ (r ) + UC (r )ψ (r ) = Eψ (r ) UC (r + a) = UC (r ) 2m 0

    ik ir    ψ (r ) = u  r e u  (r + a) = u  (r ) k: Brillouin zone k ( ) k k “Bloch wave”

  p ≠ k but…. crystal momentum ! ! ! E k bandstructure (dispersion) E k = "ω k ( ) ( ) ( ) Lundstrom ECE-656 F17 7 Crystal structure

Silicon

N ≈ 5 × 1022 cm-3 atoms

4 nearest 5.43 A neighbors

8 Band structure (electron dispersion) bottom of E(k) for electrons in Si conduction band

top of valence band

[100] [111]

DFT calculations by Dr. J. Maassen, Purdue 9 Constant energy surfaces

Si conduction band z

y 2 2 2 2 2 2 ! " k " ky " k E k = x + + z ( ) 2m* 2m* 2m* xx yy zz m* 0.9m  = 0 x * mt = 0.19m0 “Valley degeneracy” = 6

Valence band is “warped”.

10 Lundstrom ECE-656 F17 outline

1) Electron dispersion (band structure)

2) Simple band structures

3) Phonon dispersion

4) Simple phonon dispersions

5) Electrons vs. phonons

11 Lundstrom ECE-656 F17 Bandstructure basics Electrons in a solid behave as both particles (quasi-particles) E and as waves.

Electron waves are described   BW by a “dispersion:” E(k ) = ω (k )

k Because the crystal is periodic, −π a π a the dispersion is periodic in k (Brillouin zone).

Bandwidth on the order of eV. 12 12 Bandstructure basics

Particles described by a E “wavepacket.” ! ΔxΔp ≥ 2

The “group velocity” of a wavepacket is determined by k the dispersion: −π a k0 π a ! ! 1 ! υg (k ) = ∇k E(k ) "

13 Lundstrom ECE-656 F17 13 Bandstructure basics

E E 2k 2 2m* E = C +  n Near a band minimum or maximum, E(k) is a parabola.

The curvature of the parabola is the “effective mass.” k −π a π a

E = E −2k 2 2m* V p

14 Effective mass model for electrons

E E = 2k 2 2m* As long as the BW >> kBT, the effective mass model often works fine.

BW Typically, only states near the band edge matter, and k these regions can be −π a π a described by an effective mass approximation.

15 Lundstrom ECE-656 F17 15 Complex band structures (Bi2Te3)

2k 2 E(k) ≠ * 2m

DFT calculations of Bi2Te 3 by Dr. J. Maassen, Purdue 16 Lundstrom ECE-656 F17 First order correction for non-parabolicity

E “Kane bands” 2 2 * E =  k 2m !2k 2 E(1+αE) = 2m* 0 ( )

-1 BW α = 0.5 eV Si

k α = 0.64 eV-1 Si −π a π a

17 Lundstrom ECE-656 F17 17 Surfaces of constant energy

2 2 2 2 2 2 2 2 2 2 ! k ! ky ! k ! k ! k x z E = ⎡1∓ g θ,φ ⎤ E = E = * + * + * * ⎣ ( )⎦ * 2m 2m 2m 2m 2m xx yy zz - Heavy hole + Light hole

See L. Reggiani, “Chapter 2: General Theory,” pp.7-86, in Hot Electron Transport in Semiconductors, Springer-Verlag, New York, 1985 18 Lundstrom ECE-656 F17 18 Model bandstructure: Si

2k 2 E(k) = EC + 2m* E n m* 0.92m  = 0 * mt = 0.19m0

L 1.12eV X  k < 111 > < 100 > Eso hh

lh 2 2 so  k E(k) = EV − E 0.044 eV * so = 2mp 19 19 Model bandstructure: GaAs

* ≈ 0.30eV m = 0.063m0 1.42eV L X  k < 111 > < 100 > Eso hh

lh E 0.34 eV so = so

Lundstrom ECE-656 F17 20 E(k) for graphene

Recall:  1 dE(k) E υ k = g ( ) dk  For graphene: k x ! υ k = υ ≈ 108 cm/s g ( ) F ky Also recall: g 2 V = 2 −1 * ⎛ 1 d E(k)⎞ m = ⎜ 2 dk 2 ⎟ 2 2 ⎝ ! ⎠ E(k) = ±υF kx + ky = ±υF k For graphene: m* = ? 21 Lundstrom ECE-656 F17 Outline

1) Electron dispersion (band structure)

2) Simple band structures

3) Phonon dispersion

4) Simple phonon dispersions

5) Electrons vs. phonons

22 Lundstrom ECE-656 F17 mass and spring

1 2 U = ks (x − x0 ) 2

k dU s F = − = −ks (x − x0 ) dx d 2 x M 2 = −ks (x − x0 ) dt M iωt x(t)− x0 = Ae ⎛ 1⎞ En = n + ω ω = k M ⎝⎜ ⎠⎟ s 2

Lundstrom ECE-656 F17 23 Elastic waves: acoustic modes

! ! ! i q r t ! ! u(r ) A eˆ e ( i −ω ) q = i i β →

Longitudinal wave: displacement in the direction of propagation. Transverse waves: displacement transverse to the direction of propagation.

LA phonons: a near q = 0:

Lundstrom ECE-656 F17 24 Elastic oaves: optical modes

! ! ! i q r t u(r ) A eˆ e ( i −ω ) = i i

Atoms in the unit cell oscillate against each other.

Longitudinal and transverse optical modes.

x unit cell near q = 0:

Lundstrom ECE-656 F17 25 Lattice (phonon) dispersion

phonons in Si

DFT calculations by Dr. J. Maassen, Purdue

2 Lundstrom ECE-656 F17 6 outline

1) Electron dispersion (band structure)

2) Simple band structures

3) Phonon dispersion

4) Simple phonon dispersions

5) Electrons vs. phonons

27 Lundstrom ECE-656 F17 Phonon dispersion basics

Lattice vibrations behave both ω as particles (quasi-particles) optical phonons and as waves.

Lattice vibrations are described   ω q = E q by a “dispersion:” ( ) ( )  acoustic phonons BW Because the crystal is periodic, the dispersion is periodic in k q (Brillouin zone). −π a π a

Lundstrom ECE-656 F17 28 Phonon dispersion basics

Particles described by a ω “wavepacket” (phonon).

The “group velocity” of a phonon is determined by the dispersion:

   υ q = ∇ ω q g ( ) q ( ) q

−π a q0 π a

Lundstrom ECE-656 F17 29 Phonon dispersion basics

Optical phonons have a flat ω dispersion (near the zone center). Einstein Near zero group velocity.

Einstein approximation Debye

Acoustic modes have a linear dispersion (near the zone center). q

Constant sound velocity. −π a π a

Debye approximation. Lundstrom ECE-656 F17 30 General features of phonon dispersion

low group velocity ω ω 0 LO (1) TO (2) LA (1) ω = υSq υ = c ρ TA (2) S ℓ

q π a

31 Lundstrom ECE-656 F17 Outline

1) Electron dispersion (band structure)

2) Simple band structures

3) Phonon dispersion

4) Simple phonon dispersions

5) Electrons vs. phonons

32 Lundstrom ECE-656 F17 Bandstructure (dispersion) electrons in Si phonons in Si

note the different energy scales! DFT calculations by Dr. J. Maassen, Purdue 3 Lundstrom ECE-656 F17 3 Real dispersions

BW k T BW ~ k T >> B B 3 0.06 TO (2) 2 cb LO (1) 1 0.04 LA (1) (eV) (eV) l h e 0 p E E -1 0.02 hh TA (2) -2 lh 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 k (x /a ) q (x π/a ) kxx (×π a0) qxx(×π a)0 note the different energy scales! electrons in Si (along [100]) phonons in Si (along [100]) 34 Lundstrom ECE-656 F17 34 References

For a though treatment of electrons and phonons in crystals, see:

N.W. Ashcroft and N.D. Mermin, Solid State Physics, Saunders College, Philadelphia, 1976.

For an introduction to lattice waves and phonons, see:

T.S. Fisher, Thermal Energy at the Nanoscale, Chapter 1, World Scientific, 2014.

Lundstrom ECE-656 F17 35 Summary

1) Electron dispersion (band structure)

2) Simple band structures

3) Phonon dispersion

4) Simple phonon dispersions

5) Electrons vs. phonons

Lundstrom ECE-656 F17 36