6.4 Lattice dynamics

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General Physics part 1 / PhysPraktikum / Woan G. The Cambridge Handbook of Physics Formulas (CUP, 2000)(ISBN 0521573491)(230s)_PRef_.pdf

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6.4 Lattice dynamics	129

6.4	Lattice dynamics
Phonon dispersion relationsa
	m m			m1		m2 (> m1)
	a					2a
2(α/m)1/2				(2α/µ)1/2
2(α/m)1/2				(2α/m1)1/2		ω
		ω		(2α/m1)1/2		ω
				(2α/m2)1/2
	π		π	−	π		π
	− a	0 k	a	−	2a	0 k	2a
		monatomic chain				diatomic chain

= 4

sin

(6.34)

phonon angular frequency

spring constantb

Monatomic

= a

1/2

(6.35)

atomic mass

linear chain

vp =

sinc

sinπx )

λ!

phase speed (sincx

1/2

group speed

≡ πx

∂ω

cos

vg =

= a

(6.36)

phonon wavelength

∂k

wavenumber (= 2π/λ)

Diatomic

± α

(ka)

1/2

atomic separation

−

linear chainc

sin

(6.37)

µ i

reduced mass

µ2

m1m2

atomic masses (m

> m

)

[= m1m2/(m1 + m2)]

Identical

α1 + α2

1/2

αi

alternating spring constants

masses,

(α1

+ α2

+ 2α1α2 coska)

alternating

(6.38)

spring

0, 2(α

+ α2)/m

if k = 0

α1 α2

α1

constants

= "2α1/m,

2α2/m

if k = π/a

(6.39)

aAlong inﬁnite linear atomic chains, considering simple harmonic nearest-neighbour interactions only. The shaded region of the dispersion relation is outside the ﬁrst Brillouin zone of the reciprocal lattice.

bIn the sense α = restoring force/relative displacement.

cNote that the repeat distance for this chain is 2a, so that the ﬁrst Brillouin zone extends to |k| < π/(2a). The optic and acoustic branches are the + and − solutions respectively.

130 Solid state physics

Debye theory

Mean energy

mean energy in a mode at ω

¯hω

¯h

(Planck constant)/(2π)

modea

E =

(6.40)

Boltzmann constant

2 ¯hω + exp[¯hω/(kBT )] − 1

per phonon

phonon angular frequency

temperature

ωD

Debye (angular) frequency

ωD = vs(6π2N/V )1/3

(6.41)

Debye

e ective sound speed

frequency

where

longitudinal phase speed

(6.42)

vs3

vl3

vt3

transverse phase speed

Debye

number of atoms in crystal

θD = ¯hωD/kB

(6.43)

crystal volume

temperature

θD

Debye temperature

3V ω2

g(ω) density of states at ω

Phonon

g(ω) dω =

dω

(6.44)

heat capacity, V constant

2π2v3

density of

thermal phonon energy

states

(for 0 < ω < ωD, g = 0 otherwise)

within crystal

D(x) Debye function

Debye heat			T 3	θD/T	x4ex		3NkB
capacity	CV =	9NkB θD3		0	(ex − 1)2 dx	(6.45)	CV
Dulong and		3NkB	(T θD)			(6.46)	CV
Dulong and
Petit’s law
Debye T 3		12π4		T 3	(T θD)	(6.47)
law		5 NkB θD3			(T θD)	(6.47)	0	1	T /θD	2

U(T ) =

ωD

¯hω3

dω ≡ 3NkBT D(θD/T )

(6.48)

Internal

ωD3

exp[¯hω/(kBT )]

−

thermal

x y3

energyb

where

D(x) =

(6.49)

ey − 1

aOr any simple harmonic oscillator in thermal equilibrium at temperature T . bNeglecting zero-point energy.

6.4 Lattice dynamics	131

Lattice forces (simple)

3 αp2 ¯hω

φ(r)

two-particle potential

Van der Waals

φ(r) = −

(6.50)

energy

interactiona

particle separation

(4π 0)2r6

αp

particle polarisability

φ(r) = −

(6.51)

¯h

(Planck constant)/(2π)

r12

Lennard–Jones

= 4

permittivity of free space

6-12 potential

−

(6.52)

polarised orbital

(molecular

σ = (B/A)1/6 ;

= A2

angular frequency of

crystals)

/(4B)

A,B

constants

21/6

,σ

Lennard–Jones

φmin

r =

(6.53)

parameters

De Boer

de Boer parameter

Λ =

(6.54)

Planck constant

parameter

σ(m )1/2

particle mass

lattice Coulomb energy

Coulomb

per ion pair

αM

Madelung constant

interaction

UC = −αM 4π 0r0

(6.55)

−e

electronic charge

(ionic crystals)

nearest neighbour

separation

aLondon’s formula for ﬂuctuating dipole interactions, neglecting the propagation time between particles.

Lattice thermal expansion and conduction

Gruneisen¨

∂lnω

Gruneisen¨ parameter

γ =

(6.56)

parametera

− ∂lnV

normal mode frequency

volume

linear expansivity

Linear

∂p

γCV

isothermal bulk modulus

(6.57)

expansivityb

α =

3KT ∂T V

= 3KT V

temperature

pressure

lattice heat capacity, constant V

Thermal

thermal conductivity

conductivity of

λ =

vsl

(6.58)

e ective sound speed

a phonon gas

3 V

phonon mean free path

Umklapp mean

lu exp(θu/T )

(6.59)

umklapp mean free path

θD/2)

free pathc

θu

umklapp temperature (

aStrictly, the Gruneisen¨ parameter is the mean of γ over all normal modes, weighted by the mode’s contribution to

CV .

bOr “coe cient of thermal expansion,” for an isotropically expanding crystal.

cMean free path determined solely by “umklapp processes” – the scattering of phonons outside the ﬁrst Brillouin zone.

132	Solid state physics

6.5Electrons in solids

Free electron transport properties

Current density

J = −nevd

(6.60)

Mean electron

vd =

−

eτ

(6.61)

drift velocity

d.c. electrical

σ0 =

ne2τ

(6.62)

conductivity

a.c. electrical

σ(ω) =

σ0

(6.63)

conductivitya

1 − iωτ

1 CV

λ =

c2 τ

(6.64)

Thermal

conductivity

π2nkB2 τT

(T TF)

3me

(6.65)

Wiedemann–

π2k2

Franz lawb

= L =

(6.66)

σT

3e2

Hall coe cientc

RH = − ne = JxBz

(6.67)

Hall voltage

BzIx

(rectangular

= RH

(6.68)

strip)

Jcurrent density

nfree electron number density

−e electronic charge

vd mean electron drift velocity

τmean time between collisions (relaxation time)

me electronic mass

Eapplied electric ﬁeld

σ0 d.c. conductivity (J = σE )

ωa.c. angular frequency

σ(ω) a.c. conductivity

CV total electron heat capacity, V constant

Vvolume

c2 mean square electron speed

kB Boltzmann constant

Ttemperature

TF Fermi temperature

LLorenz constant ( 2.45 × 10−8 WΩ K−2)

λ	thermal conductivity		Jx
RH	Hall coe cient	Ey Bz		w
Ey	Hall electric ﬁeld
Ey	Hall electric ﬁeld
Jx	applied current density
Bz	magnetic ﬂux density	+
VH	Hall voltage	VH
Ix	applied current (= Jx × cross-sectional area)

wstrip thickness in z

aFor an electric ﬁeld varying as e−iωt. bHolds for an arbitrary band structure.

cThe charge on an electron is −e, where e is the elementary charge (approximately +1.6 × 10−19 C). The Hall coe cient is therefore a negative number when the dominant charge carriers are electrons.

6.5 Electrons in solids	133

Fermi gas

3/2

g(E) =

2 e

E1/2

(6.69)

Electronadensity

2π2

¯h2

of states

g(EF) =

3 nV

(6.70)

2 EF

Fermi

kF = (3π2n)1/3

(6.71)

wavenumber

Fermi velocity

vF = ¯hkF/me

(6.72)

Fermi energy

¯h2k2

¯h2

2/3

(T = 0)

EF =

(3π

(6.73)

2me

Fermi

TF =

(6.74)

temperature

Electron heat

π2

)k2 T

V e

g(E

(6.75)

capacityb

(T TF)

π2k2

(6.76)

2EF

Total kinetic

U0 =

nV EF

(6.77)

energy (T = 0)

Pauli

M = χHPH

(6.78)

paramagnetism

(6.79)

µ0µBH

2EF

Landau

χHL = −

χHP

(6.80)

diamagnetism

Eelectron energy (> 0)

g(E) density of states

V“gas” volume

me	electronic mass
¯h	(Planck constant)/(2π)
kF	Fermi wavenumber

nnumber of electrons per unit volume

vF	Fermi velocity
EF	Fermi energy
TF	Fermi temperature
kB	Boltzmann constant

CV e heat capacity per electron

Ttemperature

U0	total kinetic energy	6
χHP	Pauli magnetic susceptibility	6

Hmagnetic ﬁeld strength

Mmagnetisation

µ0	permeability of free space
µB	Bohr magneton

χHL Landau magnetic susceptibility

aThe density of states is often quoted per unit volume in real space (i.e., g(E)/V here). bEquation (6.75) holds for any density of states.

Thermoelectricity

Thermopower	a	J		E	electrochemical ﬁeldb
				J	current density
		E = σ + ST T	(6.81)	J	current density
		E = σ + ST T	(6.81)	σ	electrical conductivity
				σ	electrical conductivity
				ST	thermopower
		H = ΠJ − λ T		ST	thermopower
Peltier e ect		H = ΠJ − λ T	(6.82)	T	temperature
				H	heat ﬂux per unit area
				Π	Peltier coe cient
Kelvin relation		Π = T ST	(6.83)	Π	Peltier coe cient
Kelvin relation		Π = T ST	(6.83)	λ	thermal conductivity
				λ	thermal conductivity

aOr “absolute thermoelectric power.”

bThe electrochemical ﬁeld is the gradient of (µ/e) − φ, where µ is the chemical potential, −e the electronic charge, and φ the electrical potential.

134 Solid state physics

Band theory and semiconductors

electron eigenstate

Bloch’s theorem

Ψ(r + R) = exp(ik · R)Ψ(r)

(6.84)

Bloch wavevector

lattice vector

position vector

electron velocity (for wavevector

Electron

¯h

(Planck constant)/2π

velocity

vb(k) = ¯h k Eb(k)

(6.85)

band index

Eb(k)

energy band

E ective mass

∂2Eb(k) −1

mij

e ective mass tensor

tensor

mij = ¯h2

(6.86)

components of k

∂ki∂kj

Scalar e ective

∂2E (k)

−1

scalar e ective mass

massa

m = ¯h2

(6.87)

∂k2

= |k|

particle mobility

mean drift velocity

Mobility

µ =

|vd|

(6.88)

applied electric ﬁeld

kBT

−e

electronic charge

di usion coe cient

temperature

current density

Net current

J = (neµe + nhµh)eE

(6.89)

ne,h

electron, hole, number densities

density

µe,h

electron, hole, mobilities

Semiconductor

nenh =

(kBT )

(me mh)3/2e−Eg /(kBT )

Boltzmann constant

band gap

equation

2(π¯h )3

(6.90)

me,h

electron, hole, e ective masses

current

− 1

saturation current

I = I0

(6.91)

intrinsic carrier concentration

exp kBT

bias voltage (+ for forward)

area of junction

p-n junction

I0 = eni

(6.92)

LeNa

LhNd

De,h

electron, hole, di usion

Le = (Deτe)1/2

(6.93)

coe cients

Le,h

electron, hole, di usion lengths

τ )1/2

= (D

(6.94)

τe,h

electron, hole, recombination

times

Na,d

acceptor, donor, concentrations

aValid for regions of k-space in which Eb(k) can be taken as independent of the direction of k.

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