3. LATTICE VIBRATIONS • Atoms in lattice are not stationary even at T =

0K.

• They vibrate about particular equilibrium positions at T = 0K ( zero-point energy).

• For T > 0K, vibration amplitude increases as atoms gain thermal energy.

3.1 Sound Waves • Sound waves travel through solids with typical

speeds ~ 5(km/s).

• At low frequencies (f < 1THz), λ~50Ao, one can treat solid as continuum, ie. ignore discrete nature. At any frequency & in a given direction in crystal, 3 sound waves possible: One longitudinal and two transverse.

3.2 Continuum Model Propagation of an elastic wave in a sample of long bar follows the dispersion relation :

qvs=ω Where vs is the wave velocity (sound velocity) which equal in this case to ρ/Yvs = Y & ρ are Young’s modulus and density of the sample.

3.3 Discrete Lattice As the wavelength decreases and q increases, the atoms begin to scatter the wave, and hence to decrease its velocity. Thus, one expects the dispersion relation to be as shown ω

Continuum

Discrete

q 0

A- Monatomic Chain • Consider 1D chain of identical atoms each of mass

M, where atoms joined to nearest neighbors by 'spring' of force constant α, and spacing a apart.

• Look for wavelike solution:

Where Xn =na is the equilibrium position . Substitute into eq. of motion get:

) ( tqXin

nAeu ω−=

qaM 2

122 sin4αω =

Hence, the Dispersion relation , which Determines how wave vector q is related to ω,in 1D is :

Where ωm =(4α/M)1/2 .

sin 21 qamωω =

• Note:

∗ ω is periodic in q & all values of ω contained in range: -π/a < q < π/a. * Range of q related to 1st Brillouin zone. Values of q outside range have no physical meaning. * N does not appear in dispersion relation, ie. equations of motion of all atoms lead to the same algebraic relation between ω and q.

1.

Long wavelength limit: As q →0, sinθ =θ, and then

qam

2ωω =

Or Hence,

2.

3.

2av m

sω

=

Ya=α

short wavelength limit: As q increases , the dispersion relation begins to deviate from the straight line. It will saturate at q=π/a with maximum frequency

Mmαω 4

=

Phase & Group velosity : defines the velocity of a pure wave with frequency ω and wave vector q. defines the velocity of a wave pulse with average frequency ω and average wave vector q. Thus, at q →0 , phase velocity = group velocity. defines velocity of sound, which can be identified with the appropriate Young’s modulus. And, at q=π/a , group velocity is zero, ie. standing waves.

qpv ω=

qv ∂

=g ∂ω

B- Diatomic Chain • Consider 1D chain of two different atoms of

masses M1 and M2, where the distance between two neighbouring atoms is a.

Dispersion relation ,that determines how wave vector q is related to ω, is

)(sin4)11()11( 21

22

2121

2

MMqa

MMMM−+±+= ααω

3.5. Lattice Vibrations in 3D • If we Extend properties of 1D chain to 3D

crystals:

1- For crystal lattice with unit cell containing only one atom, * 3 dispersion relations, one in each direction of vibration. * All three branches are acoustic.

2- Crystals with unit cell containing 2 different atoms, * 3 acoustic branches & 3 optical branches. 3- Generalize to crystals with s different atoms in unit cell. * 3 acoustic branches & 3(s - 1) optical branches.

3.6 Phonons • Define:

Phonons: quantum of vibrational (elastic) energy. photons: quantum of electromagnetic energy. * The energy of phonon is: With a momentum equal to:

ωε h=

qp h=

ωhKTn =

* The number of phonons is given by Bose-Einstein distribution: This number depends on temperature , at T=0 ⇒n=0. At very high temperature,

Fact:

The number of phonons in a system is not conserved.

Interaction of phonons with other excitation (or with themselves) governed by conservation laws.

11

/ −= KTe

n ωh

3.6. Density of States In calculating physical properties of solids, the no.

of modes in a given frequency (or energy, or q-space) range is required.

1D density of States

A 'better' way of obtaining this is to apply the periodic boundary condition:

u(x=L) = u(x=0)

eiqL =1

qL = n2π, ie. q = n2π/L

Thus, 'length' occupied by one mode in q-space is 2π/L . Hence the No. of modes in the range dq (i.e. the density of states in q-space) is:

L/2π dq

To obtain density of states in frequency space, need dispersion relation. Since there is the same No. of states in g(q)dq & g(ω)dω, ie.

dqLqdqgdgπ

ωω2

)()()()( ==

or 2

) (ωπ

ωd

=dqLg

Since the modes lying in( –ve) q-region must be included as well, then:

(1)

/)(

dqdωπω =

1Lg

Note the relation between the density of states and the group velocity.

* At long wavelength limit: ω = vsq, ie. dω = vsdq

and hence, (2) )(

vπω =

1

s

Lg

* At Dispersive region:

sin 21 qamωω =

or

)(sin 2 1

maq

ωω−=

Differentiating:

)1( 2 1/2-22 ωωω −

=mad

dq Hence density of states in ω-space is: (3)

) )( 22ω =g 1(2 1/2-

ωωπ −aL

m

3D density of States Volume occupied by one mode in q-space is:

(2π/L)3 = (8π3)/V

Thus, No. of modes contained in a small spherical shell in q-space is: 4πq2dq / 8π3/V

Hence, density of states in ω-space is

48

)( 23 ω

ππ

ωddqqVg =

or (4)

2) 2( 2

ωπω

dq=

dqVg This is the density of state if we assume that each value of q associated with one single mode. But, in fact for each q there are 3 modes (1 longitudinal & 2 transverse ), then (5)

2 2

3)( 2

ωπFor non-dispersive limit;

ω dqqVg =d

1)(

23)( 2

2ss vv

Vg ωπ

ω = or,

23)( 3

2

2sv

Vg ωπ

ω =

3.7. Specific Heat of Lattice • General definition of specific heat is;

TQC

ΔΔ

=

Where ΔQ is the heat required to increase the temperature of one mole by ΔT.

• At constant volume all internal energy E converts to heat, i.e. ΔQ= ΔE , then one can define the specific heat at constant volume as;

v

v TEC ⎟

⎠⎞

⎜⎝⎛

∂∂

= (1)

Cv

Experimentally, it was found that there is a relation between Cv & T as is shown in the figure.

Theoretically, three different approaches were used to determine Cv:

T

A- Classical Theory: • Vibration of lattice may be treated as that of a

harmonic oscillator.

• Thus, the average energy for 1D oscillator is

kT=ε

Therefore, the total thermal energy per mole in 3D is

RTkTNE A 33 == where NA & R are Avogadro’s no. and the universal gas constant respectively. Substituting into (1), we find

RCv 3= (2) This relation predicts a constant value to Cv whatever the temperature is. Although it is in agreement with experiment at high T, it fails totally at low T.

B- Einstein model: * Considering the atoms as independent oscillators, vibrating with a common frequency. * Treating the oscillator quantum mechanically, where the average energy for 1D oscillator is

1/ −= kTe ω

ωεh

h

• At high T , kT=ε (and the classic value is reached)

• At low T ,total energy is;

13 / −

= kTE

EeNE ω

ωh

h

Where the common frequency of the oscillators, ωE , is known as Einstein frequency.

• Substituting into (1), we find

( )2/

/2

13

−⎟⎠⎞

⎜⎝⎛=

kT

kTE

vEe

ekT

RCω

ωωh

hh

Introducing Einstein temperature, kEE /ωθ h= , the above expression can be reduced to:

( )2/

/2

13

−⎟⎠⎞

⎜⎝⎛=

T

TE

vE

E

ee

TRC

θ

θθ (3)

This formula is in a good agreement with the experimental results as it shown in the following figure:

• At very low temperature, T<< θE , and hence,

TEv

EeT

RC /2

3 θθ −⎟⎠⎞

⎜⎝⎛=

Which indicates that Cv approaches zero exponentially. But careful measurements show that it does that slower.

Although the fact that (Cv → 0 as T → 0 ) agrees with experiments, the manner in which this is approached is

not.

C- Debye model:

* Treating the atoms not as independent oscillators, but as coupled oscillators vibrating collectively as sound wave follows the dispersion relation: ω = vsq

• Total energy in such system is ,

∫= ωωωε dgE )()( where g(ω) is the density of state, and hence the integration is taken over all allowed frequencies. Or;

ωωωπ ω d

evVE kT

s 123

/2

32 −∫ h

h

Debye stated the integral limit : (from 0 to ωD), where ωD is Debye frequency wich given by

( ) 3/126 nvsD πω = Specific heat Cv then can be found to be,

( )dx

eexTRC

T

x

x

Dv

D

∫−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

/

02

43

19

θ

θ

With kTx /ωh= and kDD /ωθ h= Here θD is Debye temperature.

At very low temperature, T<< θD , and hence

34

512

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Dv

TRCθ

π

4. FREE ELECTRON THEORY OF METALS � Classical theory : Drude & Lorentz (based on free-

electron model) gave explanations for conductivity and optical properties in metal.

� Quantum theory: Summerfield included Pauli's exclusion principle into classical theory.

4.1 Conduction electrons

• A valence electron in a free atom, becomes a conduction electron in a solid.

• The core electrons do not contribute anything to the electric current.

• Hence, No. of conduction electrons = No. of atoms x atomic valence

M

NZN Am

v ′=

ρ Where M`is the molar mass of the element (gm/mole).

4.2 The Free-Electron Gas • Assumptions:

a. Electrons free to move in a background of uniform (+ve) charge provided by ions (jellium model), ie. electrons behave like a gas.

b. No interactions between electrons, because of: 1- Pauli exclusion.

2- To minimize the system energy, electrons tend to stay away of each other.

The two above reasons lead to the hole concept: Each electron is surrounded by a spherical region which

is lacking of other electrons Known as a hole.

4.3 Electrical Conductivity

• Apply external electric field (ε) on a wire has a

length L and cross section A, we find:

AL and

LV ρε === R

AIJ

Where J is the current density, and ρ is the electrical resistivity =1/σ (σ is the electrical conductivity).

• According to Ohm’s Law: (1) σε=J

• When external electric field is applied two main things happen:

1. Coulomb force causes electrons to accelerate, ie. , where vd is the drift velocity .

2. Scattering (with other electrons, ions, impurities etc) produces a decay of vd back to zero. This retardation is represented by: -vd/τ, where τ is the relaxation time.

Thus, adding up total force on electron get:

v medt

dvm dd

τε ∗∗ −−=

In steady state, dv/dt=0, and hence

ετ m

evd ∗−= But since J represents the amount of charge crossing a unit area per unit time, then

(2) ( ετ==

NeJ )2

∗−m

vNe d Comparing (1) with (2) leads to;

∗=m

Ne τσ2

Since τ is the time between two following collisions, it can be expressed as;

( ~10-14 s in metals ) Where l is distance between two following collisions (electron’s mean free path) and vr is the random velocity. Using these terms (σ) can be rewritten as In metals : N ~1029 m-3 & vr ~106 m.s-1 ⇒ σ ~ 107 Ω-1.m-1

In semiconductors: N ~1020 m-3 & vr ~104 m.s-1 ⇒ σ ~ 1 Ω-1.m-1

The mean free path, l , for electrons in metals may be estimated via: vrτ = l . This give l ~ 10-8 m ~ 102 Ao, which is 20 times larger than the interatomic spacing. i.e., electrons can move freely in a perfect periodic structure. scattering occurs when periodicity is interrupted by phonons or impurities.

rv=τ l

rvmlNe

∗=2

σ

4.4 Temperature dependence of Electrical Resistivity In metals, the temperature dependence of ρ is determined by τ, according to the relation:

τρ 1

2Nem∗

= But total scattering rate τ given by addition of two individual scattering rates:

iph τττ111

+= Which known as Matthiesens' Rule. Therefore;

phiphi Ne

mNem

ττρρρ 11 22

∗∗

+=+=

Impurity Scattering ⇒ ρi , the residual resistivity, which is independent of T. It is determined by distance between impurities. Electron-phonon scattering⇒ ρph , the ideal resistivity, which is dependent of T. * At very low temperature, scattering by phonons is negligible. So τph ⇒∞ & ρph ⇒0, and hence ρ = ρi

* At high temperature, no. of phonons increases as T. Hence scattering increases as T . τ proportional to 1/T (true for T > ¼θD).

4.5. Heat Capacity of Conduction Electrons Based on Drude-Lorentz model, the average energy of electrons per mole is : RTkTNE A 2

323 ==

Hence, the electrons heat capacity is RCe 2

3= The total heat capacity in metals should then be; eph CCC += This means that at high temperature C=4.5 R ~ 9 cal/mole K But, experiments show that C~3R , and Ce is smaller than the classical value by a factor of 10-2.

Quantum explanation * Pauli's exclusion principle implies that only 2 electrons of opposite spin orientation are allowed in each state. * At T = 0K, electrons occupy all energy levels from the ground state upward. The energy of the highest filled state is known as the Fermi energy. * At T = 0K, electrons distribution function is

{ FEE,

FEE,)E(f><=

0 1

Thermal Distribution * For T > 0, electrons with energy ~ EF can gain thermal energy and occupy energy states >EF. * In many cases, it is the electrons near EF that dominate the properties of the metal. * For T > 0, electrons distribution function is

( ) 11)( / +

= KTE-E FeEf

i.e. Fermi-Dirac distribution. * Using this distribution function, one can find energy of electron gas, hence the heat capacity of the electrons. electron per mole as

)E/kT(N FA

Which leads to an approximate thermal energy

F

A

E)kT(NE

2

=

Hence, the electrons heat capacity is,

F

e EkTRC 2=

Or F

e TTRC 2=

With EF = kTF, where TF is Fermi temperture. * The exact value of electrons heat capacity is then,

Fe E

kTRC2

2π=

4.6. Fermi Surface Because of the random motion, the energy of an electron is mainly kinetic. Hence:

221

rvmE ∗= • In the velocity space, all conduction electrons confined in a sphere of radius vF which is called Fermi velocity. Energy at the surface of this sphere is; vy

• Fermi Surface

2

21

FF vmE ∗= v

vvx

F

6 vF is very large ~ 10 m.s-1. • Fermi surface (FS) is temperature

Independent, and so the Fermi velocity. • The value of EF is determined by the electron concentration N according to :

( ) 3/22

2

32

Nm

EF π∗=h

~ 5 eV

4.7. Electrical Conductivity (with respect to FS):

F

F

vmlNe

∗=2

σ

Although the formula of σ is the same as the classical case, the actual picture is different.

Lorntz Model Summerfield model No. of electrons

responsible about conductivity

N N(vd /vF)

Velocity of those electrons

v vd F

4.8 Hall Effect in Metals

• Applying a magnetic field B normally on a wire

carries a current of density Jx will produce an electric field normal to both Jx and B known as Hall field εH.

• In steady state;

Force of Hall field = Lorentz force -e εH = -e vx B εH = vx B Or,

BJNe xH1 −=ε

Since Jx = N(-e)vx ,where N is the electron density. The proportionality constant, i.e. ,is known as Hall constant R , therefore,

BJxH / εH

Ne

RH1 −=

4.9 Failure of Free-Electron Model 1- The model suggests that electrical conductivity

is proportional to electron concentration, but it is found that the divalent metals (Zn, Cd,..) and even trivalent metals (Al, In) are less conductive than the monovalent metals such as (Cu,Na,Ag,..) .

2- Some metals show positive Hall constant (Zn, Cd,..).

3- Fermi surface was found to be often non-spherical in shape.

5.1. Energy Bands in Solids * Solving Schrodiger equation for a free atom gives a series of discrete energy levels. * When two atoms come together to form a molecule, each of the atomic levels will split into two closely spaced levels. * The amount of splitting depends strongly on:

1- interatomic distance; the closer the atoms, the greater the splitting.

2- atomic orbital; the higher the energy, the greater the splitting.

This is an eigenvalue equation which leads to many solutions. Therefore, for each value of k there are a large number of solutions (i.e. a set of discrete energies E1,k, E2,k, ….).

* Each band covers a certain range of energy, extending from the lowest to the highest value it takes when plotted in k-space. * Within each band the electron states can be classified according to their momentum (p=ħk). 3-The crystal potential V(r) is the crystal potential “seen” by the electron. The most important property of this potential is its periodicity. This potential is composed of two parts; ionic & electronic: (4) ((( VVV )) rrr ei += )

* When N atoms come together to form a solid, each of the atomic levels will split into N very closely spaced sublevels; i.e. energy band. * The energy spectrum in a solid is composed of a set energy bands. Regions separating these bands are energy gaps, i.e. regions of forbidden energy.

• The wave function describing the electronic states in a band, known as the crystal orbital, can extend throughout the solid, unlike the atomic orbital which is localized.

5.2. The Bloch Theorem

The wavelike properties of an electron in a crystalline solid can be described using the following form of Schrodinger equation: (1)

)(()( EV ψψ =⎤

⎢⎡

+∇h )2

2

rrrm ⎥−

2 ⎦⎣ Where V(r) is the crystal potential and ψ(r) & E are the state function and energy of this electron.

1-The Bloch Function The solution of (1) for a periodic potential is given by (2) (( f )()) rurr =ψ Where both the function u(r) and the probability density ψ(r) 2 must have the same translational symmetry as the lattice , i.e,

22 )()(

)()(

rfRrf

ruRru

=+

=+and The only function that satisfies this is eikr, hence (2) can be written as (3) )()( ek =ψ . rur k

rik

Which known as Bloch function. Note: *This state function has the form of a traveling plane wave so it propagates through the crystal like a free particle. *This function represents the crystal orbital of the solid. 2-The Energy Bands Substituting from (3) into (1) gives

)()()()(2

22

ruErurVikm kkk =⎥

⎦

⎤⎢⎣

⎡++∇−

h

The first term represents the interaction with the ion cores, and given by Since electron-electron interaction is very weak and has some difficulties to be calculated, the second term in (4) can be dropped out. The major effect of this interaction is to screen the ions from other electrons and hence to weaken the elecrton-ion interaction.

∑ −=j

jii RrvrV )()(

Thus,

∑ −==j

jsi RrvrVrV )()()( Where vs(r-Rj) is the potential of the screened ion located at Rj.

5.3. Number of States in the Band Since each Bloch function ψ(r)n,k defines a specific electron state, let see how many states there are in each band. For 1D case: )()( . xuex k

xikk =ψ

Applying the periodic boundary condition on this function shows that the allowed number of k is ; For the first zone, the number of states is

Lnk π2

=

N

aL

La==)2/()2( ππ

Where N is the number of unit cells. Hence, we end with these two important facts: 1- In each band the number of states (orbitals) inside the first zone is equal to the number of unit cells in the crystal N. 2- In each band the maximum number of electrons is equal to 2N .