3. Binding Energy in Crystals

3. Binding Energy in Crystals

3.1

. Bonding in Solids

Crystalline materials exhibit the complete spectrum of bond types: ionic, covalent, and metallic.

3.2. Ionic Bonding

Structure with high symmetry and coordination number as high as possible, such that the electrostatic attractive force is maximized. Alkali and alkaline earth elements usually form ionic structures, especially in combination with small electronegative anions such as 𝑜2−_and𝐹− _.

3.3. Covalent Bonding

Covalent bonding is chemical bonding which arises out of electrons being equally shared between the bonding partners. It’s bonding due to a strong overlap of the atomic-like wave functions of the different atoms. The valence electron density is therefore increased b/n the atoms; in contrast to ionic bonding. It’s intuitive that such an overlap will also depend on the orbital character of the wave function involved, i.e. in which directions the bonding partners lie.

The covalent bond is usually formed from two electrons, one from each atom participating in the bond. The electrons forming the bond tend to be partly localized in the region b/n the two atoms joined by the bond. The spin of the two electrons in the bond are antiparallel.

 Covalent bond is a strong bond: the bind b/n two carbon atoms in diamond wrt separated neutral atoms is comparable with the bond strength in ionic crystals.

 The covalent bond has strong directional properties. Thus carbon, silicon and germanium have the diamond structure, with atoms

joined to four nearest neighbors at tetrahedral angles. The tetrahedral bond allows only four nearest neighbors.

 Highly directional bonds irrespective of other atoms that are present,

(a) small atoms with high valence which, in the cationic state, would be highly polarizing , e.g. 𝐵3+_{, 𝑆𝑖}4+_,𝑃4+_{, 𝑆}4+_{, etc.}

(b) Large atoms which in the anionic state are highly polarizable

o

Roughly, a covalent bond is a bond where electrons are shared

equally between two atoms.

Figure 1: Perfectly Covalent Germaniun. Four electrons per unit cell are

identically distributed about the 𝐺𝑒4+_{ion cores. The electron density is large in} certain directions in the interstitial region.

Diamond is typical of the crystal structures formed by the elements from column IV of the periodic table: C, Si, Ge, and (grey) Tin. These elements all crystallize in the tetrahedrally coordinated diamond structure.

Covalent crystals are not as good insulators as ionic crystals; this is consistent with the delocalization of charge in the covalent bond. The semiconductors are all covalent crystals, sometimes (as in the III-V compounds) with a small touch of ionic bonding.

3.4. Metallic Bonding

It is sometimes hard to distinguish metallic bonding from covalent

bonding. Roughly, however, one defines a metallic bond to be the bonding that occurs in metal. These bonds are similar to covalent bonds in the sense that electrons are shared between atoms, but in this case the electrons

We should think of the delocalized free electrons as providing the glue that holds together the positive ions that they have left behind.

Figure 2: Metallic bonding in Potassium. Electrons are completely delocalized in to an electron gas which holds together the ions.

3.5. Properties of Metallic Crystals

It is possible to regard metallic bonding as an extreme case of bonding in which the electrons are accumulated b/n the ion cores. However, in contrast to covalent bonding, the electrons now have wave functions that are much extended in comparison to the separation b/n atoms.

Since the electrons are completely delocalized, the bonds in metals tend not to be directional. Metals are thus often ductile and malleable. Since the electrons are free, metals are good conductors of heat and electricity. In this sense, the metallic bond is a feature that is peculiar to solids, i.e. to aggregates of many atoms.

The electrical conductivity of metals is in the order of the ‘Acronym’: SiCo GoAl i.e. Silver (Ag) is the best conductor, Copper (Cu) is the 2nd _best conductor, Gold (Au) is the 3rd _{best & Aluminum (Al) is the 4}th _best conductor.

3.6. Calculation of cohesive energy

The cohesive energy of a crystal is defined as the energy that must be added to the crystal to separate its components in to neutral free atoms at rest, at infinite separation, with the same electronic configuration. The term lattice energy is used in the discussion of ionic crystals and is defined as the energy

that must be added to the crystal to separate its component ions in to free ions at rest at infinite separation.

Cohesive energy is the value of energy needed to move an atom completely away from its equilibrium position. It’s the measure of binding energy. Cohesive energy is the potential energy which is resulting due to attractive and repulsive forces b/n atoms. Hence, it depends on the distance b/n atoms.

Let’s consider two atoms 1 and 2:

The attractive force b/n these two atoms bring them close together until the strong repulsive force arises b/n these two atoms. Due to the overlap of electron shell, when the two atoms are approaching each other, the negatively charged electron shell come closer than the positive nucleus.

At certain separation, attractive and repulsive forces are equal. This separation is called equilibrium separation (𝒓𝟎). Expression for resultant force (general expression for bonding force b/n two atoms) is given by:

𝐹 =

𝐴

𝑟𝑀

−

𝐵

𝑟𝑁

(1)

At equilibrium, 𝐹 = 0 & 𝑟 = 𝑟_𝑜. The equation can be rewritten as:

0 =

𝐴

𝑟

_𝑜𝑀

−

𝐵

𝑟

_𝑜𝑁 𝐴

𝑟_𝑜𝑀

is the attractive force and 𝐵

𝑟_𝑜𝑁 is the repulsive force.

Since the attractive forces in inter-atomic bonds are largely electrostatic, ‘𝑀’ is usually 2, following Coulombs inverse-square law of electrostatics. The value of ‘𝑁’ is not easy to approximate, but it usually takes values [7 − 10) for metallic bond and [10 − 12] for ionic and covalent bonds.

From equation (1), we know that

𝐹 =

𝐴

𝑟𝑀

−

𝐵 𝑟𝑁

,

At equilibrium, 𝐹 = 0 & 𝑟 = 𝑟𝑜; substituting in (1) gives:

0 =

𝐴

𝑟

_𝑜𝑀

−

𝐵

𝑟

_𝑜𝑁 𝑟_𝑜𝑁 𝑟_𝑜𝑀

=

𝐵 𝐴

⇒ 𝑟

𝑜

= (

𝐵 𝐴

)

1 𝑁−𝑀

……..

formula to calculate the inter-atomic separation b/n two atoms at equilibrium position

The expression for potential energy can be obtained by integrating equation (1) wrt : 𝑈(𝑟) = ∫ 𝐹(𝑟)𝑑𝑟 = ∫ (

𝐴

𝑟

𝑀

−

𝐵

𝑟

𝑁) 𝑑𝑟 = ∫(𝐴𝑟−𝑀 − 𝐵𝑟−𝑁) 𝑑𝑟

𝑈(𝑟) = (

−𝐴 𝑀−1

)

1 𝑟𝑀−1

+(

𝐵 𝑁−1

)

1 𝑟𝑁−1

+c,

c

is integration constant

Setting 𝑎 = _𝑀−1𝐴

, 𝑏 =

_𝑁−1𝐵

, 𝑚 = 𝑀 − 1 and 𝑛 = 𝑁 − 1

; the whole equation can be rewritten as :

𝑈(𝑟) = −𝑎

𝑟𝑚 + 𝑏 𝑟𝑛 + 𝑐 If 𝑟 ⟶ ∞, 𝑈(𝑟) → 0, which implies 𝑐 = 0.

Then,

𝑈(𝑟) =

−𝑎

𝑟𝑚

+

𝑏

𝑟𝑛

(2)

i.e.

PE is the sum of attractive energy and repulsive energies.

Hence, the PE

𝑈(𝑟)

is minimum on cohesion.

For minimum PE:

(

𝑑𝑈

𝑑𝑟

)

_𝑟=𝑟 𝑜

= 0.

𝑑 𝑑𝑟( −𝑎 𝑟𝑚 + 𝑏 𝑟𝑛) 𝑟=𝑟𝑜 = 0 𝑎𝑚𝑟₀−(𝑚+1) = 𝑛𝑏𝑟₀−(𝑛+1) ⇒ 𝑛𝑏 𝑚𝑎 = 𝑟_𝑜𝑛+1 𝑟_𝑜𝑚+1 ⇒ 𝑟_𝑜𝑛 = 𝑛𝑏 𝑚𝑎𝑟𝑜 𝑚 _(*)

Substituting (*) in eq.(2) we get the dissociation energy/ Cohesive energy:

𝑈

_𝑚𝑖𝑛

=

−𝑎 𝑟_𝑜𝑚

+

𝑏 𝑟_𝑜𝑛

=

−𝑎 𝑟_𝑜𝑚

+

𝑏 (𝑛𝑏 𝑚𝑎)𝑟𝑜𝑚

𝑈

_𝑚𝑖𝑛

=

−𝑎

𝑟

_𝑜𝑚

+

𝑚𝑎

𝑛𝑟

_𝑜𝑚

=

−𝑎

𝑟

_𝑜𝑚

[1 −

𝑚

𝑛

]

This is the expression of cohesive energy.

4. Thermal properties of solids

Introduction

Chapter 3 was concerned with the binding forces in crystals and with the

manner in which atoms were arranged. In this chap

ter we discuss about the internal energy excitation modes of the crystals. The quanta of these modes are the “particles” of the solid. This chapter is primarily devoted to a particular type of internal mode – the lattice vibrations.

4.1. Crystal vibration

Atomic motions in molecules and crystals are organized in to vibrational modes. In crystals these modes are called Phonons. A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter (solids).

A Phonon is the quantum unit of crystal vibration. Phonons play vital role in many of the physical properties of solids such as thermal conductivity and electrical conductivity.

4.1.1. Vibration of Crystals with Monatomic Basis (Classical Treatment) Consider the elastic vibration of a crystal with one atom in the primitive cell. We want to determine the frequency of an elastic wave interms of the wave vector that describes the wave and interms of the elastic constants.

In order to simplify the analysis needed for a 3D lattice of atoms, it is convenient to model a 1D lattice or linear chain .This model is complex enough to display the salient features of phonons.

Consider a rigid regular crystalline lattice composed of N particles. Since the lattice is rigid, the atoms must be exerting forces on one another to keep each atom near its equilibrium position. The forces b/n the atoms are assumed to be linear and nearest-neighbor, and they are represented by an elastic spring. Each atom is

assumed to be a point particle and the nucleus and electrons move in step (adiabatic theory):

Figure 3: Monatomic linear chain

Where 𝑛 denotes the nth _{atom out of a total of 𝑁, ‘a’ is the distance b/n atoms when} the chain is in equilibrium, and 𝑈_𝑛 is the displacement of the nth atom from

equilibrium.

The force on atom n is due to the two springs on the right and left sides of the atom:

𝐹_𝑛 = 𝐶[(𝑈_𝑛+1 − 𝑈_𝑛) − (𝑈_𝑛− 𝑈_𝑛−1)] (4.1) 𝐹_𝑛 = 𝐶[𝑈_𝑛+1+ 𝑈_𝑛−1 − 2𝑈_𝑛]

The right spring is compressed more than the left one. Thus the force on atom n is to the left. If 𝐶 is the elastic constant of the spring and 𝑚 is mass of the atom, then the equation of motion of nth _{atom is}

𝐹

_𝑛

= 𝐶[𝑈

_𝑛+1

+ 𝑈

_𝑛−1

− 2𝑈

_𝑛

] = 𝑚

𝑑2𝑈𝑛

𝑑𝑡2

(4.2)

Time dependence: Let 𝑈_𝑛(𝑡) = 𝑈_𝑛𝑒𝑥𝑝(−𝑖𝜔𝑡), then eqn. (4.2) becomes:

−𝑚𝜔

2

𝑈

_𝑛

=

𝐶

[

𝑈

_𝑛+1

+ 𝑈

_𝑛−1

− 2𝑈

_𝑛

] (4.3)

How to solve?

Looks complicated

Since the equation is the same at each atom 𝑛, the solution must have the same form at each 𝑛 differing only by a phase factor. This is most easily written as

𝑈

_𝑛

= 𝑈𝑒𝑥𝑝(𝑖𝑘𝑛𝑎) (4.4)

Where ‘a’ is the spacing b/n the planes and ‘k’ is the wave vector (𝑘 =2𝜋

𝜆);’ 𝑛’

denotes the atom

Now, using equations (4.4) and (4.3), we have:

−𝑚𝜔

2𝑈 = 𝐶[exp

(

𝑖𝑘𝑎

)

+ exp

(

−𝑖𝑘𝑎

)

− 2]𝑈 (4.5) Or

−𝜔

2 = 𝐶

𝑚[2 cos

(

𝑘𝑎

)

− 2] A more convenient form is

𝜔

2

= 4

𝐶

𝑚

[𝑠𝑖𝑛

2

₍

𝑘𝑎

2

)

] (4.6)

How? Show this!!

Finally,

𝜔 =

2

√

𝐶

𝑚

|

sin(

𝑘𝑎

2

)

| .

The solution is an infinite set of independent

oscillations, each labled by 𝑘(wave vector) and having a frequency

𝜔

_𝑘

=

2

√

𝐶

𝑚

|

sin(

𝑘𝑎

2

)

| (4.7)

The relation

𝜔

_𝑘 as a function of 𝑘 is called a dispersion curve/dispersion relation.

Figure 4: Phonon dispersion curve Brillouin Zone

Q. What range of K is physically significant for elastic waves?

Ans. Only those in the first Brillouin zone.Consider K ranging over all reciprocal space.

The expression for

𝜔

_𝑘 is periodic.

Figure 5: Brillouin Zone in 1D

All the information is in the first Brillouin Zone – the rest is repeated with periodicity 2𝜋

reciprocal lattice vector G = integer times 2π/a • What does this mean?

In fact the motion of atoms with wave vector K is identical to the motion with wave vector K + G

• All independent vibrations are described by K inside BZ.This is a general result valid in all crystals in all dimensions.

• The vibrations are an example of excitations. The atoms are not in their lowest energy positions but are vibrating.

• The excitations are labeled by a wave vector K and are periodic functions of K in reciprocal space.

All the excitations are counted if one considers only K inside the Brillouin zone (BZ). The excitations for K outside the BZ are identical to those inside and are not independent excitations.

Group velocity of vibration wave

The wave 𝑈_𝑛 = 𝑈exp(𝑖𝑘𝑛𝑎 − 𝑖𝜔𝑡) is a traveling wave. The transmission velocity of a wave packet is the group velocity, given by

𝑉

_𝑔

=

𝑑𝜔

𝑑𝐾

(4.8)

Or 𝑉⃗ = 𝑔𝑟𝑎𝑑_𝐾 𝜔(𝐾) – slope of 𝜔_𝑘 𝑣𝑠 𝑘. This is the velocity of energy propagation in the medium.Now,

𝜔

_𝑘

=

2

√

𝐶

𝑚

|

sin(

𝑘𝑎 2

)

|

, so

𝑉

_𝑔

= √

𝐶𝑎2 𝑚

cos

(𝑘𝑎) 2

(4.9)

N.B: At the zone boundary where

𝐾 = ±

𝜋

𝑎

, 𝑉

𝑔

= 0.

Here the wave is a standing

wave, hence zero net transmission velocity for a standing wave.

Long Wavelength Limit

When

𝐾𝑎 ≪ 1,from Eqn. (4.5) we have

𝜔

2

= 2

𝐶

𝑚

[1 − cos(𝐾𝑎)]

(4.5)

cos 𝑥 = ∑(−1)

𝑛 ∞ 𝑛=0

𝑥

2𝑛

(2𝑛)!

= 1 −

𝑥

2

2!

+ ⋯

Now, using Taylor expansion for cosine and ignoring hegher order terms, Cos(𝐾𝑎)

≅ 1 −

(𝐾𝑎)₂ 2

(4.10)

Subbing Eqn.(4.10) in Eqn.(4.5) gives

𝜔

2

=

𝐶

𝑚

𝑘

2

_𝑎

2

_(4.11)

In the long wavelength limit, frequency is directly proportional to the wave vector (𝜔 ∝ 𝑘), i.e. the velocity of sound is independent of frequency in this limit.

Quantization of Elastic Waves

The energy of a lattice vibration is quantized. The quantum of energy is called a Phonon in analogy with the photon of the EMW. The energy of an elastic mode of angular frequency 𝜔 is

𝜖 = (𝑛 +

1

2

)ℏ𝜔 (4.12)

when the mode is excited to quantum number 𝑛 (i.e. when the mode is occupied by 𝑛 phonons). The term 1

2

ℏ𝜔

is the zero point energy of the mode. It occurs for

both phonons and photons as a consequence of their equivalence to a quantum of harmonic oscillator of frequency 𝜔.

4.2. Lattice Specific heat

4.2.1. Heat capacity

The heat capacity, C, of a system is the ratio of the heat added to the system, or withdrawn from the system, to the resultant change in the temperature: 𝐶 = ∆𝑄_∆𝑇 =𝑑𝑄

𝑑𝑇 Unit of C: (𝐽/𝑑𝑒𝑔).

This definition is only valid in the absence of phase transitions. Usually Cis given as specific heat capacity, c, per gram or per mol. Heat capacity can be measured under conditions of constant temperature or constant volume. Thus, two distinct heat capacities can be defined:

𝐶

_𝑉

= (

𝜕𝑈

𝜕𝑇

)

_𝑉 heat capacity at constant volume

(4.13)

𝐶

_𝑃

= (

𝜕𝑈

𝜕𝑇

)

_𝑃 heat capacity at constant pressure.

Where U is the energy and T is the temperature. By heat capacity we usually mean the heat capacity at constant volume. The contribution of the phonons to the heatcapacity of a crystal is called the lattice heat capacity (𝑪_𝒍𝒂𝒕).

Heat capacity is a measure of the ability of the material to absorb thermal energy.

Thermal energy = kinetic energy of atomic motions + potential energy of distortion of interatomic bonds.Thermal energy is the energy of all phonons (or all vibrational waves) present in the crystal at a given temperature.

Q. 𝑪𝑷 is always greater than 𝑪𝑽 - Why?

Hint: The difference between 𝑪𝑷 and 𝑪𝑽is very small for solids and liquids,

but large for gases.

4.2.2. Specific Heat

The specific heat of any material is defined as the quantity of heat that raises the temperature of unit mass by one degree. The main contributions to heat capacity are atomic vibrations (phonons) and thermal excitations of electrons.

The units of specific heat capacity are: 𝐽

𝑘𝑔𝐾 or 𝑐𝑎𝑙 𝑔℃

4.3. Classical theory (Dulong-Petit law)

The classical theory predicts constant specific heat which deviates from reality. This temperature independent specific heat is determined only by number of degrees of freedom. Consider motion of harmonic crystal is

For a vibrating atom:

𝐸₁ = 𝐾 + 𝑈 (4.14)

Classical statistical mechanics — equipartition theorem: in thermal equilibrium each quadratic term in the E has an average energy 1

2 𝑘𝐵𝑇 , so:

𝐸̅₁ = 6 ( 1

2 𝑘𝐵𝑇) = 3𝑘𝐵𝑇 (4.15)

𝒌_𝑩 → Boltzmann constant (𝒌_𝑩 ≈ 𝟏. 𝟑𝟖𝒙𝟏𝟎−𝟐𝟑𝑱/𝑲) and 𝑻 is absolute

temperature

For a solid composed of 𝑁 such atomic oscillators:

𝐸̅ = 𝑁𝐸̅

₁

= 3𝑁𝑘

_𝐵

𝑇 (4.16)

Giving a total energy per mole of sample: 𝐸̅

𝑛

=

3𝑁𝑘_𝐵𝑇

𝑛

= 3𝑁

𝐴

𝑘

𝐵

𝑇 = 3𝑅𝑇

(4.17)

So the heat capacity at constant volume per mole is

𝐶

_𝑉

=

𝑑

𝑑𝑇

(

𝐸

̅

𝑛

) =

𝑑

𝑑𝑇

(

3𝑅𝑇

) = 3𝑅 ≈ 25𝐽/𝑚𝑜𝑙𝐾

This is known as the Dulong-Petit law: the lattice specific heat capacity is independent of material properties and temperature This holds for higher temperatures (> 300𝐾) but, for lower temperatures 𝐶𝑣 is no longer a constant anddecreases with temperature. This

behavior can be explained by quantum mechanicalprinciples.

Q. What is (are) the limitation(s) of Dulong and Petit law of specific heat?

4.4. Einstein’s theory of specific heat

The first quantum mechanical theory to explain the lattice specific heat was the Einstein model of specific heat for solids. It postulates that all atoms are vibrating independently with same frequency ω. In this theory the correlation between the

motions of neighboring atoms were ignored. Einstein Uses Planck’s Work to show temperature dependence of specific heat of solids.

Planck (1900): vibrating oscillators (atoms) in a solid have quantized energies:

𝐸

_𝑛

= 𝑛ℏ𝜔

𝑛 = 0,1,2, …

(

Later QM showed that

𝐸

_𝑛

= (𝑛 +

1

2

)

ℏ𝜔

)

Einstein (1907): model a solid as a collection of 3𝑁 independent 1D

oscillators, all with constant , and use Planck’s equation for energy levels. Probability of oscillator being in level n (occupation of energy level n):

(4.18)

Average total energy of solid:

(4.19)

Using Planck’s equation (

𝐸

_𝑛

= 𝑛ℏ𝜔

):

(4.20) Now letting

𝑥 =

ℏ𝜔

𝑘𝑇

,

equation (4.20) can be rewritten as,





     



0 / 0 /

3

n kT n n kT n

e

e

n

N

U

 



 





   



0 / /

)

(

n kT E kT E n n n

e

e

E

f







       







0 / 0 / 0

3

)

(

3

n kT E n kT E n n n n n n

e

e

E

N

E

E

f

N

U

E

(4.21)

Again Eqn. (4.21) can be rewritten as

(4.22)

Now using the infinite sum:

We can evaluate the summation as, So we obtain: (4.23)

1

1

1

0









 

x

for

x

x

n n





     



0 0

3

n nx n nx

e

e

n

N

U





 

 









           

_







0 0 0 0

₃

3

n n x n n x n nx n nx

e

e

dx

d

N

e

e

dx

d

N

U









 

1

1

1

0









_   



x x x n n x

e

e

e

e

1

3

1

3

1

1

3

_/









































_{x kT} x x x x

e

N

e

N

e

e

e

e

dx

d

N

U









_





Using the definition for 𝐶𝑉:

Differentiating yields:

Now it is traditional to define an “Einstein temperature”:

𝜃

_𝐸

=

ℏ𝜔

𝑘 (4.24) So we obtain the time dependence prediction 𝐶𝑉(𝑇):

(4.25)

 Limiting Behavior of 𝑪𝑽(𝑻)

1. High Temperature limit: 𝜃𝐸

𝑻 ≪ 𝟏

2. Low Temperature limit: 𝜃𝐸

𝑻 ≫ 𝟏































1

3

/ kT A V V

e

N

dT

d

n

U

dT

d

C

_





 











 

_/



2 / 2 2 / /

1

3

1

3

2











 kT kT kT kT kT kT A V

e

e

R

e

e

N

C

     



     



   





R

R

T

C

T T T V E E E

3

1

1

1

3

)

(

₂ 2

















 



_/



2

/

2

1

3

)

(





T

T

T

V

E E E

e

e

R

T

C







These predictions are qualitatively correct: CV 3R for large T and

CV 0 as T  0:

But, from comparison of experimental values of the heat capacity of

diamond and values calculated on the Einstein model (using 𝜃𝐸 =

1320𝐾), it was observed that:

i) High T behavior reasonably agree with exp’t

ii) Low T behavior “fails” (i.e. 𝐶𝑉 → 0 too quickly as 𝑇 → 0 )

Q. What is (are) the limitation(s) of Einstein’s law of specific heat?

4.5. Debye’s theory

Despite its success in reproducing the approach of CV 0 as T  0, the

Einstein model is clearly deficient at very low T. What might be wrong with the assumptions it makes?

 3𝑁 independent oscillators, all with the same frequency



 Discrete allowed energies:

𝐸

_𝑛

= 𝑛ℏ𝜔

𝑛 = 0,1,2, …

 

 

_T

 

T

T

T

T

V

E E E E E

e

R

e

e

R

T

C

₂

2

/

/

/

2

3

3

)

(

















At low temperatures, the optical branch phonons have energies higher than 𝐾_𝐵𝑇, and therefore, optical branch waves are not excited. Only acoustic waves

contribute to the heat capacity. For an acoustic branch 𝜔 → 0, as 𝑘 → 0 and the Einstein model fails to include this feature in its model. The Debye model assumes that the lattice waves are elastic waves (one longitudinal branch and two transverse branches) and the frequency here is not a constant but a specific distribution with a cut off frequency of 𝜔_𝐷 above which no shorter wave phonons are excited.

In the Debye model, the density of states 𝑔(𝜔) can be defined as

𝑔(𝜔) = {

3𝜔2 𝜔_𝐷3

,

𝑜 ,

𝜔 < 𝜔_𝐷

𝜔 < 𝜔

_𝐷 (4.26)

We may define 𝜃_𝐷 the Debye temperature as

𝜃

_𝐷

=

ℏ𝜔𝐷

𝑘

(4.27)

We get the Debye specific heat capacity as

(4.28)

Where, 𝑓_𝐷(θ𝐷

𝑇) is called the Debye specific heat capacity function. When𝑇 >> 𝜃𝐷,

𝐶_𝑣approaches the value 3𝑁𝑘. But, when 𝑇 << 𝜃_𝐷, i.e. at low temperatures theDebye

specific heat equation becomes as

(4.29)

Here, 𝐶_𝑣 is proportional to 𝑇3 this is known as the Debye 𝑇3 law. At lower temperatures the Debye approximation is best suited because almost all excited phonons belong to the long-wavelength waves in the acoustic branches and the crystal behaves like a continuum.

4.6. Thermal conductivity

When the two ends of the sample of a given material are at two different temperatures, say 𝑇₁ and 𝑇₂ such that 𝑇₂ > 𝑇₁ , heat flows down the thermal gradient, i.e. from the hotter to the cooler end. Observations show that the heat current density 𝐣 (amount of heat flowing across unit area per unit time) is proportional to the temperature gradient (𝑑𝑇/𝑑𝑥). That is,

𝑗 = −𝐾

𝑑𝑇

𝑑𝑥

(4.30)

The proportionality constant 𝐾, known as the thermal conductivity, is a measure of the ease of

transmission of heat across the bar (the minus sign is included so that 𝑲 is a positive quantity).

Heat may be transmitted in the material by several independent agents. In metals, for example, heat is carried both by electrons and phonons, although the

contribution of the electrons is muchlarger. In insulators, on the other hand, heat is transmitted entirely by phonons, since there are nomobile electrons in these

substances. Hence, it isconvenient to consider the formation of phonon gas, which moves randomly in all thedirections corresponding to the wave vectors in the Brillouin zones.

The thermal conductivity is given by

𝐾 =

1

3

𝐶

𝑣

𝑙𝑣

(4.31)

where, 𝑪_𝒗is the specific heat per volume, 𝒗 is the velocity, 𝒍 is the mean freepath. Here

𝒗 and 𝒍 are the average quantities overall occupied modes and the Brillouinzones.Since mean free path (𝒍) depends strongly on temperature, thermal conductivity depends on temperature.

Questions on the explained topic

1. What is a phonon?

2. Explain classic description of oscillations of atoms.

3. What determines the Brillouin zone in oscillations of atoms?

4. Explain the models for specific heat capacity with their strengths and drawbacks.

5. How do phonon processes affect the thermal conductivity?

Chapter 5: Dielectric Properties of Solids

Introduction

Based on their electrical property materials may be classified in to two major groups: i. Conductors: have “unlimited” supply of charges that are free to move about

through the material. They have high conductivity (𝜎 ≫ 1).

ii. Nonconductors: Nonconducting materials are usually called insulators (dielectrics). They have low conductivity (𝜎 ≪ 1). In dielectrics, all charges are attached to specific atom or molecule loosely and move a bit within the atom or molecule.

In general, the main difference b/n a conductor and a dielectric is the number (availability) of free electrons in the outermost shell.

5.1. Review of Basic Formula

Electric field can exist in free space and material medium. There are two fundamental laws governing electrostatic fields:

i) Coulomb’s law

The electrostatic force b/n two charges is given by

𝐹 = 𝑘

𝑞1𝑞2

𝑟2

(5.1)

Where

𝒌 =

𝟏

𝟒𝝅𝝐_𝟎 is the electrostatic constant.

ii) Gauss’s law

Gauss’s law, states that the net flux through any closed surface is equal to the charge enclosed by the surface.

𝜙_𝐸 = ∮ 𝑫⃗⃗ . 𝑑𝑨⃗⃗ = 𝑞 (5.2)

Electric Field Induction (or) Flux density (or) Displacement Vector is defined as the number of electric lines of force received by a unit area (A).

𝐷 = 𝑞

𝐴 (5.3)

The electric flux density 𝑫⃗⃗ is related to the electric field intensity (in free space) as 𝑫⃗⃗ =𝝐𝟎𝑬⃗⃗⃗ (5.4)

The electric flux density is also called electric displacement. The unit of 𝑫⃗⃗ is 𝑪

𝒎𝟐 .

A dielectric material is a nonconducting substance whose bound charges are polarized under the influence of an external electric field. In a dielectric there are no free charges present, but bound charges, i.e. electric dipoles are present.

Figure 6: A dipole- a pair of two equal and opposite charges separated by a distance 'd’

The dipole moment of the dipole is the product of magnitude of either charge and their separation.

𝝁

⃗⃗ = 𝑞𝒅⃗⃗ (5.5) direction:

( − → + )

A dipole placed in an electric field will experience torque and hence rotate in the field.

Torque felt

𝝉

⃗ = 𝝁⃗⃗ 𝒙𝑬⃗⃗ (5.6)

Potential energy

𝑈 = −𝝁⃗⃗ . 𝑬⃗⃗ (5.7)

Polarization: The process of producing electric dipoles out of neutral atoms or molecules is known as polarization.The total dipole moment per unit volume is also defined as the dielectric polarization (𝑷⃗⃗ ).

𝒑

⃗⃗ = 𝑁𝝁⃗⃗ (5.8)

This is nothing but induced charge per unit area.Hence polarization is also defined as induced surface charge per unit area.

𝑃 =

𝑞𝑖𝑛

𝐴 (5.9)

The dipole moment density has unit 𝐶

𝑚2

.

Relation among E, D & P: Figure below shows a parallel plate capacitor with a dielectric, ‘𝑞’ is charge on each plate and ‘𝑞1 _{= 𝑞}

𝑖𝑛’ is the induced

charge on the surface of the dielctric.The net charge enclosed in the gaussian surface is (𝑞 − 𝑞_𝑖𝑛)

Figure 8: A parallel plate capacitor filled with a dielectric

From Gauss’ law,

∮ 𝑬

⃗⃗ . 𝑑𝑨

⃗⃗ =

𝑞𝑒𝑛𝑐_𝝐 𝟎

= (

𝑞−𝑞_𝑖𝑛𝑑 𝝐_𝟎

)

𝐸𝐴 = 𝑞 𝝐_𝟎− 𝑞_𝑖𝑛𝑑 𝝐_𝟎 (5.10)

Using, eqn. (5.3) and (5.9) in eqn. (5.10), we obtain

𝜖

₀𝐸 = 𝐷 − 𝑃 ; Hence

𝐷 =

𝜖

₀𝐸 + 𝑃 (5.11)

Again the electric displacement in a material can be related to the electric field strength

𝐷 =

𝜖

₀

𝐸 + 𝑃 = 𝜖𝐸 =

𝜖

₀

𝜖

_𝑟

𝐸

where

𝜖

_𝑟

=

𝜖

𝜖₀ is

relative permittivity/ dielectric constant.

Polarizability: The strength of the induced dipole moment is directly proportional to the strength of the external applied field, i.e. μ ∝ E.

𝜇 = 𝛼𝐸 (5.12)

where 𝛼 is known as dielectric polarizability.

Succeptibility(𝝌-'chi'): It is defined as polarization per unit electric field. The

electric susceptibility of the material, is more or less a measure of how susceptible

(or sensitive) a given dielectric is to electric fields.

𝜒

_𝑒

=

𝑃

𝜖₀𝐸 (5.13)

𝐷 = 𝜖

₀

𝐸 + 𝑁𝛼𝐸 = 𝜖

₀

(1 +

𝑁𝛼

𝜖₀

)𝐸

(5.14)

Since

𝐷 = 𝜖

₀

𝜖

_𝑟

𝐸

,

it follows that

𝜖

_𝑟

= (1 +

𝑁𝛼

𝜖₀

)

(5.15)

Define electrical susceptibility 𝜒 such that

𝑃 = 𝜖

₀

𝜒𝐸 = 𝑁𝛼𝐸

which gives

𝜒 =

𝑁𝛼

𝜖₀

(5.16)

Finally from eqn. (5.15) and (5.16), we can infer that

𝜖

_𝑟

= (1 + 𝜒)

(5.17)

5.2. The microscopic concept of polarization

The Electric field which appears in the Maxwell’s equation is macroscopic electric field 𝑬⃗⃗ . The value of the field that acts at the site of the atom in the crystal, called local (internal) field, is significantly d/t from that of the macroscopic field.

The total field acting at the atom site or dipole in a system is due to external sources and other dipoles present in the specimen. The local field

determines the dipole moment of the atom.

Consider a spherical cavity of radius r within a dielectric, which is placed between the two charged parallel plates.The total field experienced by it is

Figure 9: The internal electric field on an atom in a crystal is the sum of the external applied field 𝐸⃗⃗⃗⃗ and of the field due to the other atoms in the crystal. 0

𝐸⃗

_{𝑙𝑜𝑐𝑎𝑙}

= 𝐸⃗

₀

+ 𝐸⃗

₁

+ 𝐸⃗

₂

+ 𝐸⃗

₃

(5.18)

Here

𝐸⃗

₀

=

field produced by fixed charges external to the body

𝐸⃗

₁=depolarization field, from a surface charge density 𝒏.̂ 𝑷⃗⃗ on the outer surface of the specimen

𝐸⃗

₂

=

Lorentz cavity field: field from polarization charges on inside of a spherical cavity cut out of the specimen with the reference atom as center, as in Fig.4

𝐸⃗

₃

=

field of atoms inside cavity

Depolarization field,

𝑬

⃗⃗

_𝟏=−𝑷⃗⃗

3𝜖₀

Lorentz Field,

𝑬

⃗⃗

_𝟐= 𝑷⃗⃗

3𝜖₀

The field

𝑬

⃗⃗

_{𝟐, due to the polarization charges on the surface of the fictitious cavity is the} negative of the depolarization field

𝑬

⃗⃗

_{𝟏 in a polarized sphere, so that}

𝑬

⃗⃗

_𝟏

+ 𝑬

⃗⃗

_𝟐

= 𝟎

for a sphere.

Field of Dipoles inside Cavity, 𝑬⃗⃗ _𝟑

The field 𝑬⃗⃗ _𝟑, due to the dipoles within the spherical cavity is the only term that

depends on the crystal structure. But, for a reference site with cubic surroundings in a sphere 𝐸⃗ ₃ = 0. The total local field at a cubic site is then,

𝑬

⃗⃗

_{𝑙𝑜𝑐𝑎𝑙}

_{= 𝑬}

_{⃗⃗ +}

𝑷⃗⃗

3𝜖₀

(5.19)

This is the Lorentz relation: the field acting at an atom in a cubic site is the macroscopic field 𝑬 plus 𝑷⃗⃗

3𝜖₀ , from the polarization of the other atoms in the

specimen.

Where 𝐸⃗ _{𝑀𝑎𝑥𝑤𝑒𝑙𝑙} =

𝑬

⃗⃗

= 𝐸

⃗⃗ 0

+ 𝐸

⃗⃗ 1

𝑬

⃗⃗ _{𝑙𝑜𝑐𝑎𝑙= microscopic field which fluctuates within the medium}

5.3. Langevin’s theory of polarization in polar dielectrics

Polar molecules in liquids or gases such as 𝐻2𝑂 𝑜𝑟 𝑁𝐻3, have a permanent

dipole moment which can be partially aligned with an applied field. The alignment of polar molecules in an electric field is upset by random thermal motion, so that Maxwell-Boltzmann statistics can be used to find the average

polarization. The expression for the polarization can be obtained from

Langevin-Debye theory as

< 𝑃⃗ >=

𝜇2𝑬

3𝑘𝑇

(5.20)

This is the average moment per molecule in the direction of the applied field. The magnitude of the mean orientational polarizability is

𝛼

_𝑜

=

𝑃

𝐸

=

𝜇2

3𝑘𝑇

(5.21)

5.4. Clausius-Mosotti relation

Clausius-Mosotti equation gives the relation between the macroscopic quantity (dielectric constant of an insulator) and the microscopic quantity (polarization of atoms or molecules) of an insulator.

We know that, from Eqn. (5.11)

𝐷 =

𝜖

₀

𝐸 + 𝑃

. But, again

𝐷 = 𝜖𝐸

. Therefore,

𝜖𝐸 =

𝜖

₀

𝐸 + 𝑃

→ solving for 𝐸 yields

𝐸 =

𝑃

Substituting Eqn. (5.22) in (5.19) we obtain,

𝐸

_{𝑙𝑜𝑐𝑎𝑙}

=

𝑃

𝜖 − 𝜖

_𝑜

+

𝑃

3𝜖

₀

𝐸

_𝐿

=

_3𝜖𝑃 0

(

𝜖+2𝜖₀

𝜖−𝜖

𝑜

)

(5.23)

If 𝑁 is the number of molecules per unit volume, 𝛼 the molecular polarizability then

𝑃 = 𝑁𝛼

𝐸

_𝐿

→ 𝐸

_𝐿

=

𝑃

𝑁𝛼

(5.24)

From equations, (5.23) and (5.24)

𝑃

𝑁𝛼

=

𝑃

3𝜖

₀

(

𝜖 + 2𝜖

₀

𝜖 − 𝜖

_𝑜

)

Which upon rearrangement gives,

𝑁𝛼

3𝜖

₀= (

𝜖 − 𝜖

_𝑜

𝜖 + 2𝜖

₀)

Using the relation

𝜖

_𝑟

=

𝜖

𝜖₀ the above equation can be simplified to give

𝑁𝛼

3𝜖

₀

= (

𝜖

_𝑟

−1

𝜖

_𝑟+2

) (5.25)

Equation (5.25) is known as Clausius-Mosotti relation, which relates the macroscopic dielectric constant (

𝜖

_𝑟) with the microscopic polarizability (

𝛼

).

Terminal Questions and Problems

1. Distinguish b/n polar and nonpolar dielectrics.

2. Identify the different types of polarizabilities (electronic, ionic, etc…) 3. What ate the merits of using a dielectric in a capacitor?

4. Can 𝜖𝑟 attain a value less than 1?

5. A slab of dielectric material has a 𝜖𝑟 = 3.8 and contains a uniform

electric flux density of 8𝑛𝐶

𝑚2. If the material is lossless; find

a) 𝐸 b) 𝑃

6. Show that the potential energy of a dipole 𝝁 in an electric field 𝑬 may be written as −𝝁⃗⃗ . 𝑬⃗⃗ . Also show that if 𝛼 is the polarizability of an atom,

the energy of the atom in an electric field 𝐸 is given by −(𝛼/2)𝐸2.