Recommended Textbooks for PHY3012 Solid State Physics at Queen’s University Belfast, Study notes of Solid State Physics

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Queen’s University Belfast

School of Mathematics and Physics

PHY3012 SOLID STATE PHYSICS

A T Paxton, November 2012

Books

The primary textbook for this course is

H Ibach and H L¨uth, Solid State Physics, Springer, 4th Edition, 2009

Additionally, I recommend

J R Hook and H E Hall, Solid State Physics, Wiley, 2nd Edition, 2000 S Elliott, The Physics and Chemistry of Solids, Wiley, 1998 C Kittel, Introduction to Solid State Physics, Wiley, 8th Edition, 2005

Slightly more advanced texts that I recommend are

M P Marder, Condensed Matter Physics, Wiley, 2000 N Ashcroft and N D Mermin, Solid State Physics, Holt-Sanders, 1976

• 1. Lagrangian and Hamiltonian mechanics
• 1. Lattice vibrations and phonons
  • 1.1 Vibrations in the monatomic lattice
  • 1.2 Notes on section 1.1
  • 1.3 Boundary conditions and density of states in three dimensions
  • 1.4 A useful identity
  • 1.5 Vibrations in the diatomic lattice
  • 1.6 Phonons
    • 1.6.1 The simple harmonic oscillator
    • 1.6.2 Quantisation of the lattice waves
    • 1.6.3 The Einstein model
    • 1.6.4 The Debye model
  • 1.7 Particle–particle interactions—phonon scattering
  • 1.8 Anharmonic effects
    • 1.8.1 Phonon–phonon scattering
• 2. Electrons
  • 2.1 The hydrogen molecule
    • 2.1.1 The H+ 2 molecular ion
    • 2.1.2 The hydrogen molecule
      • 2.1.2.1 The molecular orbital approach
      • 2.1.2.2 The valence bond approach
      • 2.1.2.3 Molecular orbital and valence bond pictures compared
  • 2.2 Free electron gas
  • 2.3 Effects of the crystal lattice
  • 2.4 The tight binding picture
    • 2.4.1 Case study—Si and other semiconductors
    • 2.4.2 Case study—transition metals
  • 2.5 Magnetism
    • 2.5.1 The localised picture
    • 2.5.2 The itinerant picture
    • 2.5.3 Pauli paramagnetism
    • 2.5.4 The free electron band model of ferromagnetism and Stoner enhancement
• 3. Thermal and electrical conductivity—transport
  • 3.1 Scattering
  • 3.2 Electrical conduction in semiconductors
    • 3.2.1 Fermi level and number of carriers in a semiconductor
    • 3.2.2 Conductivity of a semiconductor
    • 3.2.3 Comparison between a metal and a non degenerate semiconductor
  • 3.3 Electron dynamics in metals
    • 3.3.1 Wavepackets
    • 3.3.2 Velocity of free electrons
    • 3.3.3 Equation of motion of a wavepacket
  • 3.4 Scattering using the “kinetic method”—relaxation time
    • 3.4.1 Relaxation time
    • 3.4.2 The kinetic method
  • 3.4.3 Mean free path
  • 3.4.4 Thermal conductivity
  • 3.4.5 The role of Umklapp processes in phonon thermal conductivity
• 3.5 Boltzmann formulation of electrical conductivity in metals
  • 3.5.1 Derivation of the Boltzmann equation
  • 3.5.2 Solution of the linearised Boltzmann equation
• 3.6 The Wiedemann–Franz law
• 3.7 Plausibility argument for the linearised Boltzmann equation
• 3.8 Thermoelectric effects
  • 3.8.1 Estimate of the thermopower in metals and semiconductors
  • 3.8.1 The Peltier effect
• 3.9 Onsager relations

There follows a list of the most commonly used quantities, with occasional comparison with the usage in the textbooks, Ibach and L¨uth (IL), Kittel (K), Hook and Hall (HH) and Elliott (E). I will always put vectors in bold type face, and their magnitude in normal typeface, e.g., p = |p|. The complex conjugate of a complex number is indicated by a bar over it, for example, ψ¯.

a spacing between lattice planes, lattice constant, acceleration ai basis vectors of the direct lattice a, b, c (HH, E) a, a+^ destruction and creation operators for phonons A force constant matrix, cubic coefficient in heat capacity, exchange integral bi basis vectors of the reciprocal lattice gi (IL), a∗, b∗, c∗^ (HH, E) B magnetic induction c speed of sound C spring constant, heat capacity Ce heat capacity due to electrons ce heat capacity per unit volume due to electrons Cph heat capacity due to phonons cph heat capacity per unit volume due to phonons e charge on the electron is −e e base of the natural logarithms E eigenvalue, especially of an electron state E (IL),  (K), ε (HH), E (E) EF Fermi energy Ec energy at conduction band edge Ev energy at valence band edge Eg semiconductor energy gap E electric field E (IL), E (HH, K, E) f (E), f (k), fFD Fermi–Dirac function F force g reciprocal lattice vector G (IL, K, E), Ghkl (HH) ge electron g-factor ¯h reduced Planck constant ¯h = h/ 2 π h hopping integral H hamiltonian, magnetic field i

i, j, l labels of atoms I Stoner parameter Je electric current density JQ heat current density k wavevector, vector in reciprocal space kB Boltzmann constant kF Fermi wavevector L Lagrangian, length `, m, n direction cosines m mass of the electron, magnetic moment per atom / μB m = n↑ − n↓ m∗^ effective mass M mass of an atom

PHY3012 Solid State Physics Page 1

Introduction

We are concerned with the physics of matter when it is in the solid, usually crystalline, state. You are faced with the complex problem of describing and understanding the properties of an assembly of some 10^23 − 1024 atoms, closely bonded together. The subject can be divided into two sub disciplines.

1. The nature of the bonding (mechanical properties etc.)
2. The physics of solids (spectroscopy, energy bands, transport properties etc.)

We will deal largely with the second. Much of the physics of solids can be understood by regarding the solid as matter containing certain elementary excitations. These inter- act with externally applied fields (including electromagnetic radiation) as well as each other to produce the observable phenomena which we measure. Examples of elementary excitations are

electrons and holes phonons plasmons magnons and spin waves

We will study electrons and phonons. Generally we will describe the mechanics of a single particle at a time. Many particle physics is usually beyond the scope of an undergraduate physics programme.

PHY3012 Solid State Physics, Section 0 Page 2

1. Lagrangian and Hamiltonian mechanics

A body with mass M moves in one dimension. The position coordinate at time t is denoted q. By Newton’s second law, we have

M

d^2 q dt^2

≡ M¨q = F (q)

where F (q) is the force exerted on the body when it is at position q. We take it that this force arises from the fact that the body is moving in some potential W. For example, here is a simple harmonic potential, W ∝ q^2 :

q

potential energy W

The force is, therefore,

F (q) = −

dW dq

and the kinetic energy is

K =

mv^2 =

M

dq dt

M q˙^2

where v is the velocity of the body.

We define a function called the Lagrangian as

L(˙q, q) = K − W (0.1)

the difference between the kinetic and potential energies. If we differentiate L with respect to ˙q we get ∂L ∂ q˙

d d˙q

M q˙^2

= M q˙ (0.2)

from which it follows that

d dt

∂L

∂ q˙

= M¨q (mass × acceleration)

PHY3012 Solid State Physics, Section 1.1 Page 4

1. Lattice vibrations and Phonons

Phonons are quantised lattice vibrations. We will pursue the following steps.

1. Set up the problem of lattice waves using classical, Newtonian mechanics and the elementary properties of waves.
2. Solve the problem for a simple model. This is a typical approach in solid state physics.
3. Describe the important consequences of crystal periodicity.
4. Quantise the waves.

1.1 Vibrations in the monatomic lattice

Vibrations of the atoms in a crystal take the form of waves, standing or travelling, whose amplitude is the largest displacement of an atom from its equilibrium position. In addition to its amplitude a wave is characterised by its wavevector, k, and angular frequency, ω = 2π × frequency.

The atoms’ official, crystallographic positions are denoted rj. j runs from 1 to Nc, where Nc is the number of unit cells in the crystal; and to make the notation simple, we consider the case where there is only one atom per unit cell. The atomic displacements are denoted δrj (‘δ’ meaning ‘a small change in... ’) and can be written

δrj =

Nc

q ˜k ei(k·rj^ −ωt)^ (1. 1 .1)

q˜k is called the polarisation vector or vector amplitude; 1/

Nc is only for normalisation. The wavelength is

λ =

2 π k

The angular frequency is a function of the wavelength, thus

ω = ω(k)

and it is essentially this “dispersion relation” we shall be looking for; t is the time.

These waves are excited by temperature or externally applied fields or forces. The crystal has lowest potential energy when all the δrj are zero. We call this energy

W = W (r 1 , r 2 ,... , rNc )

and then expand the potential energy of the crystal with lattice waves in a Taylor series,

Upot = W +

j

∂W

∂rj

δrj +

ij

δri

∂^2 W

∂ri∂rj

δrj +... (1. 1 .2)

PHY3012 Solid State Physics, Section 1.1 Page 5

where the partial derivatives are evaluated at the equilibrium positions of the atoms.† Now the first term can be taken as zero, and this defines our zero of energy as the potential energy of the crystal at rest. The second term is also zero because the crystal is in equilibrium with respect to small atomic displacements. If we truncate the Taylor series after the second order terms we are said to be working within the harmonic approximation.

Let us write down the hamiltonian in this approximation,

H = kinetic energy + potential energy

=

j

p^2 j 2 M

ij

Aij δriδrj (1. 1 .3)

Aij = ∂ (^2) W ∂ri∂rj is called the force constant matrix. It is the change in potential energy when atom i is moved to ri + δri and atom j is simultaneously moved moved to rj + δrj. M is the mass of each atom. (Note, Aij really should have further indices indicating the directions in which atoms i and j are moved. A typical matrix element is Aix,jy for example. I can leave these out for simplicity and anyway we’ll only end up dealing with one-dimensional cases.)

The force constant matrices can only be determined if we have a complete knowledge of the bonding in the solid.

Next, we want the equation of motion for atom j. In classical, hamiltonian mechanics, we have d dt

pj = −

∂H

∂rj

= M

d^2 dt^2

δrj

rate of change of momentum = force acting upon atom j = mass × acceleration

Differentiating equation (1.1.3) we get

M

d^2 dt^2

δrj = −

∂H

∂rj

i

Aij δri

We can easily differentiate equation (1.1.1) twice with respect to time:

d^2 dt^2

δrj = −ω^2

Nc

˜qk ei(k·rj^ −ωt)

† (^) As a matter of notation, when I write, say,

∑ i

∂W ∂ri^ δri

I really mean ∑ iα

∂W ∂riα δriα

and α = 1, 2 , 3 (or x, y, z) is an index to each component of the vector.

PHY3012 Solid State Physics, Section 1.1 Page 7

Inserting this into equation (1.1.1), I immediately get

δrj+1 = δrj eika^ (1. 1 .5)

which is a mathematical statement of the same assertion.

The simple model that we can solve requires us to suppose that there are only forces between neighbouring planes of atoms. This is a lousy approximation, but it will serve to illustrate much of the essential physics of lattice waves. It’s as if the atoms were connected by springs with a spring constant (stiffness) C:

δrj δrj+

j − 1 j j + 1 j + 2

In this diagram, each solid dot represents a whole plane of atoms.

Considering only neighbouring planes, there will be contributions to the potential energy, proportional to the difference in displacement squared, from each spring.

U (^) potj,j+1 = C (δrj − δrj+1)^2

We sum this over all planes and divide by two (otherwise we’ve counted each plane twice)

Upot =

C

j

(δrj − δrj+1)^2

C

j

δr j^2 + δr^2 j+1 − 2 δrj δrj+

= C

j

(δrj δrj − δrj δrj+1)

You can see how I got from the second to the third line by noticing that the first two terms in the second line are identical—a sum of squares of all the δrj. Now compare the above equation with equation (1.1.3) which states, in one dimension,

Upot =

ij

Aij δri δrj

You see that we can identify all the Aij for our simple model:

Aii = 2C Ai,i+1 = Ai,i− 1 = −C all other Aij = 0

PHY3012 Solid State Physics, Section 1.2 Page 8

so we can solve the equation of motion (1.1.4) for longitudinal modes

M ω L^2 =

i

Aij eik(rj^ −ri)

There are only three non zero terms on the right hand side; k is parallel to r, and let us write the distance between neighbouring atoms as rj − ri = a, the lattice constant. We get

M ω^2 L = C

2 − eika^ − e−ika

and therefore by elementary trigonometry, and remembering that eiθ^ = cos θ + i sin θ

ωL =

C

M

∣sin

ka

This is the dispersion relation for longitudinal waves in our model crystal. The vertical bars denote “absolute value,” and we don’t expect ω to be negative.

Transverse waves will have exactly the same solution but with a different, presumably smaller, spring constant C′

ωT =

C′

M

∣sin

ka

We can sketch our dispersion relation, roughly like this:

T

√√ √√ C′ M

√√ √√ C M

−π a 0 π a

L

k

PHY3012 Solid State Physics, Section 1.2 Page 10

Both waves carry the same information about the atom displacements. Only wave- lengths longer than 2a are needed to represent the motion.

4. Only lattice waves with wavevectors in the first Brillouin zone are physically mean- ingful. Are all values of k in this range allowed? No. There is a finite number that depends on the size of the crystal, but not on the boundary conditions. What does this mean?

Suppose our crystal has length L in the direction of k and L = Nata with Nat = Nc the number of planes. We are essentially considering a one dimensional line of atoms.

What shall we say about the displacements at the surfaces of the crystal, i.e., atoms 1 and Nat? We could clamp them (i.e., set δr 1 = δrNat = 0) rather like the boundary conditions on the wavefunctions in an infinite potential well (i.e., ψ(r 1 ) = ψ(rNat ) = 0).

Or, we could impose cyclic, or periodic, or Born–von K´arm´an boundary conditions; i.e., set δr 1 = δrNat not necessarily zero According to eq (1.1.5) δrj+1 = δrj eika therefore δrj+2 = δrj+1 eika^ = δrj ei2ka and if I keep this up, we get

δrNat = δr 1 eiNatka^ = δr 1 eikL = δr 1 according to our boundary conditions

Thus eikL^ = 1 which is only true if kL = 2πn (1. 1 .6)

PHY3012 Solid State Physics, Section 1.3 Page 11

where n is any integer, and so the allowed values of k are those for which

k =

2 πn L

lying between −π/a and π/a.

They form a dense set of points separated by an interval 2π/L which decreases as the size of the crystal increases. But there is a finite number of allowed values of k (allowed states).

We can write down the number of states per unit k from (1.1.6)

dn dk

L

2 π

and the total number of states is this times the allowed range of k,

dn dk

×

2 π a

L

2 π

×

2 π a

L

a

= Nat

The number of allowed states is equal to the number of atoms in the line.

Very often we want to know the number of states in a range of frequency, or the density of states, n(ω) =

dn dω

dn dk

×

dk dω =

L

2 π

×

dω/dk dω/dk is the group velocity and is only known once the we know the dispersion relations.

1.3 Boundary conditions and density of states in three dimensions

Density of states is a very important concept in solid state physics for all elementary excitations as is the evaluation of the number of states, so let’s do the thing again, now in 3-D.

What do we imply by imposing periodic boundary conditions? In general we deal with a piece of crystal of some shape, bounded by a number of crystal surfaces. But bulk prop- erties must be independent of the details of the surface and its environment. Therefore

PHY3012 Solid State Physics, Section 1.3 Page 13

in which δij is the “Kronecker delta” (equal to one if i = j and zero otherwise).

To satisfy the three conditions, k must be some fraction of a reciprocal lattice vector

k =

m 1 N 1

b 1 +

m 2 N 2

b 2 +

m 3 N 3

b 3

You can see that this does indeed meet the three conditions—take the scalar product of k with any of the direct lattice vectors:

k · aj = 2π

i

mi Ni

δij

Thus the allowed values of k form a fine mesh of points in the reciprocal lattice.

But we also know that k must remain within the first Brillouin zone and this has the same volume in k-space as the parallelepiped formed from the vectors b 1 , b 2 and b 3.

Hence all the allowed values of m 1 , m 2 and m 3 are given by the conditions

0 < m 1 ≤ N 1 0 < m 2 ≤ N 2 0 < m 3 ≤ N 3

Therefore the number of allowed states is

N 1 N 2 N 3 = Nc

the number of unit cells in the crystal. We already got this resut in the 1-D case.

Let us denote the volume of the unit cell by vc. The volume of the “embedded” crystal is

V = Ncvc

and the volume of the first Brillouin zone is

(2π)^3

vc

PHY3012 Solid State Physics, Section 1.3 Page 14

If there are Nc k vectors allowed in the first Brillouin zone the “volume per state” in k-space is

1 Nc

(2π)^3 vc

(2π)^3 V

The “spottiness of k-space” is such that the allowed states are a mesh of points each occupying a volume

(2π)^3 V

(Compare this with the result ∆k = 2π/L that we got in 1D.)

The number of allowed states having wavevector magnitude smaller than some kmax is the number of states enclosed by a sphere of radius kmax.

That is,

number of allowed states with wavevector less than kmax =

volume of sphere

× number of states per unit volume

Nst =

4 π 3

kmax^3

V

(2π)^3

=

π^2

V k^3 max (1. 3 .1)

We define the density of states for lattice waves in terms of the number of states in a narrow range of frequencies dω about ω.