Condensed Matter Physics - Appunti

Condensed Matter Fundamentals

Tomarchio Luca

April 13, 2019

2

Contents

1 Atomic and Molecular Physics [1] 7

1.1 One-electron atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 Lifetimes of excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1.2 Fine and Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.1.3 Stark and Zeeman effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.4 Periodic system of the Elements . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Hybridization and Chemical bonds [14] . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Solids with Many Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Diatomic Molecular Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Solid State Physics [2] 17

2.1 The Drude-Sommerfeld (free electrons) model . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Classic coupling with EM fields [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Anomalous skin effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Classification of Bravais lattices . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3.2 Classification of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 Defects in Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Bloch Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5.1 Energy bands construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.2 Semiclassical model of electron’s dynamics . . . . . . . . . . . . . . . . . . . . 31

2.5.3 Semiclassical motion in a uniform electromagnetic field . . . . . . . . . . . . . 32

2.6 The Relaxation-Time Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6.1 DC and AC electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6.2 Thermal Conductivity & Thermopower . . . . . . . . . . . . . . . . . . . . . 35

2.7 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7.1 Lattice Specific Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.7.2 Measuring the phonon dispersion relations . . . . . . . . . . . . . . . . . . . . 39

2.7.3 Lattice Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.7.4 Phonons in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.8 Dielectric Properties of ionic Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.9 Homogeneous Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.10 Inhomogeneous Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.11 Diamagnetism and Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.11.1 Susceptibility of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.11.2 Quantum Oscillations of Electrons . . . . . . . . . . . . . . . . . . . . . . . . 51

2.12 Magnetic Properties of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3

4 CONTENTS

2.12.1 Insulating Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.12.2 Spin Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.12.3 Exchange in Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.13 Ferromagnetism [10] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.13.1 Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.14 Physics at Surfaces [4] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.14.1 Thermodynamics of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 Statistical Mechanics and Thermodynamics [5] [3] 61

3.1 Equilibrium theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2 Averages and Static Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Ordered Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 Second Order Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.1 Corrections to the Correlation Functions . . . . . . . . . . . . . . . . . . . . . 67

3.4.2 Frustration and Randomness . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.4.3 Landau-Ginsburg Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Magnetism: An annoying friend [10] 69

4.1 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.1 The Magnetization and H-field . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.2 Fields equations for Metals in Equilibrium [11] . . . . . . . . . . . . . . . . . 72

4.1.3 The Demagnetizing Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2 Magnetostatics Energies and Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Linear Optics [8] [9] 75

5.1 Ray Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1.1 Optical Systems Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.1.2 Matrix Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1.3 Aberrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.2.1 Gaussian Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3 Fourier Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3.1 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4 Electromagnetic Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4.1 Dielectric Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.4.2 Polarization of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.3 Optical Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.4.4 Polarizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.4.5 Anisotropic Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.5 Statistical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5.1 Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.5.2 Theory of Partial Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.6 Photon Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.6.1 Photon Streams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

CONTENTS 5

6 Non-linear Optics [16] 101

6.1 Frequency Domain Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

6.1.1 Time Domain Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.2 Wave-Equation Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2.1 Sum-Frequency Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.2.2 Upconversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2.3 Second Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.2.4 Optical Parametric Amplification . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.2.5 Optical Parametric Oscillators (OPO) . . . . . . . . . . . . . . . . . . . . . . 110

6.3 Non-linear Optical Interaction with Focused Gaussian Beams . . . . . . . . . . . . . 111

6.4 Quantum Theory of non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.4.1 Local Fields Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.5 The Intensity-Dependent Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . 115

6.5.1 Propagation through Isotropic non-linear Media . . . . . . . . . . . . . . . . 115

6.5.2 Self-Focusing of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.5.3 Self-Phase Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.6 Ultrafast Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7 Computational Methods [7] 121

7.1 Error Analysis and Probability Concepts [15] . . . . . . . . . . . . . . . . . . . . . . 122

7.1.1 Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.1.2 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.1.3 Limits of Sequences of Random Variables . . . . . . . . . . . . . . . . . . . . 124

7.1.4 Maximum Likelihood Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.2 Discrete Stochastic Processes [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.2.1 Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.3 Introduzione al Quantum Monte-Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . 126

7.3.1 Campionamento per importanza . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.3.2 Catene di Markov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

7.3.3 Algoritmo di Metropolis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6

7.4 T = 0: Path Integral Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7.4.1 Approssimazione primitiva . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.4.2 Stima dell’energia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

7.4.3 Applicazione a particelle indistinguibili . . . . . . . . . . . . . . . . . . . . . . 133

7.5 Ground State: Variational Quantum Monte-Carlo . . . . . . . . . . . . . . . . . . . . 137

7.5.1 Ottimizzazione della funzione d’onda: Correlated Sampling . . . . . . . . . . 140

7.6 Projection Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.6.1 Sistemi Fermionici: Problema del segno . . . . . . . . . . . . . . . . . . . . . 143

7.7 Diffusion Monte Carlo (DMC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.7.1 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.8 Ab-Initio Approaches to the Treatment of periodic Systems [17] . . . . . . . . . . . . 150

7.8.1 Lattice Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.8.2 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

7.8.3 Richiami teoria degli elettroni di Bloch . . . . . . . . . . . . . . . . . . . . . . 153

7.9 Teoria dello pseudo-potenziale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.9.1 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7.10 Density-Functional Perturbation Theory [18] . . . . . . . . . . . . . . . . . . . . . . 160

7.11 Somme di Ewald . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6 CONTENTS

8 Tables 165

Chapter 1

Atomic and Molecular Physics [1]

Before embarking in the description of atomic and molecular wave functions, it’s better to recall

some quantum mechanical fundamentals, such as the angular momentum. The quantum orbital

angular momentum maintains its classical shape, but with a redefined momentum operator

 

−i}(y∂ −

L = z∂ ) [L , L ] = i}L

x z y x y z

 

  2

× −i}(z∂ −

L = r p [L , L ] = 0

L = x∂ ) [L , L ] = i}L z

y x z y z x

 

−i}(x∂ −

L = y∂ ) [L , L ] = i}L

 

z y x z x y

2

hence, only a contemporary exact representation of an axis and L are possible. The simulta-

neous eigenfunctions are called the spherical harmonics, satisfying the eigenvalue equations

2 2 ∈ 3 ∈

L Y (θ, φ) = l(l + 1)} Y (θ, φ) L Y (θ, φ) = m}Y (θ, φ) l m [−l, l]

N Z

z

lm lm lm lm

Due to their dependence on trigonometric functions in θ, the functions can be re-expressed through

the Legendre polynomials 1/2

−

(2l + 1)(l m)! ∗

m imφ m

m ≥

Y (θ, φ) = (−1) P (cos θ)e Y (θ, φ) = (−1) Y (θ, φ) m 0

lm l,−m

l lm

4π(l + m)! 1

l

m Z

d 1 d 2

m 2 m/2 2 l

− −

P (x) = (1 x ) P (x) P (x) = (x 1) P (x)P (x)dx = δ

0 0

l l l l ll

l m l l

dx 2l + 1

2 l! dx −1

We define the raising and lowering operators as 1/2

± |lmi − ± |l(m ±

L = L iL L = + 1) m(m 1)] 1)i

}[l(l

± ±

x y 2

In some applications it’s convenient to use an alternative set of eigenfunctions of L , which are the

2

real forms of the spherical harmonics (figure below). These are eigenfunctions of L , but not L

z

z

(except for m = 0) and, due to their simple behaviour in the Cartesian space, they are well suited

for describing chemical bonds. 7

8 CHAPTER 1. ATOMIC AND MOLECULAR PHYSICS [?]

All the same properties can be exploited for the spin operator

2 2 ±

S χ = s(s + 1)} χ S χ = m S = S iS

}χ ±

s,m s,m z s,m s s,m x y

s s s s

If the system is an electron, we have only two possible spin states, up and down, and we can rep-

resent the states through spinors and square matrices

3 1 0 1 0

0

1 }

2 2 S =

S =

β = χ =

α = χ = } z

1/2,−1/2

1/2,1/2 −1

0 1 0

1

0 4 2

−i 1 0

0

0 1

}

S = σ =

σ =

σ σ = z

y

x −1

0

i 0

1 0

2

When a spin (or an angular momentum) is purely directed along z, the averages of the other

hS i hS i

axes goes to zero: = 0, = 0. Sometimes, is best to use a total angular momentum

x y

J = L + S, where all the operators of spin and angular momentum commute, obtaining

2 2 |l − |l −

J ψ = j(j + 1)} ψ J ψ = m j = s|, s| + 1, ..., l + s

}ψ

j,m j,m z j,m j j,m

j j j j

Considering a sum of any two angular momentums, the quantum numbers we can know in par-

2

allel are just the total J and M , and the components j and j . Through this knowledge, we

1 2

can build the total eigenfunction, using the Clebsh-Gordan coefficients, as the decomposition into

eigenfunctions for the summed systems

1.1. ONE-ELECTRON ATOMS 9

1.1 One-electron atoms

The Hamiltonian of the hydrogen atom can be splitted into three terms in Gaussian units

2 2 2

P p Ze mM

−

H = + µ =

2(M + m) 2µ r m + M

where P identifies the motion of the centre of mass, which behave like a free particle and can be

omitted. The central potential can be easily resolved through the decomposition of angular and

radial wave functions, obtaining 1/2

( )

3

− −

2Z (n l 1)! −ρ/2 2l+1

l

−

Ψ (r, θ, φ) = R (r)Y (θ, φ) R (r) = e ρ L (ρ)

nlm nl lm nl n+l

3

na 2n[(n + l)!]

µ

n−l−1 k 2

2 ρ 2Z m

[(n + l)!] }

X

2l+1 k+1

= ρ = r a = a a =

L (−1) µ 0 0

n+l 2

− − −

(n l 1 k)!(2l + 1 + k)! k! na µ me

µ

k=0

The three quantum numbers define the energy levels. The first, n, can assume every natural value,

and a single electron atom will have energies that depend only upon it

n−1

2

Z µ X 2

− ∈ −

E = degeneracy = (2l + 1) = n l [0, n 1]

n 2

2n m l=0

An atom with high principal quantum number is known as a Rydberg atom. Using the square

modulus of the wave functions, we can calculate the expectation values for different properties, for

instance 2

1 l(l + 1) Z

n −1

− hr i

hri 1+ 1 =

= a µ nlm

nlm 2 2

Z 2 n a n

µ

The easiest way to describe the interaction between radiation and a single electron atom is using

a semi-classical approach coupled with a time-dependent perturbative theory, due to a small inter-

action. Using an incoherent monochromatic radiation, we exploit the cross section for absorption

and emission between two states a and b as

2 2 2 2

4π α} 4π e }

spon

abs 2 stim abs 2

|M |M | −

σ = (ω )| σ = σ W = δ(ω ω )

ba ba ba ba

ab ab ba ab

2 2

m ω m 4π V ω

0

ba ba

where the latter relation is the rate of transition for spontaneous emission, and where M =

ba

hψ | · · ∇|ψ i.

exp(ik r)ˆ

This last contribution can be approximated through the dipole approx-

a

b

imation, i.e, an expansion of the exponential where just the order zero in maintained. This gives

rise to an electric dipole transition, while the first order causes the introduction of a magnetic

dipole and an electric quadrupole. The dipole approximation is easily written as

mω

ba

D · −ehψ |r|ψ i

ˆ D D =

M = a

ba ba b

ba }e

Under this approximation, we obtain the following selection rules

10 CHAPTER 1. ATOMIC AND MOLECULAR PHYSICS [?]

( k

∆m = 0 if

ˆ ẑ

±1

∆n = whatever ∆l = ±1 k

∆m = if k ẑ

The spin of the electron, instead, remains constant, while the conservation of angular momentum

is obtained through the helicity of the photons, resembling circular polarized radiation. Thus, the

spectrum of the single electron atom is 2

2

1 e

1 µ m

2 −

ν = Z R(M ) R(M ) = R(∞)