Patterson, Bailey - Solid State Physics Introduction to theory

			4.1 Particles and Interactions of Solid-state Physics (B)	215
			Table 4.2. Solid-state particles and related quantities
	Bogolon		Elementary energy excitations in a superconductor. Linear combina-
	(or	Bogoliubovtions of electrons in (+k, +), and holes in (−k, −) states. See Chap. 8.
	quasiparticles)		The + and − after the ks refer to “up” and “down” spin states.
	Cooper pairs		Loosely coupled electrons in the states (+k, +), (−k, −). See Chap. 8.
	Electrons		Electrons in a solid can have their masses dressed due to many inter-
			actions. The most familiar contribution to their effective mass is due
			to scattering from the periodic static lattice. See Chap. 3.

Mott–Wannier andThe Mott–Wannier excitons are weakly bound electron-hole pairs Frenkel excitons with energy less than the energy gap. Here we can think of the binding as hydrogen-like except that the electron–hole attraction is screened by the dielectric constant and the mass is the reduced mass of the effective electron and hole masses. The effective radius of this exciton is the Bohr radius modified by the dielectric constant and effective

reduced mass of electron and hole.

Since the static dielectric constant can only have meaning for dimensions large compared with atomic dimensions, strongly bound excitations as in, e.g., molecular crystals are given a different name Frenkel excitons. These are small and tightly bound electron–hole pairs. We describe Frenkel excitons with a hopping excited state model. Here we can think of the energy spectrum as like that given by tight binding. Excitons may give rise to absorption structure below the bandgap. See Chap. 10.

Helicons	Slow, low-frequency (much lower than the cyclotron frequency),
	circularly polarized propagating electromagnetic waves coupled to
	electrons in a metal that is in a uniform magnetic field that is in the
	direction of propagation of the electromagnetic waves. The frequency
	of helicons is given by (see Chap. 10)

	ωH =	ωc (kc)2
		.
		ω2p
Holes	Vacant states in a band normally filled with electrons. See Chap. 5.
Magnon	The low-lying collective states of spin systems, found in ferromag-
	nets, ferrimagnets, antiferromagnets, canted, and helical spin arrays,

whose spins are coupled by exchange interactions are called spin waves. Their quanta are called magnons. One can also say the spin waves are fluctuations in density in the spin angular momentum. At very long wavelength, the magnetostatic interaction can dominate exchange, and then one speaks of magnetostatic spin waves. The dispersion relation links the frequency with the reciprocal wavelength, which typically, for ordinary spin waves, at long wavelengths goes as the square of the wave vector for ferromagnets but is linear in the wave vector for antiferromagnets. The magnetization at low tempera-

tures for ferromagnets can be described by spin-wave excitations that reduce it, as given by the famous Bloch T3/2 law. See Chap. 7.

216 4 The Interaction of Electrons and Lattice Vibrations

	Table 4.2. (cont.)
Neutron	Basic neutral constituent of nucleus. Now thought to be a composite
	of two down quarks and one up quark whose charge adds to zero.
	Very useful as a scattering projectile in studying solids.
Acoustical phonons	Sinusoidal oscillating wave where the adjacent atoms vibrate in phase
	with the frequency, vanishing as the wavelength becomes infinite. See
	Chap. 2.
Optical phonons	Here the frequency does not vanish when the wavelength become
	infinite and adjacent atoms tend to vibrate out of phase. See Chap. 2.
Photon	Quanta of electromagnetic field.
Plasmons	Quanta of collective longitudinal excitation of an electron gas in
	a metal involving sinusoidal oscillations in the density of the electron
	gas. The alkali metals are transparent in the ultraviolet, that is for
	frequencies above the plasma frequency. In semiconductors, the
	plasma edge in absorption can occur in the infrared. Plasmons can be
	observed from the absorption of electrons (which excite the plasmons)
	incident on thin metallic films. See Chap. 9.
Polaritons	Waves due to the interaction of transverse optical phonons with trans-
	verse electromagnetic waves. Another way to say this is that they are
	coupled or mixed transverse electromagnetic and mechanical waves.
	There are two branches to these modes. At very low and very high
	wave vectors the branches can be identified as photons or phonons but
	in between the modes couple to produce polariton modes. The cou-
	pling of modes also produces a gap in frequency through which radia-
	tion cannot propagate. The upper and lower frequencies defining the
	gap are related by the Lyddane–Sachs–Teller relation. See Chap. 10.
Polarons	A polaron is an electron in the conduction band (or hole in the valence
	band) together with the surrounding lattice with which it is coupled.
	They occur in both insulators and semiconductors. The general idea is
	that an electron moving through a crystal interacts via its charge with
	the ions of the lattice. This electron–phonon interaction leads to
	a polarization field that accompanies the electron. In particle lan-
	guage, the electron is dressed by the phonons and the combined parti-
	cle is called the polaron. When the coupling extends over many lattice
	spacings, one speaks of a large polaron. Large polarons are formed in
	polar crystals by electrons coulombically interacting with longitudinal
	optical phonons. One thinks of a large polaron as a particle moving in
	a band with a somewhat increased effective mass. A small polaron is
	localized and hops or tunnels from site to site with larger effective
	mass. An equation for the effective mass of a polaron is:
	mpolaron m	1			,
		1 −	α
			6

where α is the polaron coupling constant. This equation applies both to small and large polarons.

		4.1 Particles and Interactions of Solid-state Physics (B)	217
		Table 4.2. (cont.)
	Polarons summary	(1) Small polarons: α > 6. These are not band-like. The transport
		mechanism for the charge carrier is that of hopping. The electron
		associated with a small polaron spends most of its time near a particu-
		lar ion.
		(2) Large polarons: 1 < α < 6. These are band-like but their mobility is
		low. See Chap. 4.
	Positron	The antiparticle of an electron with positive charge.
	Proton	A basic constituent of the nucleus thought to be a composite of two up
		and one down quarks whose charge total equals the negative of the

charge on the electron. Protons and neutrons together form the nuclei of solids.

Table 4.3. Distinctions that are sometimes made between solid-state quasi particles (or “particles”)

1.	Landau	quasi	Quasi electrons interact weakly and have a long lifetime provided
	particles		their energies are near the Fermi energy. The Landau quasi electrons
			stand in one-to-one relation to the real electrons, where a real electron
			is a free electron in its measured state; i.e. the real electron is already
			“dressed” (see below for a partial definition) due to its interaction
			with virtual photons (in the sense of quantum electrodynamics), but it
			is not dressed in the sense of interactions of interest to solid-state
			physics. The term Fermi liquid is often applied to an electron gas in
			which correlations are strong, such as in a simple metal. The normal
			liquid, which is what is usually considered, means as the interaction is
			turned on adiabatically and forms the one-to-one correspondence, that
			there are no bound states formed. Superconducting electrons are not
			a Fermi liquid.
2.	Fundamental		Quasi particles (e.g. electrons): These may be “dressed” electrons
	energy	excita-	where the “dressing” is caused by mutual electron–electron interac-
	tions	from	tion or by the interaction of the electrons with other “particles.” The
	ground state of		dressed electron is the original electron surrounded by a “cloud” of
	a solid		other particles with which it is interacting and thus it may have
			a different effective mass from the real electron. The effective interac-
			tion between quasi electrons may be much less than the actual interac-

tion between real electrons. The effective interaction between quasi electrons (or quasi holes) usually means their lifetime is short (in other words, the quasi electron picture is not a good description) unless their energies are near the Fermi energy and so if the quasi electron picture is to make sense, there must be many fewer quasi electrons than real electrons. Note that the term quasi electron as used here corresponds to a Landau quasi electron.

218 4 The Interaction of Electrons and Lattice Vibrations

		Table 4.3. (cont.)
2.	(cont.)	Collective excitations (e.g. phonons, magnons, or plasmons): These
		may also be dressed due to their interaction with other “particles.” In
		this book these are also called quasi particles but this practice is not
		followed everywhere. Note that collective excitations do not resemble
		a real particle because they involve wave-like motion of all particles in
		the system considered.
3.	Excitons	and Note that excitons and bogolons do not correspond either to a simple
	bogolons	quasi particle (as discussed above) or to a collective excitation. How-
		ever, in this book we will also call these quasi particles or “particles.”
4.	Goldstone	Quanta of long-wavelength and low-frequency modes associated with
	boson	conservation laws and broken symmetry. The existence of broken
		symmetry implies this mode. Broken symmetry (see Sect. 7.2.6)
		means quantum eigenstates with lower symmetry than the underlying
		Hamiltonian. Phonons and magnons are examples.

Once we know something about the interactions, the question arises as to what to do with them. A somewhat oversimplified viewpoint is that all solid-state properties can be discussed in terms of fundamental energy excitations and their interactions. Certainly, the interactions are the dominating feature of most transport processes. Thus we would like to know how to use the properties of the interactions to evaluate the various transport coefficients. One way (perhaps the most practical way) to do this is by the use of the Boltzmann equation. Thus in this chapter we will discuss the interactions, the Boltzmann equation, how the interactions fit into the Boltzmann equation, and how the solutions of the Boltzmann equation can be used to calculate transport coefficients. Typical transport coefficients that will be discussed are those for electrical and thermal conductivity.

The Boltzmann equation itself is not very rigorous, at least in the situations where it will be applied in this chapter, but it does yield some practical results that are helpful in interpreting experiments. In general, the development in this whole chapter will not be very rigorous. Many ideas are presented and the main aim will be to get the ideas across. If we treat any interaction with great care, and if we use the interaction to calculate a transport property, we will usually find that we are engaged in a sizeable research project.

In discussing the rigor of the Boltzmann equation, an attempt will be made to show how its predictions can be true, but no attempt will be made to discover the minimum number of assumptions that are necessary so that the predictions made by use of the Boltzmann equation must be true.

It should come as no surprise that the results in this chapter will not be rigorous. The systems considered are almost as complicated as they can be: they are interacting many-body systems, and nonequilibrium statistical properties are the properties of interest. Low-order perturbation theory will be used to discuss the interactions in the many-body system. An essentially classical technique (the Boltzmann equation) will be used to derive the statistical properties. No precise statement of the errors introduced by the approximations can be given. We start with the phonon–phonon interaction.

4.2 The Phonon–Phonon Interaction (B) 219

4.2 The Phonon–Phonon Interaction (B)

The mathematics is not always easy but we can see physically why phonons scatter phonons. Wave-like motions propagate through a periodic lattice without scattering only if there are no distortions from periodicity. One phonon in a lattice distorts the lattice from periodicity and hence scatters another phonon. This view is a little oversimplified because it is essential to have anharmonic terms in the lattice potential in order for phonon–phonon scattering to occur. These cause the first phonon to modify the original periodicity in the elastic properties.

4.2.1 Anharmonic Terms in the Hamiltonian (B)

From the Golden rule of perturbation theory (see for example, Appendix E), the basic quantity that determines the transition probability from one phonon state (|i ) to another (|f ) is the matrix element | i|H1|f |2, where H 1 is that part of the Hamiltonian that causes phonon–phonon interactions.

For phonon–phonon interactions, the perturbing Hamiltonian H 1 is the part containing the cubic (and higher if necessary) anharmonic terms.

H	1	α,β,γ	α	β γ	(4.1)
		= ∑lbl′b′l′′b′′Ulbl′b′l′′b′′xlb xl′b′xl′′b′′ ,

α,β,γ

where xα is the αth component of vector x and U is determined by Taylor’s theorem,

α,β,γ

1

∂

3

U

≡

V

,

(4.2)

lbl′b′l

′′b′′

3!

∂xαlb∂x

β′

′∂xγ′′ ′′

l b

l b

all xlb =0

and the V is the potential energy of the atoms as a function of their position. In practice, we generally do not try to calculate the U from (4.2) but we carry them along as parameters to be determined from experiment.

As usual, the mathematics is easier to do if the Hamiltonian is expressed in terms of annihilation and creation operators. Thus it is useful to work toward this end by starting with the transformation (2.190). We find,

	H 1 =	1		∑q,b,q′,b′q′′,b′′∑l ,l′,l′′exp[−i(q l + q′ l′ + q′′ l′′)]
		N 3 2
				α,β,γ			(4.3)
		α,β,γ			α	β	γ
	×Ulbl′b′l′′b′′Xq′,b Xq′′,b′Xq′′′,b′′.
	In (4.3) it is convenient to make the substitutions l′ = l + m′, and l″ = l + m″:
	H ′ =			1	∑q,b,q′,b′,q′′,b′′∑l exp[−i(q + q′ + q′′) l]
			N 3 2
					α,β,γ		(4.4)
				α	β	γ	α,β,γ

× Xq′,b Xq′′,b′Xq′′′,b′′Dq,b,q′,b′,q′′,b′′.

220 4 The Interaction of Electrons and Lattice Vibrations

where

Dα,β,γ q,b,q′,b′,q′′,b′′

could be expressed in terms of the U if necessary, but its fundamental property is that

				α,β,γ			,			(4.5)
				Dq,b,q′,b′,q′′,b′′ ≠ f (l)
because there is no preferred lattice point.
We obtain
	1		1	G	α	β	γ		α,β,γ
H		=		∑q,b,q′,b′,q′′,b′′δq+nq′+q′′Xq′,b Xq′′,b′Xq′				′′,b′′Dq,b,q′,b′,q′′,b′′ .		(4.6)
			N1/ 2
				α,β,γ

In an annihilation and creation operator representation, the old unperturbed Hamiltonian was diagonal and of the form

H 1 =	1	∑q, p (aq†, paq, p +	1	) ωq, p .	(4.7)
	N1/ 2
		2

The transformation that did this was (see Problem 2.22)

Xq′,b = −i∑ p eq,b, p 2m ω	(aq†, p − a−q, p ) .	(4.8)
b	q, p

Applying the same transformation on the perturbing part of Hamiltonian, we find

	H 1 = ∑q, p,q′, p′,q′′, p′′δqG+nq′+q′′	(aq†, p − a−q, p )(aq†′, p′ − a−q′, p′)	(4.9)
	× (aq†′, p′ − a−q′, p′)M q, p,q′, p′,q′′, p′′,
	where
	M q, p,q′, p′,q′′, p′′ =	α,β,γ	(4.10)
		f (Dq,b,q′,b′,q′′,b′′) ,

i.e. it could be expressed in terms of the D if necessary.

4.2.2 Normal and Umklapp Processes (B)

Despite the apparent complexity of (4.9) and (4.10), they are in a transparent form. The essential thing is to find out what types of interaction processes are allowed by cubic anharmonic terms. Within the framework of first-order timedependent perturbation theory (the Golden rule) this question can be answered.

4.2 The Phonon–Phonon Interaction (B) 221

In the first place, the only real (or direct) processes allowed are those that conserve energy:

Etotal	= Etotal .	(4.11)
initial	final

In the second place, in order for the process to proceed, the Kronecker delta function in (4.9) says that there must be the following relation among wave vectors:

q + q′+ q′′ = Gn .

(4.12)

Within the limitations imposed by the constraints (4.11) and (4.12), the products of annihilation and creation operators that occur in (4.9) indicate the types of interactions that can take place. Of course, it is necessary to compute matrix elements (as required by the Golden rule) of (4.9) in order to assure oneself that the process is not only allowed by the conservation conditions, but is microscopically probable. In (4.9) a term of the form a†q,pa–q′,p′a–q″,p″ occurs. Let us assume all the p are the same and thus drop them as subscripts. This term corresponds to a process in which phonons in the modes −q′ and −q″ are destroyed, and a phonon in the mode q is created. This process can be diagrammatically presented as in Fig. 4.1. It is subject to the constraints

q = −q′ + (−q′′) +Gn and ωq = ω−q′ + ω−q′′ .

If Gn = 0, the vectors q, −q′, and −q″ form a closed triangle and we have what is called a normal or N-process. If Gn ≠ 0, we have what is called a U or umklapp process.2

−q ′

q

−q ″

Fig. 4.1. Diagrammatic representation of a phonon–phonon interaction

Umklapp processes are very important in thermal conductivity as will be discussed later. It is possible to form a very simple picture of umklapp processes. Let us consider a two-dimensional reciprocal lattice as shown in Fig. 4.2. If k1 and k2 together add to a vector in reciprocal space that lies outside the first Brillouin zone, then a first Brillouin-zone description of kl + k2, is k3, where kl + k2 = k3 − G. If kl and k2 were the incident phonons and k3 the scattered phonon, we would call such a process a phonon–phonon umklapp process. From Fig. 4.2 we

2Things may be a little more complicated, however, as the distinction between normal and umklapp may depend on the choice of primitive unit cell in k space [21, p. 502].

222 4 The Interaction of Electrons and Lattice Vibrations

k1	k2
k3

Fig. 4.2. Diagram for illustrating an umklapp process

see the reason for the name umklapp (which in German means “flop over”). We start out with two phonons going in one direction and end up with a phonon going in the opposite direction. This picture gives some intuitive understanding of how umklapp processes contribute to thermal resistance. Since high temperatures are needed to excite high-frequency (high-energy and thus probably large wave vector) phonons, we see that we should expect more umklapp processes as the temperature is raised. Thus we should expect the thermal conductivity of an insulator to drop with increase in temperature.

So far we have demonstrated that the cubic (and hence higher-order) terms in the potential cause the phonon–phonon interactions. There are several directly observable effects of cubic and higher-order terms in the potential. In an insulator in which the cubic and higher-order terms were absent, there would be no diffusion of heat. This is simply because the carriers of heat are the phonons. The phonons do not collide unless there are anharmonic terms, and hence the heat would be carried by “phonon radiation.” In this case, the thermal conductivity would be infinite.

Without anharmonic terms, thermal expansion would not exist (see Sect. 2.3.4). Without anharmonic terms, the potential that each atom moved in would be symmetric, and so no matter what the amplitude of vibration of the atoms, the average position of the atoms would be constant and the lattice would not expand.

Anharmonic terms are responsible for small (linear in temperature) deviations from the classical specific heat at high temperature. We can qualitatively understand this by assuming that there is some energy involved in the interaction process. If this is so, then there are ways (in addition to the energy of the phonons) that energy can be carried, and so the specific heat is raised.

The spin–lattice interaction in solids depends on the anharmonic nature of the potential. Obviously, the way the location of a spin moves about in a solid will have a large effect on the total dynamics of the spin. The details of these interactions are not very easy to sort out.

More generally we have to consider that the anharmonic terms cause a temperature dependence of the phonon frequencies and also cause finite phonon lifetimes.

4.2 The Phonon–Phonon Interaction (B) 223

We can qualitatively understand the temperature dependence of the phonon frequencies from the fact that they depend on interatomic spacing that changes with temperature (thermal expansion). The finite phonon lifetimes obviously occur because the phonons scatter into different modes and hence no phonon lasts indefinitely in the same mode. For further details on phonon–phonon interactions see Ziman [99].

4.2.3 Comment on Thermal Conductivity (B)

In this Section a little more detail will be given to explain the way umklapp processes play a role in limiting the lattice thermal conductivity. The discussion in this Section involves only qualitative reasoning.

Let us define a phonon current density J by

Jph = ∑q′, p q′ Nq′p ,

(4.13)

where Nq,p is the number of phonons in mode (q, p). If this quantity is not equal to zero, then we have a phonon flux and hence heat transport by the phonons.

Now let us consider what the effect of phonon–phonon collisions on Jph would be. If we have a phonon–phonon collision in which q2 and q3 disappear and ql appears, then the new phonon flux becomes

J ′ph = q1	( Nq p +1) + q2 ( Nq			2	p −1)
	1									(4.14)
+ q3( Nq			p −1) + ∑q(≠q ,q				,q		), p q Nq, p.
		3				2		3
				1

Thus

Jph′ = q1 − q2 − q3 + Jph .

For phonon–phonon processes in which q2 and q3 disappear and ql appears, we have that

q1 = q2 + q3 +Gn ,

so that

Jph′ = Gn + Jph .

Therefore, if there were no umklapp processes the Gn would never appear and hence J′ph would always equal Jph. This means that the phonon current density would not change; hence the heat flux would not change, and therefore the thermal conductivity would be infinite.

The contribution of umklapp processes to the thermal conductivity is important even at fairly low temperatures. To make a crude estimate, let us suppose that the temperature is much lower than the Debye temperature. This means that small q are important (in a first Brillouin-zone scheme for acoustic modes) because these are the q that are associated with small energy. Since for umklapp processes q + q′ + q″ = Gn, we know that if most of the q are small, then one of the phonons involved in a phonon–phonon interaction must be of the order of Gn, since the wave vectors in the interaction process must add up to Gn.

224 4 The Interaction of Electrons and Lattice Vibrations

By use of Bose statistics with T << θD, we know that the mean number of phonons in mode q is given by

				1
	Nq =					exp(−	ωq / kT ) .	(4.15)
			exp(	ωq / kT ) −1

Let ωq be the energy of the phonon with large q, so that we have approximately

		ωq kθD ,	(4.16)
so that
		q exp(−θD / T ) .	(4.17)
	N

The more N¯qs there are, the greater the possibility of an umklapp process, and since umklapp processes cause Jph to change, they must cause a decrease in the thermal conductivity. Thus we would expect at least roughly

		q K −1 ,	(4.18)
	N

where K is the thermal conductivity. Combining (4.17) and (4.18), we guess that the thermal conductivity of insulators at fairly low temperatures is given approximately by

K exp(θD / T ) .

(4.19)

More accurate analysis suggests the form should be Tnexp(FθD/T), where F is of order 1/2. At very low temperatures, other processes come into play and these will be discussed later. At high temperature, K (due to the umklapp) is proportional to T−1. Expression (4.19) appears to predict this result, but since we assumed T << θD in deriving (4.19), we cannot necessarily believe (4.19) at high T.

It should be mentioned that there are many other types of phonon–phonon interactions besides the ones mentioned. We could have gone to higher-order terms in the Taylor expansion of the potential. A third-order expansion leads to three phonon (direct) processes. An N th-order expansion leads to N phonon interactions. Higher-order perturbation theory allows additional processes. For example, it is possible to go indirectly from level i to level f via a virtual level k as is illustrated in Fig. 4.3.

k

i

f

Fig. 4.3. Indirect i → f transitions via a virtual or short-lived level k