Patterson, Bailey - Solid State Physics Introduction to theory

4.3 The Electron–Phonon Interaction 225

There are a great many more things that could be said about phonon–phonon interactions, but at least we should know what phonon–phonon interactions are by now.

The following statement is by way of summary: Without umklapp processes (and impurities and boundaries) there would be no resistance to the flow of phonon energy at all temperatures (in an insulator).

4.3 The Electron–Phonon Interaction

Physically it is easy to see why lattice vibrations scatter electrons. The lattice vibrations distort the lattice periodicity and hence the electrons cannot propagate through the lattice without being scattered.

The treatment of electron–phonon interactions that will be given is somewhat similar to the treatment of phonon–phonon interactions. Similar selection rules (or constraints) will be found. This is expected. The selection rules arise from conservation laws, and conservation laws arise from the fundamental symmetries of the physical system. The selection rules are: (1) energy is conserved, and (2) the total wave vector of the system before the scattering process can differ only by a reciprocal lattice vector from the total wave vector of the system after the scattering process. Again it is necessary to examine matrix elements in order to assure oneself that the process is microscopically probable as well as possible because it satisfies the selection rules.

The possibility of electron–phonon interactions has been introduced as if one should not be surprised by them. It is perhaps worth pointing out that electron– phonon interactions indicate a breakdown of the Born–Oppenheimer approximation. This is all right though. We assume that the Born–Oppenheimer approximation is the zeroth-order solution and that the corrections to it can be taken into account by first-order perturbation theory. It is almost impossible to rigorously justify this procedure. In order to treat the interactions adequately, we should go back and insert the terms that were dropped in deriving the Born–Oppenheimer approximation. It appears to be more practical to find a possible form for the interaction by phenomenological arguments. For further details on electron–phonon interactions than will be discussed in this book see Ziman [99].

4.3.1 Form of the Hamiltonian (B)

Whatever the form of the interaction, we know that it vanishes when there are no atomic displacements. For small displacements, the interaction should be linear in the displacements. Thus we write the phenomenological interaction part of the Hamiltonian as

H ep = ∑l,b xl,b [ xl ,bU (re )]all xl,b =0 ,

(4.20)

where re represents the electronic coordinates.

226 4 The Interaction of Electrons and Lattice Vibrations

As we will see later, the Boltzmann equation will require that we know the transition probability per unit time. The transition probability can be evaluated from the Golden rule of time-dependent first-order perturbation theory. Basically, the Golden rule requires that we evaluate f|Hep|i , where |i and f| are formal ways of representing the initial and final states for both electron and phonon unperturbed states.

As usual it is convenient to write our expressions in terms of creation and destruction operators. The appropriate substitutions are the same as the ones that were previously used:

		xl,b =		1	∑q xq′,be−iq l ,
				N
x′	= −i	∑ p	e		2m ω	(a†	− a	−q, p	) .
q,b			q,b, p			q, p
					b	q, p

Combining these expressions, we find

xl,b = −i∑q, p 2Nm ω	e−iq l eq,b, p (aq†, p − a−q, p ) .	(4.21)
b	b

If we assume that the electrons can be treated by a one-electron approximation, and that only harmonic terms are important for the lattice potential, a typical matrix element that will have to be evaluated is

Tk,k′ ≡ nq, p | ∫ψk (r)H epψk′(r)dr | nq, p −1 ,

(4.22)

where |nq,p are phonon eigenkets and ψk(r) are electron eigenfunctions. The phonon matrix elements can be evaluated by the usual rules (given below):

n

p

−1 | a

q′, p′

| n

q, p

=

n

p

δ q′δ p′

,

(4.23a)

q,

q,

q

p

and

n

+1| a†

| n

p

=

n

+1 δ q′δ p′

.

(4.23b)

q, p

q′, p′

q,

q, p

q

p

Combining (4.20), (4.21), (4.22), and (4.23), we find

Tk,k′ = −i∑l,b

nq, p

e−iq l ∫

ψk

(r)eq,b, p [ x

U (r)]0ψk′(r)d3r .

l ,b

2Nmbωq,b

all space

(4.24)

Equation (4.24) can be simplified. In order to see how, let us consider a simple problem. Let

G = ∑l e−iql ∫L	f (x)Ul (x)dx ,	(4.25)
−L

4.3 The Electron–Phonon Interaction	227
where
f (x + la) = eikl f (x) ,	(4.26)

l is an integer, and Ul(x) is in general not a periodic function of x. In particular, let us suppose

	∂U
		,
Ul (x) ≡	∂x
	l xl =0

where

U (x, xl ) = ∑l exp[−K (x − dl )2 ] ,

and

dl = l + xl .

U(x, xl) is periodic if xl = 0. Combining (4.27) and (4.28), we have

Ul = +2K exp[−K (x −l)2 ](x −l)

≡F(x −l).

Note that Ul(x) = F(x − l) is a localized function.

Therefore we can write

G = ∑l e−iql ∫−LL f (x)F(x −l)dx .

In (4.31), let us write x′ = x − l or x = x′ + l. Then we must have

G = ∑l e−iql ∫−LL−−l l f (x′ + l)F(x′)dx′. Using (4.26), we can write (4.32) as

G = ∑l e−i(q−k)l ∫−LL−−l l f (x′)F(x′)dx′ .

(4.27)

(4.28)

(4.29)

(4.30)

(4.31)

(4.32)

(4.33)

If we are using periodic boundary conditions, then all of our functions must be periodic outside the basic interval −L to +L. From this it follows that (4.33) can be written as

G = ∑l e−i(q−k)l ∫L	f (x′)F(x′)dx′.	(4.34)
−L

The integral in (4.34) is independent of l. Also we shall suppose F(x) is very small for x outside the basic one-dimensional unit cell Ω. From this it follows that we can write G as

G (∫Ω f (x′)F(x′)dx′)(∑l e−i(q−k )l ).

(4.35)

228 4 The Interaction of Electrons and Lattice Vibrations

A similar argument in three dimensions says that

∑l,b e−iq l ∫all spaceψk (r)eq,b, p[ xl ,bU (r)]0ψk′(r

∑l,b e−i(k′−k −q) l ∫Ωψk (r)eq,b, p[ xl ,bU (r)]0ψ

Using the above, and the known delta function property of (4.24) becomes

Tk,k′ = −i	nq, p	N	δ	Gn	∫		∑	1
		2ωq,b				ψk		b m	eq,b, p [ x
				k′-k -q Ω
								b

)d3r

k′(r)d3r.

∑l eik·l, we find that

l,bU ]0ψk′d3r . (4.36)

Equation (4.36) gives us the usual but very important selection rule on the wave vector. The selection rule says that for all allowed electron–phonon processes; we must have

k′− k −q = Gn .

(4.37)

If Gn ≠ 0, then we have electron–phonon umklapp processes. Otherwise, we say we have normal processes. This distinction is not rigorous because it depends on whether or not the first Brillouin zone is consistently used.

The Golden rule also gives us a selection rule that represents energy conservation

Ek′ = Ek + ωq, p .

(4.38)

Since typical phonon energies are much less than electron energies, it is usually acceptable to neglect ωq,p in (4.38). Thus while technically speaking the electron scattering is inelastic, for practical purposes it is often elastic.3 The matrix element considered was for the process of emission. A diagrammatic representation of this process is given in Fig. 4.4. There is a similar matrix element for phonon absorption, as represented in Fig. 4.5. One should remember that these processes came out of first-order perturbation theory. Higher-order perturbation theory would allow more complicated processes.

It is interesting that the selection rules for inelastic neutron scattering are the same as the rules for inelastic electron scattering. However, when thermal neutrons are scattered, ωq,p is not negligible. The rules (4.37) and (4.38) are sufficient to map out the dispersion relations for lattice vibration. Ek, Ek′, k, and k′ are easily measured for the neutrons, and hence (4.37) and (4.38) determine ωq,p versus q for phonons. In the hands of Brockhouse et al [4.5] this technique of slow neutron diffraction or inelastic neutron diffraction has developed into a very powerful modern research tool. It has also been used to determine dispersion relations for magnons. It is also of interest that tunneling experiments can sometimes be used to determine the phonon density of states.4

3This may not be true when electrons are scattered by polar optical modes.

4See McMillan and Rowell [4.29].

4.3 The Electron–Phonon Interaction 229

k

k′

q

Fig. 4.4. Phonon emission in an electron–phonon interaction

k

k′

q

Fig. 4.5. Phonon absorption in an electron–phonon interaction

4.3.2 Rigid-Ion Approximation (B)

It is natural to wonder if all modes of lattice vibration are equally effective in the scattering of electrons. It is true that, in general, some modes are much more effective in scattering electrons than other modes. For example, it is usually possible to neglect optic mode scattering of electrons. This is because in optic modes the adjacent atoms tend to vibrate in opposite directions, and so the net effect of the vibrations tends to be very small due to cancellation. However, if the ions are charged, then the optic modes are polar modes and their effect on electron scattering is by no means negligible. In the discussion below, only one atom per unit cell is assumed. This assumption eliminates the possibility of optic modes. The polarization vectors are now real.

In what follows, an approximation called the rigid-ion approximation will be used to discuss differences in scattering between transverse and longitudinal acoustic modes. It appears that in some approximations, transverse phonons do not scatter electrons. However, this rule is only very approximate.

230 4 The Interaction of Electrons and Lattice Vibrations

So far we have derived that the matrix element governing the scattering is

T

=

n

q, p

N δ Gn

H k,k′

,

(4.39)

k,k′

2mωq, p

k′-k -q

q, p

where

Hqk,,pk′

=

∫

ψkeq, p ( x

l,b

U )0ψk′d3r

.

(4.40)

Ω

Equation (4.40) is not easily calculated, but it is the purpose of the rigid-ion approximation to make some comments about it anyway. The rigid-ion approximation assumes that the potential the electrons feel depends only on the vectors connecting the ions and the electron. We also assume that the total potential is the simple additive sum of the potentials from each ion. We thus assume that the potential from each ion is carried along with the ion and is undistorted by the motion of the ion. This is clearly an oversimplification, but it seems to have some degree of applicability, at least for simple metals. The rigid-ion approximation therefore says that the potential that the electron moves in is given by

U (r) = ∑l′va (r − xl′) ,

(4.41)

where va(r − xl′) refers to the potential energy of the electron in the field of the ion whose equilibrium position is at l′. The va is the cell potential, which is used in the Wigner–Seitz approximation, so that we have inside a cell,

		2	2 + v
	−			a	(r) ψ	k′	(r) = E	ψ	k′	(r) .	(4.42)
		2m
								k′

The question is, how can we use these two results to evaluate the needed integrals in (4.40)? By (4.41) we see that

	xl U = − r va ≡ − va .							(4.43)
What we need in (4.40) is thus an expression for va. That is,
	Hqk,,pk′		=		∫ ψk eq, p vaψk′d3r		.	(4.44)
					Ω

We can get an expression for the integrand in (4.44) by taking the gradient of (4.42) and multiplying by ψ*k. We obtain

ψ v

ψ

+ψ ( v

)ψ

=ψ

2

3ψ

+ E

ψ ψ

.

(4.45)

k′

a

k′

k′

k′

k a

k

k 2m

k′ k

Several transformations are needed before this gets us to a usable approximation: We can always use Bloch’s theorem ψk′ = eik′·r uk′(r) to replace ψk′ by

ψk′ = eik′ r uk′(r) + ik′ψk′ .

(4.46)

4.3 The Electron–Phonon Interaction 231

We will also have in mind that any scattering caused by the motion of the rigid ions leads to only very small changes in the energy of the electrons, so that we will approximate Ek by Ek′ wherever needed. We therefore obtain from (4.45), (4.46), and (4.42)

	ψk ( va )ψk′ =ψk	2	2 (eik′r uk′) −	2	( 2ψk )eik′r uk′ . (4.47)
		2m		2m

We can also write

2

2m ∫surface S{ψk [eik′r ( uk′)α ] − eik′r ( uk′)α ψk } dS

2

= 2m ∫ {ψk [eik′r ( uk′)α ] − eik′r ( uk′)α ψk }dτ

2

= 2m ∫{ψk 2[eik′r ( uk′)α ] − eik′r ( uk′)α 2ψk }dτ ,

since we get a cancellation in going from the second step to the last step. This means by (4.44), (4.47), and the above that we can write

	Hqk,,pk′		=	2	∫{ψk [eik′r (eq, p uk′)] − eik′r eq, p ( uk′) ψk } dS	.(4.48)
				2m

We will assume we are using a Wigner–Seitz approximation in which the Wigner–Seitz cells are spheres of radius r0. The original integrals in Hqk ,, pk ′ involved only integrals over the Wigner–Seitz cell (because va vanishes very far from the cell for va). Now uk′ ψk′ = 0 in the Wigner–Seitz approximation, and also in this approximation we know ( ψk′ = 0)r=r0 = 0. Since ψ0 = rˆ(∂ψ0 / ∂r), by the above reasoning we can now write

	Hqk,,pk′		=	∫ψkeik′r	2	2ψ0 (ek, p rˆ)dS	.	(4.49)
					2m

Consistent with the Wigner–Seitz approximation, we will further assume that va is spherically symmetric and that

2

2ψ

= [v

(r ) − E

]ψ

,

0

a

0

0

2m

0

which means that

Hqk,,pk′

=

[va (r0 ) − E0 ]∫ψkeik′rψ0eq, p rˆdS

]∫ψkψk′eq, p rˆdS

[va (r0 ) − E0

(4.50)

[va (r0 ) − E0

]∫

eq, p (ψkψk′)dτ

,

Ω

232 4 The Interaction of Electrons and Lattice Vibrations

where Ω is the volume of the Wigner–Seitz cell. We assume further that the main contribution to the gradient in (4.50) comes from the exponentials, which means that we can write

(ψ ψ

) i(k′ − k)ψ ψ

.

(4.51)

k k′

k k′

Finally, we obtain

Hqk,,pk′

=

eq, p (k′ − k) [va (r0 ) − E0 ]∫ψkψk′dτ

.

(4.52)

Neglecting umklapp processes, we have k′ − k = q so

Hqk,,pk′

eq, p q .

Since for transverse phonons, eq,p is perpendicular to q, eq,p · q = 0 and we get no scattering. We have the very approximate rule that transverse phonons do not scatter electrons. However, we should review all of the approximations that went into this result. By doing this, we can fully appreciate that the result is only very approximate [99].

4.3.3 The Polaron as a Prototype Quasiparticle (A)5

Introduction (A)

We look at a different kind of electron–phonon interaction in this section. Landau suggested that an F-center could be understood as a self-trapped electron in a polar crystal. Although this idea did not explain the F-center, it did give rise to the conception of polarons. Polarons occur when an electron polarizes the surrounding media, and this polarization reacts back on the electron and lowers the energy. The polarization field moves with the electron and the whole object is called a polaron, which will have an effective mass generally much greater than the electrons. Polarons also have different mobilities from electrons and this is one way to infer their existence. Much of the basic work on polarons has been done by Fröhlich. He approached polarons by considering electron–phonon coupling. His ideas about elec- tron–phonon coupling also helped lead eventually to a theory of superconductivity, but he did not arrive at the correct treatment of the pairing interaction for superconductivity. Relatively simple perturbation theory does not work there.

There are large polarons (sometimes called Fröhlich polarons) where the lattice distortion is over many sites and small ones that are very localized (some people call these Holstein polarons). Polarons can occur in polar semiconductors or in polar insulators due to electrons in the conduction band or holes in the valence band. Only electrons will be considered here and the treatment will be limited to Fröhlich polarons. Then the polarization can be treated on a continuum basis.

5 See, e.g., [4.26].

4.3 The Electron–Phonon Interaction 233

Once the effective Hamiltonian for electrons interact with the polarized lattice, perturbation theory can be used for the large-polaron case and one gets in a relatively simple manner the enhanced mass (beyond the Bloch effective mass) due to the polarization interaction with the electron. Apparently, the polaron was the first solid-state quasi particle treated by field theory, and its consideration has the advantage over relativistic field theories that there is no divergence for the self-energy. In fact, the polaron’s main use may be as an academic example of a quasi particle that can be easily understood. From the field theoretic viewpoint, the polarization is viewed as a cloud of virtual phonons around the electron. The coupling constant is:

α		=	1		1
	c
			8πε0		K (∞)

−	1		e2 2mω	L .
	K (0)		ωL

The K(0) and K(∞) are the static and high-frequency dielectric constants, m is the Bloch effective mass of the electron, and ωL is the long-wavelength longitudinal optic frequency. One can show that the total electron effective mass is the Bloch effective mass over the quantity 1 – αc/6. The coupling constant αc is analogous to the fine structure coupling constant e2/ c used in a quantum-electrodynamics calculation of the electron–photon interaction.

The Polarization (A)

We first want to determine the electron–phonon interaction. The only coupling that we need to consider is for the longitudinal optical (LO) phonons, as they have a large electric field that interacts strongly with the electrons. We need to calculate the corresponding polarization of the unit cell due to the LO phonons. We will find this relates to the static and optical dielectric constants.

We consider a diatomic lattice of ions with charges ±e. We examine the optical mode of vibrations with very long wavelengths so that the ions in neighboring unit cells vibrate in unison. Let the masses of the ions be m± and if k is the effective spring constant and Ef is the effective electric field acting on the ions we have (e > 0)

m+r+ = −k(r+ − r−) + eEf ,	(4.53a)
m−r− = +k(r+ − r−) − eEf ,	(4.53b)

where r± is the displacement of the ± ions in the optic mode (related equations are more generally discussed in Sect. 10.10).

Subtracting, and defining the reduced mass in the usual way (μ–1 = m+–1 + m––1), we have

μr = −kr + eEf ,	(4.54a)
where
r = r+ − r− .	(4.54b)

234 4 The Interaction of Electrons and Lattice Vibrations

We assume Ef in the solid is given by the Lorentz field (derived in Chap. 9)

Ef = E +	P	,	(4.55)
	3ε0

where ε0 is the permittivity of free space.

The polarization P is the dipole moment per unit volume. So if there are N unit cells in a volume V, and if the ± ions have polarizability of α± so for both ions α = α+ + α–, then

	N
P =		(er +αEf ) .
	V

Inserting Ef into this expression and solving for P we find:

	N		er +αE
P =				.
			1− (Nα / 3Vε0 )
	V

Putting Ef into Eqs. (4.54a) and (4.56) and using (4.57) for P, we find

			r = ar + bE ,
		P = cr + dE ,
where
b =					e / μ	,
			1− (Nα / 3Vε0 )
	N				e
c =							,
		V			1− (Nα / 3Vε0 )

and a and d can be similarly evaluated if needed. Note that

b = NVμ c .

(4.56)

(4.57)

(4.58a)

(4.58b)

(4.59a)

(4.59b)

(4.60)

It is also convenient to relate these coefficients to the static and high-frequency dielectric constants K(0) and K(∞). In general

	D = Kε0 E = ε0 E + P ,			(4.61)
	so
	P = (K −1)ε0 E .			(4.62)
	For the static case r¨ = 0 and
	r = −	b	E .	(4.63)
		a