Kosevich A.M. The crystal lattice (2ed., Wiley, 2005)(ISBN 3527405089)(342s)_PSa_

6.6 Quantization of Elastic Deformation Field 179

The transition from the quantum mechanics of a discrete crystal lattice to the quantum field theory of elastic deformations can be carried out using a scalar crystal model. This allows one to exclude cumbersome expressions and calculations that refer to a vector quantum field.

A classical field of elastic deformations is the field of the function of coordinates and time u(r, t). The Lagrange function of this field is given by (2.9.20) and corresponds to the following density of the Lagrange function

L =	1		∂u 2			1		∂u	2
		ρ			−		G		.	(6.6.1)
	2		∂t			2		∂xi

It is easy to write the energy of a classical elastic field as

	E =	1		∂u 2		+	1		∂u	2
			ρ					G		dV.	(6.6.2)
		2		∂t			2		∂xi

We also find the momentum of the deformation field. By definition, the field momentum

P =	−		∂L			grad	u dV	,	(6.6.3)
		∂		∂u
				∂t

and this definition is independent of a specific form of the Lagrange function density. Using (6.6.1) we obtain

P = −	ρ	∂u	grad u dV.	(6.6.4)
		∂t

We note that the field momentum (6.6.4) does not coincide with the momentum of atoms involved in motion by elastic crystal vibrations. Indeed the momentum (6.6.4), as well as the energy (6.6.2) and the Lagrange function (2.9.20) in the harmonic approximation is quadratic in the derivatives of the function u, and the momentum of the atoms is proportional to the first degree of atomic velocity and is, thus, linear in the time derivatives of the atomic displacement vector.

According to the results of quantizing crystal lattice motion in quantum mechanics, the function u(r) is replaced by an operator dependent on coordinates as on parameters. The occupation number representation is the simplest way to write this operator, since we know its expansion (6.1.15) in the lattice.

If we take explicitly into account the finite dimensions of the volume V, then the formula (6.1.15) may also be used in the case of an elastic deformation field by replacing mN → ρV and assuming the radius vector r = r(n) to be a continuously varying quantity:

u(r) =	h¯	∑	1		(ak + a†	k )eikr ,	(6.6.5)
	2ρV
				ω(k)
		k			−

where the operators ak and a†k obey the commutation relations (6.1.8).

180 6 Quantization of Crystal Vibrations

The expansion (6.6.5) is usually performed not in a finite volume, but in all coordinate space, (V → ∞), which results in a continuous k-space. In this case, instead of the sum over discrete k in (6.6.5), it is necessary to introduce an integral, using the rule (2.5.4), and to redefine the phonon creation and annihilation operators

		2π 3/2			a†		2π	3/2
	ak =			b(k),		=		b†(k)	(6.6.6)
		L					L
					k

(we have denoted V = L3 ). Simultaneously it is necessary to replace the Kronecker symbol in (6.1.8) by the δ-function

δkk → (2Vπ)3 δ(k − k ).

The operators b(k) and b†(k) have commutation relations that directly generalize (6.1.8)

[b(k), b†(k)] = δ(k − k ),				[b(k), b(k )] = [b†(k), b†(k )] = 0.						(6.6.7)
On performing the above renormalization, we obtain
							d3 r
1				h¯
u(r) =									[b(k) + b†(−k)]eikr .	(6.6.8)
	(2π)3/2			2ρ
								ω(k)

As a result of the quantization, the field function u(r, t) describing the elastic displacements of atoms and satisfying the dynamic equation (2.9.21) is represented as an integral of linear form in the operators b(k) and b†(k). Thus, it becomes an operator.

The time evolution of the operator u(r, t) is determined by the time dependence of the operators b(k) and b†(k), and the latter in a harmonic approximation is based on

the relations

bk (t) = bk (0)e−iω(k)t, b†k (t) = b†k (0)eiω(k)t, where ω(k) = sk.

Using (6.6.8) and the expressions for the energy (6.6.2), it is easy to construct the

Hamiltonian of a vibrating continuum
H = E0 + hskb¯ †(k)b(k) d3 k,	(6.6.9)

where E0 is the energy of zero vibrations.

The eigenstates of the operator (6.6.9) contain quanta (phonons) with rigorously fixed quasi-momenta. The wave functions of these states in the coordinate representation correspond to plane waves.

6.6 Quantization of Elastic Deformation Field 181

We shall clarify ultimately the quantum meaning of the field momentum operator (6.6.3) or (6.6.4). We calculate the spatial and time derivatives of (6.6.5)

grad u = i

h¯

∑

k

ak

+ a†

k eikr

;

2ρV

ω(k)

k

−

(6.6.10)

∂u

h¯

= i

∑

a†

k − ak eikr .

ω(k)

∂t

2ρV

k

−

We substitute (6.6.10) into (6.6.4)

P =

1

∑h¯ k

ω(k)

a†−k

− ak

ak + a†−k

2

ω(k )

kk

∞

1

e

i(k+k )r

dV

(6.6.11)

×V

1

1

−∞

=

∑h¯ k ak a†k + a†k ak

+

∑h¯ k a†

k a†k + ak a−k .

2

2

k

k

−

The last sum in (6.6.11) vanishes, as its terms are the odd functions of k, and the first sum transforms trivially

P = ∑h¯ ka†k ak = h¯ kb†(k)b(k) d3 k.	(6.6.12)
k

It follows from (6.6.12) that the operator of the field momentum P is the operator of the total quasi-momentum of a vibrating crystal.

Writing the operators (6.6.8), (6.6.9), (6.6.12) and also the commutation rules (6.6.7) accomplishes the quantization of a field of elastic deformations.

7

Interaction of Excitations in a Crystal

7.1

Anharmonicity of Crystal Vibrations and Phonon Interaction

In a harmonic approximation the phonons are noninteracting quasi-particles. However, the situation is changed if the anharmonicity of crystal vibrations is taken into account – the phonon gas ceases to be ideal.

To clarify the qualitative role of the anharmonicity, we consider an unbounded crystal with a monatomic spatial lattice. We assume the anharmonicity to be small and in the expansion of the potential crystal energy in power of the displacement, we restrict ourselves to cubic terms:

U = U0 +	1	∑αik(n − n )uik(n)uk (n )
	2
+	1	∑S˜ γikl(n − n , n − n ) ui(n) − ui(n )	(7.1.1)
	3
× uk(n) − uk(n ) ul (n ) − ul (n ) ,

where ˜ is the symmetrization operation in number vectors , , . This takes into

S n n n

account at once the potential energy invariance with respect to crystal motion as a whole. The coefficients γikl (n, n ) characterize the intensity of the crystal vibration anharmonicity. Generally, they are of the order of magnitude γ α/a.

After introducing normal coordinates and quantizing the vibrations, the quadratic term in (7.1.1) will belong to the Hamiltonian of the phonon ideal gas. The cubic term in (7.1.1) will be quantized by (6.1.16). Denoting the corresponding term in the

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich Copyright c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9

+ a†−k )(ak + a†−k ),

184 7 Interaction of Excitations in a Crystal

Hamiltonian by Hint, we obtain

	h¯	3/2			eie e
Hint =			∑	√	j l	Γijl(k, k , k ) aα(k) + a†α(−k)
	2mN
					ωω ω
						(7.1.2)
× aα (k ) + a†α (−k ) aα (k ) + a†α (−k ) ∑ei(k+k +k )r(n),
						n
where ω = ωα(k), ω			= ωα (k ), ω = ωα (k ), e = eα(k), e = eα (k ),

e = eα (k ). The tensor function Γijl(k, k , k ) is defined by the double sum over the sites

Γijl(k, k , k ) = S˜ ∑ γijl(n, n ) 1−e−ikr	1−e−ik r e−ik r −e−ik r ,
n,n
where r = r(n ) and the symmetrization is	performed in quasi-wave vectors

k, k , k .

If one takes the Hamiltonian (7.1.2) into account in deriving the equations of motion for the operators ak and a†k , then instead of a system of separable equations for individual phonons such as (6.1.18), we get a nonlinear system of “coupled” equations that will relate the evolution of the phonon in the state (αk), with the evolution of the remaining phonons in the other states. As a result, the operator Hint can be regarded as the phonon interaction operator. Thus, the phonon interaction displays the anharmonicity of crystal vibrations.

Let us note that the last multiplier in (7.1.2) is nonzero and equals N only for k + k + k = G, where G is any reciprocal lattice vector. Thus, (7.1.2) can be

written as				V(k, k , k )
Hint =	√1	∑
				ω(k)ω(k )ω(k )
	N k+k +k =G
			(7.1.3)

×(ak + a†−k )(ak

where for simplicity, we omit the indices that number the branches of vibrations over which the summation should also be performed.

In (7.1.3), a new notation is used

V(k, k , k )	≡ Vαα α (k, k , k )
	1			h¯	3/2	(k)ej					(7.1.4)
=					ei		(k )el	(k )Γ	ijl	(k, k	, k ).
		N		2m
					α	α	α

The summation in (7.1.3) is carried out under the condition

k + k + k = G,

(7.1.5)

7.1 Anharmonicity of Crystal Vibrations and Phonon Interaction 185

and since each of the vectors k, k , and k does not go beyond the unit cell then (7.1.5) reduces to the two alternative conditions

k + k + k = 0,	(7.1.6)
k + k + k = G0,	(7.1.7)

where G0 is a vector that can be only composed of the three fundamental vectors of a reciprocal lattice b1, b2 and b3.

If the phonon interaction takes place under the condition (7.1.6), it is called a normal collision (or N-process). If the condition (7.1.7) is satisfied, the interaction is called an anomalous collision or an umklapp process (U-process, from the German word

Umklappprozess).

Let us note a remarkable feature of the coefficients V(k, k , k ) in the interaction Hamiltonian (7.1.3), associated with their behavior at small values of quasi-wave vectors. It follows from the definition (7.1.5) that a function of three independent variables V(k, k , k ) vanishes if at least one of these variables tends to zero. For ak 1 this function can be represented in the form

Vαα α (k, k , k ) = eiα(k)eαj (k

where M is some tensor coefficient.

In other words, for k, k , k → 0 we have

V(k, k , k ) kk k

)elα (k )Mijlnpqknkpkq, (7.1.8)

ωω ω .

(7.1.9)

Such a behavior of the corresponding matrix elements in the long-wave limit is quite natural. Indeed, in view of the limit k → 0, for estimations one can make use of the results of elasticity theory. However, in elasticity theory the crystal energy is expressed through the strain tensor. The cubic anharmonicity corresponds to terms of third order in deformations in the elastic energy. The deformation tensor for a plane wave of displacements can be estimated as εij iuj kiuj. Thus, the cubic anharmonicity is characterized by an intensity (7.1.9) at small k. This property of the quantities V(k, k , k ) is used essentially in evaluating the contribution of the vibration anharmonicity to the different macroscopic characteristics of the crystal.

We denote temporarily a−(k) = a(k) and analyze the characteristic product of the phonon creation and annihilation operators that enters into the Hamiltonian (7.1.3),

just a product of the type

a±α (k)a±α (k )a±α (k ). (7.1.10)

It follows from (6.1.21) that the nonzero matrix elements of the operator (7.1.10)

are proportional to the multiplier exp(±iωt ± iω t ± iω t), whereas before, ω =

ω(α, k); ω = ω(α , k ); ω = ω(α , k ).

186 7 Interaction of Excitations in a Crystal

If we now calculate by perturbation theory the probability of a corresponding collision using the matrix element of the operator (7.1.10), then it will be proportional to the δ-function

δ(±ω ± ω ± ω ).

(7.1.11)

The signs in (7.1.11) correspond to the ± signs in (7.1.10).

Let us consider for instance the term in Hint proportional to the operator a†(α, k)a†(α1 , k1 )a(α2 , −k2 ). This term describes the two-phonon creation process in the states (α, k) and (α1 , k1) and phonon vanishing in the state (α2, −k2 ). It gives nonzero probability under the condition

ω(α, k) + ω(α1, k1 ) = ω(α2, k2 ),

which is equivalent to the energy conservation under the collision

h¯ ω(α, k) = h¯ ω(α1 , k1) + h¯ ω(α2, k2 ).

(7.1.12)

By virtue of the conditions (7.1.4) and relation (6.2.4) the collision occurs either with total quasi-momentum conservation (N-process, Fig. 7.1a)

p + p1 = p2,

(7.1.13)

or when the quasi-momentum of “the phonon center of gravity” changes by h¯ G0 (U- process, Fig. 7.1b)

p + p1 = p2 + h¯ G0.

(7.1.14)

Fig. 7.1 Three-phonon processes: (a) is the normal process, (b) is the umklapp process.

The term with the operator a†k a−k a−k in the interaction Hamiltonian has an analogous meaning.

7.2

The Effective Hamiltonian for Phonon Interaction and Decay Processes

The term Hint(3) (cubic in anharmonicity and describing the phonon interaction) present in the crystal Hamiltonian affects both the dynamics of separate quasi-particles and the

7.2 The Effective Hamiltonian for Phonon Interaction and Decay Processes 187

equilibrium properties of a phonon gas. In particular, the crystal energy cannot now be represented as (6.2.3). However, if the anharmonicity is small, it can be taken into account by perturbation theory in calculating the crystal energy. Small corrections to the energy are generally expressed through the displacements of the eigenfrequencies of crystal vibrations. However, when they are calculated consecutively, it is impossible to confine oneself only to cubic anharmonicity.

Indeed, for the Hamiltonian (7.1.3) we have

Hint(3) = 0.

(7.2.1)

Thus, when taking cubic anharmonicity into account only, the correction to the first order of perturbation theory is absent. The frequency shift can be obtained in secondorder perturbation theory only, but the neglected fourth-order terms in the expansion

of the displacement energy would lead to the interaction Hamiltonian Hint(4) involving
the products of four operators a	k	and a†	of the type a±a±a±a±	. Thus,	(4)	= 0
		k	1 2 3 4		Hint

and the corresponding anharmonicity would result in the frequency renormalization just in the first order of perturbation theory. It turns out that the corrections to the energy in first-order perturbation theory coming from Hint(4) are generally of the same second-order corrections coming from Hint(3).

In this situation, it makes no sense to calculate the eigenfrequency shift with the Hamiltonian (7.1.3). We only note that in calculating a similar shift in the secondorder perturbation theory, all terms of the Hamiltonian (7.1.3) make nearly the same contribution to the final result. This is so because the virtual transitions in a system for which the phonon energy conservation law should not necessarily be satisfied are taken into account in the second-order perturbation theory. When real collisions of phonons conserving their energy are described the situation appears to be different.

Since we have defined the frequency ω = ω(α, k) as a positive quantity, the δ- function (7.1.11) may be nonzero only if the terms in its argument have different signs. Hence it follows that to describe real phonon collisions in a crystal with small anharmonicity (7.1.3), one can introduce a simpler effective interaction Hamiltonian. Indeed, if we restrict ourselves to the basic terms of the phonon interaction energy and use the first-order perturbation theory approximation (when real scattering processes are considered), we can omit in the Hamiltonian Hint the terms involving the products of the operators ak ak ak and a†k a†k a†k . The remaining terms, by a simple replacement

of the summation indeces, can be written in the form
Hintef	1			∑	W(k, k , k )			(a†k a†k a−k + a†k a−k ak ),
	=	√			√				(7.2.2)
			N			ωω ω

where the coefficients W(k, k , k ) differ from the V(k, k , k ) in numerical multipliers only, the summation is performed over the quasi-wave vectors under the condition (7.1.4) and the indices that number the vibration branches are omitted.

188 7 Interaction of Excitations in a Crystal

The permutation of the operators a†k and ak in deriving (7.2.2) gives no additional terms in Hintef , since the latter vanish by virtue of the conservation law (7.1.6) and the property (7.1.9) of the function V(k, k , k ).

We analyze the probability of the elementary process described by the Hamiltonian (7.2.2) by focusing on the long-wave acoustic phonons (ak 1). We note that if the long-wave phonons only participate in the collision, the process is normal. Thus, we shall first be interested in the probability of normal “three-phonon” processes.

Similar processes take place when the conservation laws (7.1.6), (7.1.12) are obeyed

k1 = k2 + k3 ; ω(α1 , k1) = ω(α2, k2 ) + ω(α3 , k3).

(7.2.3)

To clear up whether all conditions (7.2.3) can be satisfied simultaneously, in particular, whether a quite definite phonon (α, k) always vanishes (decays) with an arbitrarily small cubic anharmonicity, we discuss this problem qualitatively. For long-wave phonons, it suffices to consider the approximation where the dispersion laws are almost the same as for sound

ωα(k) = sαk, α = 1, 2, 3.

(7.2.4)

In such an approximation “longitudinal phonons” (l) exist whose dispersion law is close to the isotropic one (sl =constant) and “transverse” phonons (t) whose velocities satisfy the relation

st < sl .

(7.2.5)

Indeed, the true dispersion law differs insignificantly from (7.2.4) even for small k. For the longitudinal phonons, as a rule, ωl (k) < s1 k. Thus, for the process involving only the longitudinal phonons, (7.2.3) cannot be satisfied simultaneously. For phonons of the same type in the isotropic model from (7.2.3), (7.2.4) it follows that |k1 + k2 | = k1 + k2 . Thus, when the isotropic dispersion law (7.2.4) is obeyed, the process we are interested in would occur only in the case of parallel vectors k1 , k2 , k3. We use this fact to elucidate the possibility of longitudinal phonon decay in a one-dimensional process.

The dispersion law for longitudinal phonons is given by curve 1 in Fig. 7.2. We show the point (k1 , ω1) corresponding to the state of one of the phonons after the decay. Taking this point as the reference frame origin, starting at this point we construct the curve 2 for the same dispersion law of the second phonon after the decay. To satisfy (7.2.3), curves 1 and 2 should intersect and the intersection point will determine the state of a decaying phonon (k2 , ω2). These curves, however, do not intersect at small k and hence, the process l → l + l is impossible. In a scalar model (for the same type of phonons) the dispersion law analyzed would be nondecaying.

The conclusion about the nondecaying character of the dispersion law with the property (∂2 ω/∂k2 ) < 0 is also valid for an anisotropic crystal, if the isofrequency surfaces are convex. These include isofrequency surfaces for the branch of phonons that corresponds mainly to the longitudinal polarization of vibrations. However, the

7.2 The Effective Hamiltonian for Phonon Interaction and Decay Processes 189

Fig. 7.2 The nondecaying dispersion law.

crystal also involves phonons of another type, namely, transverse ones. By examining the dispersion law of bending vibrations of a layered crystal (Section 1.2.5) it has been established that in certain directions the dispersion law for transverse vibrations has the property ωt(k) > stk (Fig. 7.3a, curve 1). Let us check that in this case the t → t + t process is possible.

We choose the required direction in reciprocal space along the kx-axis and repeat the constructions just made above. Curves 1 and 2 may not intersect along the direction Okx. We consider a two-dimensional picture on the plane kxOky and construct isofrequency lines corresponding to the frequency ω2 for the dispersion law 1 in Fig. 7.3a and to the frequencies ω2 − ω for the dispersion law 2. These lines intersect (Fig. 7.3b) and the intersection points determine the wave vectors of a phonon capable of decaying: k = k1 + k2. Thus, the dispersion law running steeper than that of sound dispersion (∂2 ω/∂k2 > 0) is the decaying one.

Fig. 7.3 Phonon decay into two phonons of the same branch of vibrations: (a) the decaying dispersion law; (b) the intersection of isofrequency curves.