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

158 5 Acoustics of Elastic Superlattices: Phonon Crystals

The periodic function U(x) should be expanded in the following Fourier series

			U(x) = ∑Ume2π mx/d,			(5.2.7)
			m
	where
		1		1	d
	Um =		U(x)e−2π imx/d dx =		U(ξ )e−2π imξ /ddξ ,	(5.2.8)
		L		d
					0

and U0 = 0 as it follows from (5.2.2). The lowest degenerated frequency corresponds to the wave vectors k = ±π/d. We seek a solution to (5.2.6) in the form of a linear combination

u = (u0+eiπ mx/d + u0−e−iπ mx/d)eiqx,

(5.2.9)

with a frequency ω slightly differing from sπ/d at small q (further q π/d). We substitute (5.2.9) into (5.2.6), multiply in turn by exp −i(π/d + q)x and exp −i(π/d − q)x and perform the integration with respect to x. Then we obtain in the main approximation the following set of equations for coefficients u0+ and

u0	:
−			πs	2		2πs2			π	2
	ω2 −				−		q u0+ +			U1 u0−	= 0,
			d			d			d
											(5.2.10)
			πs	2		2πs2			π	2
	ω2	−			+		q u0	+		U u0	= 0.
			d			d			d
							−			1 +

The condition of solvability of set (5.2.10) gives a dispersion relation removing the degeneration at the boundaries of the Brillouin zone:

		π 2			2qd	2	1/2
	ω2 =			s2 ± s4		+ |U1 |2	.	(5.2.11)
		d			π

A gap is opening at the Brillouin zone (q = 0):

δω =

π |U1 |

,

(5.2.12)

cd

where (d1 < d)

|U1 =

1

2

2

| sin

πd1

2s

|s1 − s2 | sin

πd1

.

δs1

− δs2

≈

(5.2.13)

πd

d

πd

d

The forbidden gap appears only at different sound velocities in two neighboring layers.

The higher gaps at the degeneration point can be calculated analogously if one takes pairs of waves with a difference of the wave vectors k − k = 2π(n − n ), where n and

5.3 Dispersion Relation for a Simple Superlattice Model 159

Fig. 5.3 Dispersion curves inside one Brillouin zone: (a) in the zero approximation (without gaps), (b) deformation of curves by small perturbations.

n are integers. A set of allowed bands and gaps appears in a SL. Two lowest bands of frequencies allowed are schematically shown in Fig. 5.3b. A frequency spectrum of the SL differs essentially both from the usual sound spectrum ω = sk, where s is the sound velocity and from the vibration spectrum of a crystal lattice. The number of vibration branches in the crystal is determined by the number of atoms in the crystal unit cell, but in a SL a number of vibration branches appears. The total number of vibration branches in a SL is limited only to the value d/a, that is to the number of atoms in one period of the SL (inside one unit cell of the SL).

5.3

Dispersion Relation for a Simple Superlattice Model

We return to (5.2.1) and note that each of the eigensolutions to (5.2.1) in a periodic structure with a period d is characterized by the quasi-wave number k. Natural oscillations of the field in the unit cell with the number n can be written in the form

un (x) = u0 (x)eiknd,

u0 (x + d) = u0 (x).

(5.3.1)

We write down solutions to the pair of equations (5.2.1) in the interval of one unit cell with the number n (nd < x < (n + 1)d)

un (x) = a1(n)eik1 ξ + a2(n)e−ik1 ξ ,		0 < ξ < d1;
		(5.3.2)
un (x) = b(n)eik2	ξ + b(n) e−ik2ξ ,	d1 < ξ < d,
1	2

160 5 Acoustics of Elastic Superlattices: Phonon Crystals

where ξ = x − nd and k1 = ω/s1 , k2 = ω/s2 , and ω is the frequency. Amplitudes in the neighboring cells are connected by the conditions

a(n+1) = a(n)eikd, b(n+1) = b(n)eikd. (5.3.3)

Boundary conditions at the points ξ = 0 and ξ = d lead to the set of four homogeneous algebraic equations for the amplitudes a1, a2 , b1 , and b2 . Equality of the determinant of this set to zero gives the following dispersion relation (Rytov, 1955)

cos kd = cos k1 d1 cos k2 d2 −	1		k1	+	k2	sin k1 d1 sin k2 d2 .	(5.3.4)
	2		k2		k1

A derivation of (5.3.4) using (5.3.2) and (5.3.3) can be considered as a problem exercise for this section.

Equation (5.3.4) determines in a complicated form the dependence of the frequency ω on the wave number k: ω(k). This relation coincides with an accuracy to notation with that obtained by Kronig and Penney for a quantum particle in a one-dimensional periodic potential (Kronig and Penney, 1930).

Expression (5.3.4) gives the implicit dependence of the frequency on the quasiwave number and allows us to describe readily the spectrum of long-wave vibrations (kd 1), for which the sound spectrum is naturally obtained with average elastic

modulus µ and the density ρ : ρ d = ρ1 d1 + ρ2 d2 and d/ µ = d1 /µ1 + d2/µ2 . Based on such a representation of µ, which contains only ratios dα /µα , it is interesting

to consider a limiting case, which can demonstrate the most characteristic properties of the superlattice spectrum, when d2 → 0 and µ2 → 0 for d2 /µ2 = M = const. In this case, d1 → d, k2 d2 = ωd2/s2 = ρ2 d2 ω d2 /µ2 → 0. Then the dispersion relation for the system is described by the equation

cos kd = cos z − Qz sin z,

(5.3.5)

where z = q1 d = ωd/s1 and Q = ρ2 µ1 M/(2ρ1d). Note that (5.3.5) gives the dispersion relation for an elastic SL consisting of periodic elastic blocks of length d

with the parameters µ1 and s1 under special boundary conditions. If the parameter Q is small, then the system under study represents a periodic sequence of elastic regions that are weakly connected with each other.

The allowed vibrational frequencies of a continuous spectrum of the system under consideration can be qualitatively found by analyzing graphically (5.3.5), as shown in Fig. 5.4. For the beginning we repeat our analysis concerning Fig. 1.15 in Chapter 1. If the r.h.p. of (5.3.5) runs the values between ±1, the roots of the equation have the values in the intervals shown on the abscissa.

Note that, as z increases, the allowed frequencies are localized within the narrowing intervals near the values k1 d = ±mπ, where m is a large integer. For the condition m2 Q 1, the dispersion relation for the m-th band can be readily found.

5.3 Dispersion Relation for a Simple Superlattice Model 161

Fig. 5.4 Graphical solution of (5.3.5) in the cases cosh kd > 1.

Indeed, near odd m = 2 p + 1 (see the vicinity of z = 3π) in Fig. 5.4 , we can write with sufficient accuracy (s = s1 )

cos kd = −1 + Qmπ(z − mπ) = − 1 +				mπd	Q	ω −	mπs	,
				s			d
which yields	s
ω = mω0 +		(1	+ cos kd),					(5.3.6)
	mπQd

where ω0 = πs/d. Similarly, near even m = 2 p (see the vicinity of z = 4π) in Fig. 5.4, we can write

cos kd = 1 − Qmπ(z − mπ) = 1 −				mπd	Q ω −	mπs	,
				s		d
which gives
ω = mω0 +	s	(1	− cos kd).				(5.3.7)
	mπQd

By combining (5.3.6) and (5.3.7), we can obtain the dispersion relations for the m-th

	band:				kd
				sin2			m = 2 p;
						,
					2
	ω = mω0	+	2s				(5.3.8)
			mπQd		kd
				cos2			m = 2 p + 1.
						,
					2

One can easily see that expressions (5.3.8) represent the size-quantization spectrum of phonons in a layer of thickness d, whose levels split into minibands due to a low “transparency” of the interface between layers, that is due to a weak interaction between adjacent blocks.

The frequency spectrum that we obtained is of interest because its high-frequency part has a set of narrowing allowed frequency bands in which the dispersion relation

162 5 Acoustics of Elastic Superlattices: Phonon Crystals

can be calculated analytically with good accuracy. The spectrum has a number of forbidden bands (gaps in a continuous spectrum).

Consider now the possibility of appearance of vibrational states in forbidden bands. Such vibrations correspond to solutions of the type

un exp( κnd)	for	k = iκ,
or
un (−1)n exp( κnd)	for	k = iκ + π,

exponentially decreasing (increasing) with the number n. It is obvious that such states can have a physical meaning only on the coordinate semi-axis under the condition that a solution vanishing at infinity is chosen, which reflect some boundary conditions at the coordinate origin.

For the solution of the first type (k = iκ), the frequency dependence of the parameter κ can be found from the relation

cosh(κd) = cos z − Qz sin z > 1,

(5.3.9)

while, for the solution of the second type (k = iκ +π ), it can be found from the relation

− cosh(κd) = cos z − Qz sin z > 1 < −1.

(5.3.10)

The solutions of the first type correspond to frequencies in the intervals (2 p − 1)π < z < 2 pπ, and those of the second type, in intervals 2 pπ < z < (2 p + 1)π (see Fig. 5.4). Note that such situations appear on the semi-axis, for example, at the ends of the SL.

5.3.1

Problem

Show that (5.3.5) describes the dispersion relation for an elastic SL consisting of periodic blocks of length d under the following boundary conditions on the interfaces: the normal stresses are continuous ([σ]+− = 0, i. e., [∂u/∂x]+− = 0), while elastic displacements exhibit the jump [u]+− = Q(ρ1 /ρ2 )σ.

Part 3 Quantum Mechanics of Crystals

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

6

Quantization of Crystal Vibrations

6.1

Occupation-Number Representation

To describe crystal vibrations one uses the classical equations of motion of atoms (or molecules) positioned at the lattice sites. A classical description of crystal vibrations is only a rough approximation, and from the beginning one should proceed from quantum laws. Small-amplitude vibrations of an ideal crystal, however, represent the rare case of a physical system where a quasi-classical treatment leads to the same results as those obtained in a rigorous quantum-mechanical approach. In this approximation the system of quantized vibration in the crystal is assumed to be equivalent to a system of independent harmonic oscillators. The classification of states and the calculation of the energy spectrum of a harmonic oscillator at a quasi-classical level are known to be accurate quantum mechanically.

Thus, for the majority of crystals the vibrations can be quantized at a late stage in the calculations when the vibration dispersion law is found, the vibration field is represented as a set of harmonic oscillators and the harmonic oscillator frequencies are determined. In particular, for the initial stage of quantization one may take the Hamilton function (2.11.14) written in terms of the real canonically conjugated generalizes coordinates X(k) and momenta Y(k). Since the quantum treatment is not dependent on the vector character of the displacements and the momenta corresponding to them, we begin by using a scalar model based on the Hamilton function

H = ∑H(k), H(k) =	1	Yk2 + ω2 (k)Xk2 .	(6.1.1)
	2
k

In quantum mechanics the Hamilton function is regarded as an operator (Hamiltonian) whose dynamic variables X(k) and Y(k) in (6.1.1) are replaced by the operators

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

166 6 Quantization of Crystal Vibrations

with commutative relations

[X(k), Y(k )] = ih¯ δkk ;

(6.1.2)

[X(k), X(k )] = [Y(k), Y(k )] = 0,

where [A, B] = AB − BA. We do not use special notation for the operators of physical quantities, but have to allow for the noncommutativity of these quantities in all calculations.

The simplest and most widely used way of obtaining the quantum spectrum of a multiparticle system is by writing and diagonalizing its Hamiltonian in the occupation number representation. Since the Hamiltonian (6.1.1) is already diagonalized in the k-states, the choice of new operators can be performed by following linear transformations for a given value of k:

X(k) = u(k)ak + u (k)a†k ; Y(k) = v(k)ak + v (k)a†k .

(6.1.3)

The operators ak and a†k are the Hermitian conjugated operators, u(k) and v(k) are complex functions of the vector k whose choice should satisfy the requirements (6.1.2) and reduce the Hamiltonian H(k) to a product of operators ak and a†k .

We transform an individual term in the Hamiltonian (6.1.1)

Y2 (k) + ω2 (k)X2 (k) = (|v|2 + ω2 |u|2 )(ak a†k + a†k ak ) +(v2 + ω2 u2 )ak ak + [(v )2 + ω2 (u )2 ]a†k a†k ,

and the first of the conditions (6.1.2)

[X(k), Y(k )] = u(k)v (k )[ak a†k + u (k)v(k )[a†k ak ] +u(k)v(k )[ak , ak ] + u (k)v (k )[a†k , a†k ] = ih¯ δkk .

(6.1.4)

(6.1.5)

Analyzing (6.1.4), it is easy to come to a conclusion that if we choose the functions u(k) and v(k) as

u(k) =				h¯		v(k) = −i |ω(k)| u(k),
					,		(6.1.6)
	2	|ω(k)|
the Hamiltonian of vibrations (6.1.1) will reduce to
H =			1	∑h¯ |ω(k)| (a†k ak + ak a†k ).			(6.1.7)
			2
				k

According to (6.1.5), the conditions (6.1.2) will then also be satisfied, if the operators ak and a†k obey the following commutation rules

[ak , a†k ] = δkk , [ak , ak ] = [a†k , a†k ] = 0.

(6.1.8)

6.1 Occupation-Number Representation 167

Using the first rule from (6.1.8) we simplify the Hamiltonian (6.1.7)

H = ∑h¯ |ω(k)|	a†k ak +	1	.	(6.1.9)
		2
k

Since the Hamiltonian (6.1.9) involves only the absolute frequency value, we can consider further only positive vibration frequencies.

Using the operators ak and a†k is the most efficient way in the representation, where the operator a†k ak included in the Hamiltonian is diagonal. It turns out that if ak and a†k satisfy the relations (6.1.8), the eigenvalues of the operator a†k ak are non-negative integer numbers of a natural series

a†k ak = Nk, Nk = 0, 1, 2, . . . .

(6.1.10)

This property of the operator a†k ak is proved in quantum mechanics, however, in our case it follows directly from (6.1.9). Indeed, the energy levels of a harmonic oscillator with frequency ω are known

En =	n +	1	h¯ ω, n = 0, 1, 2, . . . ,	(6.1.11)
		2

thus, the eigenvalues of the Hamiltonian (6.1.1) being the sum of the energies of independent harmonic oscillators can be represented as

E = ∑	Nk +	1	h¯ ω(k),	(6.1.12)
		2
k

where Nk are non-negative integers. Comparing the expressions (6.1.12) and (6.1.9) we are convinced of the validity of (6.1.10).

The numbers Nk are called the occupation numbers of the states k. When the systems consisting of many identical particles are studied from the point of view of quantum mechanics, it is convenient to use a mathematical method where in the occupation number representation various vibrational crystal states are characterized by different sets of numbers Nk , and the action of the operators ak and a†k changes these numbers.

In applications it is important to know not only how the Hamiltonian is written in terms of the operators ak and a†k , but also the form of the displacement operator, which is always initial.

We note that the linear transformation (6.1.3) taking into account (6.1.6) is written in the form

X(k) =	h¯	(a†k	+ ak ),	Y(k) = i	1	h¯ ω(k)(a†k	− ak ).	(6.1.13)
	2ω(k)				2

Thus, the sum of the operators a†k and ak is the coordinate, and their difference is the momentum of the corresponding harmonic oscillator.

168 6 Quantization of Crystal Vibrations

Using now the chain of transformations (6.1.13) and (2.11.15) to find a complex expression for the operators of normal coordinates, we obtain

Q(k) = uk (ak − a†−k ), Q†(k) = uk (a†k + a−k ),

(6.1.14)

where the multiplier uk is determined in (6.1.6). Generally, the complex conjugate normal coordinate Q (k) corresponds to a Hermitian conjugate operator Q†(k).

Using (6.1.14) and (1.6.3), we obtain the atomic displacement operator

		h¯	1/2	1			eikr(n) + a† e−ikr(n)
	u(n) =		∑			a
		2mN
					ω(k)	k		k
			k					(6.1.15)
		h¯	1/2	1			+ a†	k )eikr(n).
	=		∑			(ak
		2mN
					ω(k)
			k				−

We note that in replacing the displacements with the operators by (6.1.15) the de-

nominator always includes the multiplier ω(k) that accompanies the operators ak or a†k .

For a real crystal lattice when the operators aα(k) and a†α(k) belonging to different branches of the dispersion law (α = 1, 2, . . . , 3q) are introduced, the commutation relations (6.1.8) are generalized to

aα(k), a†α (k ) = δαα δkk ,

[aα(k), aα (k )] = a†α(k), a†α (k ) = 0.

The displacement operator of the (n, s)-th atom is associated with the operators aα(k) and a†α(k) by relations

	h¯		1/2	es(k, α)		[aα(k) + a†α(−k)]eikr(n).
us(n) =			∑				(6.1.16)
	2m	N
					ωα(k)
	s		kα

Finally, the Hamiltonian of crystal vibrations in a harmonic approximation is reduced to

H = ∑h¯ ωα(k)	a†α(k)aα(k) +	1	,	(6.1.17)
		2
αk

generalizing (6.1.9).

It is clear that the operators of the physical quantities can be expressed directly through the operators ak and a†k . We discuss their properties based on the Heisenberg representation when the dynamic processes are described by the time dependence of the operators of physical quantities whose equations of motion are analogous to classical Hamilton equations.

We omit the index α, i. e., return to a scalar model. From the Hamiltonian (6.1.9) there follows a very simple “equations of motion” for the operator ak and a†k . Indeed,