Kosevich A.M. The crystal lattice

Подождите немного. Документ загружается.

1.11 Heavy Defects and 1D Superlattice

such conditions allow us to use a long-wave approximation. The continuous analog of

(1.10.4) for one isolated isotope defect at the frequencies ω  ω

is the following

∂

u(x)

∂t

− s

∂

u(x)

∂x

= aU

u(0)δ(x − x

), (1.11.1)

where x

is the defect coordinate. See (1.1.4) and compare (1.11.1) with (1.10.15) for

ω −ω

 ω

The presence of the delta-function in the r.h.p. of (1.11.1) determines a jump of

the ﬁrst spatial derivative of the continuous function u(x) at the point x = x

and is

equivalent to the following boundary condition for (1.1.16)

−s



∂u(x)

∂x



−0

= aU

u(0). (1.11.2)

From the macroscopic point of view the model proposed for consideration is a 1D

acoustic superlattice consisting of elastic elements of the length d separated by set

of joints. Ex citations inside the elastic elements are described by the wave equation

(1.1.4) with the boundary conditions (1.11.2) on the all joints. From the microscopic

point of view such a system is a polyatomic 1D crystal with the unit cell including a

very large number N

of atoms. Taking into account this fact, let the large unit cells

(elements of the superlattice) be numbered by n.

Then the eigenvibration in the n-th element can be written in the form of th e Bloch

wave

(x)=v

(x)e

ikx

, v

(x + d)=v

(x), (1.11.3)

where the Bloch amplitude v(x) is a solution to the wave equation (1.1.4) and the

function u(x) obeys the boundary conditions (1.11.2).

Let the origin of coordinates coincide with one of the joints (it will have the number

n = 0). Then (1.11.3) can be written in the form

(x)=v(x − nd)e

ikx

Take the function v(x) as a general solution to (1.1.4)

v(x)=Ae

iqx

+ Be

−iq x

, q =

, (1.11.4)

and consider the solution inside the interval 0 < x < d.

Writing the boundary conditions such that at the left joint

−1

(−0)=u

(d −0)e

−ikd

= v(d)e

−ikd

and at the right joint

(d + 0)=u

(+0)e

ikd

= v(0)e

ikd

1 Mechanics of a One-Dimensional Crystal

The boundary conditions (1.11.2) and continuity of the function u(x) at the point

x = 0 or x = d turn into the following set of linear algebraic eq uations for the

coefﬁcients A and B

A − B − i



aMω



(A + B)=(Ae

iqd

− Be

−iqd

−ikd

A + B =(Ae

iqd

− Be

−iqd

−1kd

(1.11.5)

The determinant of (1.11.5) can be easy calculated and equ ality of the determinant to

zero leads to the following expression (the calculation can be considered as a p roblem

for readers)

cos kd = cos qd −



aMω



q sin qd. (1.11.6)

Introduce a new variable z = qd and rewrite (1.11.6)

cos kd = cos z − Qz sin z, Q =

2md

. (1.11.7)

It is assumed that M  m and N

 1. Since the long-wave approximation was used

the variable z can not be very large (z  N

Equation (1.11.7) gives the dispersion law for the superlattice and corresponds to a

band structure of the spectrum. The allowed vibrational frequencies of a continuous

spectrum of the system under study can be qualitatively found by analyzing graph-

ically (1.11.7) as shown in Fig. 1.15: if the expression cos z − Qz sin z has values

between ± 1, the roots of the equation have values in the intervals shown on the ab-

scissa. As a result the vibration spectrum consists of bands of two types: allowed

frequency bands and forbidden bands (gaps).

Note that, as z increases, th e allowed frequencies are localized within the narrowing

intervals near the values k

d = ±mπ where m is a large integer. For the condition

Q  1 the dispersion law for the m-th band can be found. Indeed, near odd

m = 2p + 1 (see the vicinity of z = 3π in Fig. 1.15) we can write with sufﬁcient

accuracy

cos kd = −1 + Qmπ(z − mπ)=−1 +

mπd



ω −

mπs



which yields

ω = mΩ

(1 + cos kd), (1.11.8)

where ω

= πs/d and Ω = s /(πQd). Similarly, near even m = 2p (see the vicinity

of z = 4π in Fig. 1.15) we can write

cos kd = 1 − Qmπ(z − mπ)=1 +

mπd



ω −

mπs



which gives

ω = mΩ

(1 − cos kd). (1.11.9)

1.11 Heavy Defects and 1D Superlattice

Fig. 1.15 Graphical solution of (1.11.7).Ifcos z − Qz sin z has values

between ±1, the roots of the equation have values in the intervals indi-

cated on the abscissa.

By combining (1.11.8) and (1.11.9), we obtain the dispersion relations for the m-th

band:

ω = mω







sin

(kd/2 ), m = 2p ,

cos

(kd/2 ), m = 2p + 1.

(1.11.10)

One can easily see that expressions (1.11.10) represent the size-quantization spectrum

of vibrations in a 1D elastic element of length d with levels (frequencies) split up

into minibands due to a low “transparency” of the joints dividing the 1D crystal into

elements.

Thus studying a biatomic 1D crystal in Section 1.5 and a simple continuous model

of the sup erlattice in this Section we consider two limiting cases of p olyatomic lattices:

a lattice with 2 atoms in the unit cell and a lattice with N

 1 atoms in the unit cell.

And the number of gaps in the continuous vibrational spectrum is determined by the

number of atoms in the unit cell.

General Analysis of Vibrations of Monatomic Lattices

2.1

Equation of Small Vibrations of 3D Lattice

According to the deﬁnition of a crystal structure all spatial lattice points can be repro-

duced by the integral vector n =(n

, n

) where n

(α = 1, 2, 3) are integers. If

we choose the coordinate origin at one of the sites, the position vector of an arbitrary

lattice site with “number” n will have the form

≡ r(n) =

∑

α=1

. (2.1.1)

In a monatomic lattice the sites are numbered in the same way as the atoms, so that

the integral vector n is at the same time “the number” or “the number vector” of a

corresponding atom. However, a lattice site coordinate (2.1.1) differs from the coordi-

nate of the corresponding atom when atoms are displaced relative to their equilibrium

positions. If the atom equilibrium p ositions coincide with the lattice sites (2.1.1), to

deﬁne the coordinates of an atom, it is necessary, apart from its “number”, to indicate

its displacement with respect to its own site.

We denote the displacement of an atom with “number” n from its equilibrium po-

sition by u(n).

The existence of the crystal state means that over a wide temperature range the

relative atomic displacemen ts are small compared with the lattice constant a (by the

lattice constant we mean the value whose o rder coincides with the values of funda-

mental translation vectors a

, in a cubic lattice. Therefore, we shall begin studying

crystal lattice vibratio ns with the case of small harmonic oscillations.

It is easy to derive and to write the equa tions for small lattice vibrations. We con-

sider the potential energy of a crystal whose atoms are displaced from their equilib-

rium positions and express it through the displacement u(n). The crystal energy is

1) In the following, for the coordinates of the crystal lattice sites, we shall use

the notation r(n) instead of R(n) used in the Introduction.

The Crystal Lattice: Phonons, Solitons, Dislocations, Superlattices, Second Edition. Arnold M. Kosevich

 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ISBN: 3-527-40508-9

2 General Analysis of Vibrations of Monatomic Lattices

dependent on the coordinates of all particles making up the crystal, that is, the atomic

nuclei and electrons. The latter, however, are so mobile that they manage to adapt

to nuclear motion. Thus, at every given moment the electronic state is described by a

function dependent on the positions of the nuclei. Excluding the electronic coordinates

from the crystal energy is the essence of the so-called adiabatic approximation that is

justiﬁed both in studying harmonic vibrations and in the case of small anharmonicity.

Regarding the potential energy of a crystal U as a function only o f the coordinates of

atomic nuclei that coincide with the coordinates of atomic centers of mass we expand

it in powers of u(n) and consider the ﬁrst nonvanishing expansion terms. Assuming

the crystal to be in equilibrium at u(n) = 0 we write the potential energy in the for m

U = U

∑

n, n



(n, n



(n)u



), (2.1.2)

where U

= constant and the summation is over all crystal sites. The Latin letters

i, k (also j, l, m,...) are the coordinate indices. The summation over doubly repeated

coordinate indices from 1 to 3 is assumed. To simplify the notation of the sums over

the numbers n, n



, etc., we do not indicate the summation indices under the summation

sign if the summation is over all indices involved under the summation sign.

Since we consider that the crystal at u(n) = 0 is in mechan ical equilibrium , the

matrix β

(n, n



) should be deﬁned positively; in particular, the following inequalities

must be satisﬁed

(n, n) > 0, β

(n, n) > 0.

Other important properties of the matrix will be discussed below.

The positively d e ﬁned matrix with elements β

(n, n



) is generally called the matrix

of atomic force constants or dynamical matrix of a crystal.

With a general expression for the potential energy (2.1.2) we can easily write down

the equa tion of motion of every atom in a crystal:

= −

∂U

∂u

(n)

= −

∑



(n, n



), (2.1.3)

where m is the atomic mass.

To simplify formulae and equations such as (2.1.2), (2.1.3) containing the u

(n)

displacement vectors and the β

(n, n



) dynamical force matrix or similar matrices,

we omit the coordinate indices i, k, l,... for the matrix and vector values, and the

square (3 × 3) matrices are denoted by the corresponding capital letters, e. g., the

matrix of force constants is denoted by the letter B:

B = β

, i, k = 1, 2, 3.

By u(n ) we mean the displacement vector that is in the form of a column consisting

of three elements: u = co l (u

, u

). Such a notation is only “decoded” when

possible misunderstanding may appear.

2.1 Equation of Small Vibrations of 3D Lattice

Thus, the quadratic expression for the potential energy (2.1.2) is

U = U

∑

B(n, n



)u(n)u(n



). (2.1.4)

In the new notation we write the equations of motion in the form

= −

∑



B(n, n



)u(n



). (2.1.5)

Here we easily see one important property of the coefﬁcients B(n, n



). Assume the

crystal to be displaced as a whole: u(n)=u

= constant. Then the internal crystal

state cannot change (absence of external forces). Thus, the atoms are not affected by

additional forces. Consequently,

∑



B(n, n



) ≡

∑



(n, n



∑



, n)=0. (2.1.6)

The requirement (2.1.6) follows from the conservation of total momentum in the

crystal. Another obvious requirement is that the total angular momentum conserva-

tion law be automatically obeyed in the absence of external forces. We assume rigid

rotation of the crystal effected and described by the axial vector Ω:

(n)=

ikl

Ω

(n) , (2.1.7)

where 

ikl

is a rank three unit antisymmetric tensor and x

(n) is determined by (2.1.1).

The above rotation causes no change in the internal lattice state if external forces are

absent. Hence, with (2.1.7 ) substituted into (2 .1.3), (2.1.5 ) we have



kl m

Ω

∑



(n, n



)=0. (2.1.8)

Since the vector Ω is arbitrary

∑



(n, n



∑



(n, n



). (2.1.9)

The conditions (2.1.6), (2.1.9) should be satisﬁed for any atom in the lattice (for

any vector n). The last point is especially important when not all crystal sites are

equivalent. For example, in describing the vibrations of a ﬁnite crystal specimen (the

atoms near the surface are under conditions differing from those in the crystal interior)

or a real crystal with broken translational symmetry (a crystal with defects).

A detailed discussion of constraints imposed on the elements of the matrix B is

due to a usual simpliﬁcation of crystal models when the minimum possible number of

elements of this matrix is nonzero, thus describing the interaction of only the nearest-

neighbor atoms in the lattice. But it is questionable whether such a choice of model

agrees with general physical properties of a crystal, i. e., with mechanical conservation

laws.

2 General Analysis of Vibrations of Monatomic Lattices

We now turn to (2.1.4) and discuss ﬁrstly the bulk properties of a crystal assuming

the crystal lattice to be unbounded. For an unbounded homogeneous lattice, due to its

homogeneity the matrix B has the form

B(n, n



) = A(n − n



where we introduced A = α

, i, k = 1, 2, 3 whose elements are satisﬁed by the

conditions

∑



A(n − n



) ≡

∑



(n − n



∑



(n)=0. (2.1.10)

Then (2.1.4) is written as

U = U

∑

A(n − n



)u(n)u(n



), (2.1.11)

and the equations of motion (2.1.3) take the form

u(n)

= −

∑



A(n − n



)u(n



). (2.1.12)

We substitute the condition (1.1.10) into the equation of motion (1.1.12) to obtain

u(n)

∑



A(n − n



)



u(n) −u(n



)



. (2.1.13)

The equa tion of motion (2.1.13) illustrates th e invariance of a crystal state with

respect to its displacement as a whole.

Because (2.1.4) and (2.1.11)–(2.1.13) are equations for three-dimensional displace-

ment vectors we shall use the simplest model to describe the vibration o f a crystal

lattice where all atoms are displaced in one direction. Then the atom displacement

from its equilibrium position is determined by a scalar rather than a vector one. This

model allows us to describe the main properties of a vibrating crystal using simple

formulae and to obtain the correct quantitative estimates. This model, here called

conventionally scalar, is used to illustrate the calculation scheme or the ideology of

mathematical methods that produce, in a real th ree-dimensional crystal, the same re-

sults as those obtained by employing sophisticated and cumbersome calculations.

In the scalar model the quadratic expressions for the potential energy (1.1.2),

(1.1.11) preserve their form; however, we see here no coordinate indices, and the

dynamical matrix of an unbounded crystal is reduced to a scalar function of the vector

argument α(n):

U = U

∑

α(n − n



)u(n)u(n



). (2.1.14)

We include in the properties of a scalar model the energy invariance (1.1.14) with

respect to the displacement of a crystal as a whole assuming the function α(n ) to be

satisﬁed by the condition (1.1.10), i. e.,

∑

α(n)=0. (2.1.15)

2.2 The Dispersion Law of Stationary Vibrations

Replacing (2.1.11) by (2.1.14) in the analysis of crystal lattice vibrations will in

some cases allow us to avoid cumbersome calculations.

2.2

The Dispersion Law of Stationary Vibrations

The stationary crystal vibrations for which the displacement of all atoms are time

dependent only by the factor e

−iωt

are of special interest. For such vibrations we

obtain instead of (2.1.12)

mω

u(n)=

∑



A(n − n



)u(n



). (2.2.1)

If we introduce the notation  = ω

we can rewrite (1.2.1) in a canonical form:

∑



A(n − n



)u(n



) − u(n)=0. (2.2.2)

Equation (2.2.2) together with certain boundary conditions for the function u is the

eigenvalue problem for the linear Hermitian operator (1/m)A.

Let the solution to (2.2.1), (2.2.2) has the form

u(n)=ue

ikr (n)

. (2.2.3)

The vector k is analogous to a wave vector of crystal vibrations and is regarded as

a quasi-wave vector. It is now a free parameter that determines the solution. Substi-

tuting (2.2.3) into (2.2.1), we obtain for the displacement a system of linear equations

mω

u =

∑



A(n − n



)ue

ik[r(n



)−r(n)]

From the deﬁnition (1.1.1) it follows that

r(n



) − r(n)=

∑

α=1

− n



= r(n − n



hence

mω

u − A(k)u = 0, (2.2.4)

where the matrix A( k) is given by

A(k)=

∑

A(n)e

−ikr (n)

. (2.2.5)

Here, and in what follows, we denote with the same capital letter the matrix in

various “representations” and distinguish the form of representation by the argument

only. Since throughout the book the vectors n or r refer to the real space of sites

2 General Analysis of Vibrations of Monatomic Lattices

or the coordinates in a crystal and the vectors k to reciprocal space, this generally

accepted system of notations will be quite clear. Thus, in the given case, A(n) is

the force dynamical matrix in the site representation, A(k) is the same matrix in k-

representation.

The condition for the system (2.2.4) to be compatible has the form

Det mωI − A(k) = 0, (2.2.6)

where I is the unit matrix; I ≡ δ

, i, j = 1, 2, 3.

Express now (2.2.6) as

D(ω

)=0, (2.2.7)

introducing the new notation

D()=Det



I −

A(k)



. (2.2.8)

In mechanics, the relation (2.2.6) or (2.2.8) is called the characteristic equation for

eigenfrequencies and its solution relates the frequency of possible crystal vibrations

to a quasi-wave vector k. The wave-vector dependence of frequency is called the

dispersion law or dispersion relation and the equation is referred to as a dispersion

equation. Thus, solving the dispersion equation we obtain the dispersion law

 = ω

(k)

for crystal lattice vibr ations.

In a monatomic lattice each atom is an inversion center, therefore,

A(n)=A(−n),

and (2.2.5) is reduced to the following expression, where it is obvious that the matrix

A(k) is real:

A(k)=

∑

A(n) cos kr(n)=

∑

A(n)[cos kr (n) −1] . (2.2.9)

The last part of (2.2.9) follows from the force matrix property (2.1.10).

Thus, the plane-wave vibration (2.2.3) can be traveling in a crystal if its frequency

ω is connected with the quasi-wave vector k by the dispersion law (2.2.6) or (2.2.7).

In a scalar model instead of the system of (2.2.4) there is only one equation of

vibrations, and the dispersion law can be written explicitly

(k)=

∑

α(n)e

−ikr (n)

∑

α(n)[cos kr (n) −1] . (2.2.10)

From (2.2.6) or (2.2.10) we note that the dispersion law determines the frequency

as a periodic fun ction of the quasi-wave vector with a period of a reciprocal lattice

ω(k)=ω(k + G ).

2.2 The Dispersion Law of Stationary Vibrations

This proves to be the basic distinction between the dispersion law of crystal vi-

brations and that of continuous medium vibrations, since the monotonic wave-vector

dependence of the frequency is typical for the latter. The difference between the quasi-

wave vector k and the ordinary wave vector is also observed here. Only vector k values

lying inside one unit cell of a reciprocal lattice correspond to physically nonequivalent

states of a crystal.

Recall that free space is homogeneous, i. e., invariant with respect to a translation

along an arbitrary vector including an inﬁnitely small one. A set of all similar dis-

placements forms a continuous group of translations. The operator of a translation

onto an inﬁnitely small vector is the momentum operator ( the momentum operator is

said to be the generator of a continuous group of translations).

The invariance of mechanical equations with respect to a continuous group of trans-

lations generates the momentum vector p as the main ch aracteristic of a free particle

state or the wave vector k as the main characteristic of the wave process in a vacuum.

The vectors p and k in free space are not restricted in value by any conditions and are

related by

p = ¯hk.

(2.2.11)

A crystal lattice, unlike a vacuum, has no homogeneity, but is spatially periodic. We

see that the quasi-wave vector is the result of translational symmetry of the periodic

structure to the same extent as the wave vector is the result of free-space homogeneity.

Thus, in an unbounded crystal the wave processes can be described using the concept

of a quasi-wave vector k, and the motion of particles using the concept of quasi-

momentum related to the vector k via (2.2.11). The wave function corresponding to

a quasi-momentum (or quasi-wave vector) represents a plane wave modulated with a

lattice period.

When the minimum space dimension (lattice period) tends to zero, the Brillouin

zone dimensions become unbounded and we go over to a homogeneous space and

return to the concept of momentum and its eigenfunctions in the form of plane waves.

Returning to the dispersion law as a solution of the dispersion equation (2.2.6), we

take into account that it is a cubic algebraic equation with respect to ω

Det



mω

− A

(k)



= 0, i, j = 1, 2, 3. (2.2.12)

The roots of this equation determine the three branches for monato mic crystal lattice

vibrations speciﬁed by the dispersion law:  = ω

(k) , α = 1, 2, 3, where α is the

number of a branch of vibrations.

But the characteristic equation (2.2.12) only determines the squared frequency ω

Thus, the α-branch dispersion law actualizes each value of the vector k with two

frequencies: ω = ±ω

(k) . Hence the spectrum of squared frequencies of a vibrating

crystal seems to be doubly degenerate. However, as follows from (1.2.9), (1.2.12)

the dispersion law is invariant relative to the change in sign of the quasi-wave vector:

(k)=ω

(−k). Therefore, the wave with a quasi-wave vector k and frequency