33
Progress of Theoretical Physics, Vol. 28, No. 1, July 1962
Energy Spectrum of Lattices with Defects. II*>
--Frequency Spect1-um of Lattice Vibration-Shozo T AKENO**>
Department of General Education
Osaka University of Engineering, Osaka
(Received January 17, 1962)
A systematic investigation of the vibration of a lattice contammg isotopic impurities
of a single kind is made in terms of the Green's function, and a general expression for the
squared-frequency spectrum is obtained from a solution of difference equations, which can
be applied to any number of dimensions. For a fixed spacial configuration of impurities,
the results yield the spectrum of the main continuum and of the localized modes. When
the number of isotopes is few, the frequencies of the localized modes, the criterion for their
appearance, and the distribution of in-band frequencies are studied in detail, and the results
are applied to a simple cubic lattice. The localized modes due to a substitutional impurity
atom are also studied. In the case of a random distribution of isotopes. the spectrum is
obtained in a closed form with use of a graphical method for the perturbation calculations.
It is shown that (1) the perturbation due to lighter isotopes causes an impurity band or a
tail of the spectrum into the forbidden region, and (2) when heavier isotopes are present,
the spectrum in the low frequency part is increased, while in the high frequency region
it adds a tail. The criteria for the appearance of the impurity band and for its merging
into the main continuum are obtained. For a simple cubic lattice, the construction of the
spectrum is performed.
§ I. Introduction
Considerable progress has been made in the theory of the vibration of
disordered lattices. For lattices containing isolated defects, the analysis of the
localized modes has been made by several authors. Montroll, Potts, Maradudin,
and collaborators,I>~ 4 > and Lifshitz5> developed the method of Green's function for
determining the frequencies of the localized modes. Hori and Asahi 6> presented
the method of transfer matrix for a one-dimensional problem. Teramoto and
the writer 7>investigated a. one-dimensional problem by solving integral equations
describing the time evolution of the system.
On the other hand for the problem of calculating the frequency spectra of
lattices containing isotopes of different masses in a finite ratio of concentrations,
most work has been centered on the one-dimensional problem and a number of
distinct techniques have been proposed and developed. Dyson8> and Schmide>
*l A preliminary report of this work was given in talks at The 15th Meeting of the Physical Society of Japan held in Nagoya (October. 1960).
**l Present address': Department of Physics, Faculty of Science, Hokkaido University, Sapporo.
34
S. Takeno
obtained formal expressions for the frequency spectra of disordered linear chains
in a form of functional equations. However, they did not attempt to solve them.
Practical calculations of the spectra of linear chains have recently been made
by several authors. Domb et aU 0> Hori,11 > and Fukuda12> have applied the moment
trace method to one-dimensional cases. Although this method c.an be used for
higher dimensional cases, the spectra obtained are of limited accuracy and they
converge poorly in the high frequency region. Dean13> has succeeded in devising a machine technique which enables the frequency spectrum of a linear chain
to be accurately found in a direct manner. Flinn, Maradudin, and Weiss14> applied the method of Faulkner and Korringa15> which is an ingenious modifi~ation
of the method of transfer matrix. Both methods, which seem to provide fairly
well d~scriptions of the spectra, cannot in principle be used in higher dimensions. Langer16> obtained the frequency spectrum of a linear chain with use of
the phonon propagator, which seems to be generalized so as to apply to higher
dimensional cases.
In contrast with the one-dimensional case, disordered two- and three-dimensional vibrating systems have received little attention. Systematic investigation
of the localized modes in three-dimensional lattices has not yet been made. Dean
and Martin17> gave a programming method of machine calculation for evaluating
the frequency spectrum of a two-dimensional lattice. However, the application
of this method to three-dimensional lattices seems to be a very complicated ·
task with existing machines. Lifshitz and Stepanova,18> obtained a formula which
expands formally the frequency spectra of lattices of any dimensions in terms of
the concentration of impurities. However, the method that has been suggested
is of doubtful computational value.
A number of works for one-dimensional cases possess a certain amount of
intrinsic physical interest which is instructive and suggestive also for higher
dimensional cases. However, there are some features which depend on the
dimensionality of the lattice. These are, for example, the criteria for the appearance of the localized modes and of the impurity band, the shape of the
frequency spectra, the nature of transition regions between the allowed and
forbidden bands, and so on. vVe may therefore conclude that the results reached for the one-dimensional case are not di:rectly applicable to more realistic
models in three dimensions.
The purpose of the present paper is to give a new method for determining
the frequency spectrum of a three-dimensional lattice containing isotopic impurity atoms. In the previous paper, which will hereafter be referred to as (I) / 9>
we have presented a formula for evaluating the moments of the energy spectra
of lattices with defects. The present method is a slight generalization of the
previous ones in order to calculate the frequency spectrum in a direct way, which
can easily be translated into lower dimensional cases. It is based upon the fact
that the trace of the real or the imaginary part of a Green's function is pro-
Energy Spectrum of Lattices with Defects. II
35
portional to the squared-frequency spectrum. A similar method has recently been
applied by Matsubara and Toyozawa20 > to the investigation of the impurity band
conduction in semiconductors. The Green's function used here has a similar form
to those employed by Montroll et al., Lifshitz, and Koster and Slater21 J• 22J for
studying the localized modes and impurity levels in solids. The Green's functi~n
of this type can be used extensively for other problems of disordered lattices
such as the scattering of lattice waves and the time evolution of the system,
which will be studiede lsewhere. In § 2 a formula for determining the squaredfrequency spectrum is given in terms of the solution of difference equations
satisfied by the coordinate representation of the Green's function. For a fixed
configuration of impurities, the results yield a J-function type spectrum associated
with the localized modes as well as the change of the main continuum of the
band. A brief discussion is given on the relation between the moments of the
spectrum and a random waik problem. In § 3 the frequencies of the localized
modes and the criterion for their appearance are studied in detail and the results
are applied to a simple cubic lattice. In § 4 we present general formulae for
describing the change of the main continuum of the band and for expanding
the frequency spectrum and the free energy in terms of the concentration of
impunttes. In the following two sections a random lattice is dealt with. In
§ 5 as an alternative method for solving the difference equations, a perturbation
calculation is performed, which is particularly of use when the spacial distribution of impurities is completely random, and there a graphical method is employed in order to improve approximations to any desired order. The correlation
among impurities 'is exactly taken into account up to pairs. It is shown that
the frequency spectrum of the random lattice is characterized by a shift and
damping of the spectrum of the regular lattice. Actual calculations of the
spectrum are carried out in § 6 with particular attention to the impurity band.
§ 2.
Formulation of the problem
We consider a three-dimensional harmonically coupled oscillatior system of
N atoms arranged in a Bravais lattice which contains n isotopic impurity atoms
of a single kind. Here N is assumed to be infinitely large. We assume in this
paper that each component of the displacement vector of an atom from its
equilibrium position is independent of each other. *J Then the time-independent
equation of motion can be written in the form
(2 ·1)
where u(R 1) is one of the component of the displacement vector of the atom
*l This assumption is verified, for example, for a simple cubic lattice with nearest neighbour
interaction.
36
S. Takeno
at the site R 1 and w the circular frequency. The first term of the left-hand
side represents the vibration of a regular lattice, K (Rk- R 1) being the force
constant between the atoms at the sites R 1 and Rk, and m being the mass of
the atom of the host crystal, and the second term represents the additional term
due to isotopes, i.' being given by
i.' = (m-m')/m',
(2·2)
where m' is the mass of the isotope. Let the location of n isotopes be R 1 , R 2 ,
R 3 , ···, and Rn respectively, then Eqs. (2 ·1) are reduced to an operator equation:
(La+ V)u(R) =w2 u(R),
(2·3)
where
n
V =i.' L0 ~Ll(R-R,),
(2·5)
i=l
where L1 is Chronecker's delta. w 2 (k) is obviously the square of the eigenfrequency of the regular lattice, which is characterized by the wave vector k. Then
the squared-frequency distribution G(w 2) can be obtained by the equation
(2·6)
which is connected with the frequency distribution function g (w) by the relation
g(w) =2wG(w 2) .
(2·7)
By using the well-known identity
(_!_) + irrfJ (x),
lim - 1-.- = p
x-zE
x
·~o+
(2·8)
where p denotes the principal value, we have
G(w 2)
u(iw 2 + E)},
=;r-1 Re{lim(l/N)Tr
(2·9)
E--)'0+
where Re (A) denotes the real part of A, and u (s) satisfies the following equation
u(s)
1
= - --
-
s-i(Lo+ V) '
or u(s)
1
=
s-iLo
+
z Vu(s).
s-iLo
(2 ·10)
The characteristic function of G(ai),
"'
f{J(t) =\'exp (it(i) G(w2 ) dw 2
~
0
can be given by the equation
c~ ·11)
37
Energy Spectrum of Lattices with Defects. II
rp(t) = (1/N)Trp(t), with p(t) =exp(it(Lo+ V)).
(2 ·12)
Then the moment of the squared-frequency distribution is easily obtained by Eq.
(2 ·11); the p-th moment is given by
f.lp=i-P{dPrp(t)/dtP}t~o
(! (t)
·
(2 ·13)
satisfies the equation similar to the Schrodinger equation:
op(t) =i(Lo+ V)p(t), with p(O) =1.
ot
(2 ·14)
o-(s) and ,n(t) are connecte_d with each other through the Laplace transformation:
"'
o-(s) = \exp( -st){J(t)dt, Re s>O.
(2 ·15)
ol
0
We shall apply the coordinate repe.sentation to Eq. (2 ·10); here we shall
make use of the following :?implified notations :
(Rz[<T(s) IRm)=<Tlm(s),
(R~I-1-.-IRm)=[hm(s).
s-zLo
Then, remembering Eqs. (2 · 4) and (2 · 5) and using the momentum representation, we have
n
O"zm(s) =gzm(s) +A'"L;fli(s)o-,m(s),
(2 ·16)
?.=1
where
where Q is the volume of the unit cell, and the integration should be carried
over the first Brillouin zone in k-space. Then Eq. (2 · 9) becomes
G(ai) = (Nn) -r Re {lim"L:;o-u (iw 2 + E)},
E,_O+
(2. 9')
t
where the sum extends over all lattice sites. Applying the inverse Laplace
transformation to Eqs. (2 ·16), we get the integral equations for the coordinate
representation of p (t) :
(2 ·18)
where
(2 ·19)
Noticing (,i (k) =
w 2 ( - k),
we can easily verify the relations
38
S. Takeno
glm(s) =gml(s), fim(s) =fml(s), hlm(t) =hml(t) ·
In particular we shall make use of the following simplified notations :
gll(s) =go(s), fll(s) =fo(s), hll(t) =ho(t),
gw(s) =gol(s) =gl(s), fw(s) =fol(s) =fl(s), hw(t) =hol(t) =hl(t).
Equations (2 ·10) contitute a set of simultaneous equations whose dimension 1s
equal to the squares of the number of atoms in the crystal. However, as was
first noticed by Koster and Slater, 21 > in studying the simultaneous equations of
impurity problem we may in effect consider only those IT's which belong to the
impurity sites. Using the results obtained in (I), we get the expression for
(1/N)Tru-(s):
N- 1 Tr a-(s) =g0 (s) +
~
(2. 20)
:slog D(s),
where D(s) 1s the "characteristic impurity determinant" and it 1s given by
D(s) =detla,1 -.i.'f,1 (s) I,
i=1, 2, ···, n;
(2 ·21)
j=1, 2, ···, n.
By using Eq. (2 ·17), D(s) is conveniently rewritten as
D(s) = (1+)')".fD(s), .fD(s) =detld'i 1 -.i.sg, 1 (s)
I,
(2. 22)
i=1,2,···,n; j=1,2,···,n,
where
i.= (m-m')/m.
(2. 2')
N- 1 Tr a-(s) =g0 (s) + __!__ __tl,_ log.fD(s).
(2 ·23)
Then Eq. (2·20) becomes
N
ds
Formally, Eqs. (2 · 20) and (2 · 23) have close resemblance to the contour integration formula for evalu'ating additive functions of the normal mode frequencies
originally obtained by Montroll and Potts. 1 > Remembering Eq. (2 · 9), we have
the result
(2. 24)
where Go (w 2) is the squared-frequency distribution of the regular lattice. If the
top of G0(ui) is denoted by wL\ we have Go(9) =G0 (wL2) =0. Equation (2·24)
separates the expression for G(w 2) into the unperturbed and perturbed terms due
to isotopes. Our basic assumption which will be used throughout in this paper
1s that the problem of the unperturbed latttice has been solved, so that Go (oi)
1s assumed to be known.
In the limit E---'>0, .fD(i(i+ E) approaches for the case (i): w 2>wL2*>
*> Of course, the case
w2 <w 2 (k)
w2(k) always corresponds to w2=0.
must be excluded, since the bottom of the acoustical band
39
Energy Spectrum of Lattices with Defects. 11
lim [J)(iol+ E) =det 18i1 -i.oia, 1 (oi)
I,
(2 ° 25)
{-)-0+
i=1, 2, ···, 11;
j=1, 2, ···, n,
where
(2 ° 26)
o} is within the band w 2 (k)
for the case (ii) :
lim[J)(iw 2 + E) =det IO'i 1 +i.w 2a, 1 (w 2 ) -ii.w 2{9, 1 (w 2 )
I,
(2 ° 27)
E-+0+
i=1,2,···,n;
j=1,2,···,n,
where
(2 ·28)
In both cases, the determinant lim [J)(iw 2 + E) is symmetric, since
t:-)-0+
a, 1 (w
2)
=a1i(w
2 ),
ai1 (w 2 ) =a1i(w 2 ) , (9, 1 (w 2 ) =p1i(oi).
As bdore, we shall make use of the following simplified notations :
aii (w 2) = ao (ai), ai.(ai) = ao (w 2 ) , (9ii (w 2 ) = f9o (w 2 ) ,
aio(ui) =aoi(w2 ) =a,(w 2) , aio(ai) =aoi(w2 ) =ai(w2 ) , (9iD(ai) =Poi(w2 ) '=pi(w 2 ) .
Po (w 2 ) is related to Go (w 2) as
(2. 29)
It is easily seen that the functions: g1 (s),f1 (s), h 1 (t), a 1 (a.1 2) , a 1 (ai) and p 1 (ai)
decrease monotonically as IR 1 l increases.
In the case (i), G0 (w2) vanishes and eachelement of lim[J)(iw2 +E) has no
t--7-0+
imaginary part, so that the second term of the right-hand side of Eq. (2 · 24)
vanishes except for the case
detl aij- i.w 2ai 1 Cai) I =
o,
(2. 30)
and then arg {lim [f) (iw 2 + E)} becomes
E-+0+
arg {lim[[) (iw 2 + E)} = arg {[f) (iw2 )} = n'L:}J (w2 - (v}),
~+
•
(2 ·31)
where tJ (x) is the step function, and· w. 2 ( v = 1, 2, 3, · · ·) is the solutions of Eq.
(2 · 30), which is obviously the secular equation determining the frequencies of
the modes known as localized modes.
In the case (ii), the second term of Eq. (2 · 24) is considered as a correction term to G0 (w 2) . Here aif (w 2) and PiJ (w 2) are connected with each other
by the dispersion relation
40
S. Takeno
(2. 32)
This is easily obtained by using the following relation :
~~ ____.g___~
.;z-w2
(2rr) 3 .;.,2(~l=·•
exp{il£· (R, -RJ)} dSk= __ fl__ f IJ{oi-{i(k)}
IJ7w2 (1r)l
. (2;r) 3 J
xexp{ik· (Ri-R1 )}dk,
where dS1,. is the surface element of the constant energy surface w 2 (k) = z in
k-space. Using the integral representation of the delta function, we can express
p,1 (z) as a Fourier tranform of h, 1 (t):
"'
p, 1 (z) =_!_\exp( -izt)h, 1 (t)dt, or h,1 (t) =_lfexp(izt)pi 1 (z)dz. (2•33)
.2
ir
o)
J
As a result, the expression for G(w2) 1s given by
G(o})
.
= N-1 _Eo (w 2 -
w. 2) ,
for the case (i),
G(ol) = G 0 (oi) + (Nn:) - 1 ~[arg {detiiJi, + Aw2aiJ (lu 2) - iAw2{1, 1 (w2)
dw
for the case (ii).
(2. 34)
I}],
(2 · 35)
As will be shown in the next section, the number of solutions of Eqs. (2 · 30)
does not e'xceed the number of isotopes, so that when the number of isotopes
is finite, Eq. (2 · 34) yields the spectrum which is discrete and splitted out from
the top of the main continuum of the band. The second term of the right-hand
side of Eq. (2 · 35) represents the change of the main continuum due to isotopes.
Using the results obtained above, we get the formal expression for the vibrational
contribution to the free energy F(p) as follows:
"'
F({i)=f{~nw+ ~log
(1-e-fin.,)}g(w)dw,
(2. 36)
0
where p = (kT) - 1 . From Eqs. (2 · 7) and (2 · 24), we can separate F (!'1) into the
unperturbed and perturbed terms :
"'
(1- e-.e?iVz)} __ci,__[arg{lim g) (iz +E)} ]dz,
J{!!_v;: +~log
{I
dz
F({J) = F 0 (p) + _l__f
Nrr
2
0
HO+
(2·37)
where F 0 (p) is the free energy of the unperturbed lattice, and it is assumed to
be known. Integrating partially and using Eqs. (2 · 34) and (2 · 35), we have
Energy Spectrum of Lattices with Defects. II
41
(2. 38)
The last term of the right-hand side represents the contribution of the localized modes to F((9). If the number of impurities is small, it can be neglected.
Finally we close this section with a brief discussion on tl~e moment of
G(w 2) . Applying the inverse Laplace transformation to Eq. (2 · 23) and remembering Eq. (2 ·15), we have
({J(t) =h 0 (t)
1
1
C+tco
~
+- -.1
N 2rrz J
exp(ts)d{log !n(s)},
(2. 39)
C-icn
where C is real and positive.
the integral equations (2 ·18).
This is obviously the trace of the solutions of
According to Eq. (2 ·13), fl·p is given by
(2· 40)
where
(2. 41)
p.P(O) =___g_\w 2 P(k)dk
(2n) 3 •
is the p-th moment of G 0 (w 2) . h, 1 (t) can be considered as a generating function of a random walk, starting from the site Ri and terminating at the site R 1 .
This is seen by expanding hif (t) as a power series of it:
- "' (it) p
•
•
h,1(t)- "5.:.--p.P(z-;),
p=O p!
(2. 42)
where
p.P(i-j)
=~\(v 2 P(k)exp{ik· (R,-RJ)}dl£,
(2. 43)
(2rr) "
and then inserting Eq. (2 · 4) into Eq. (2 · 43).
p.p(i-j) = i.::.m-PK(R 1 )K(R2 )
...
Then we have
K(Rp)L1(R 1 +R2 + ... +Rp+R1 -R,),
(2·44)
R 1R 2···Rp
which shows that for flp (i- j) there corresponds a random walk in which a
walker walks from the site Ri to the site RJ after p steps. In particular for
Go (w 2) , there corresponds a random walk starting from and terminating at the
same site. Considering Eqs. (2 · 33), (2 · 42), (2 · 43), and (2 · 44), we see that
the moment of (9, 1 (z) is directly connected with the random walk considered
above. Let the p-th moment of (9i 1(z) be •P(i-j), we have
•P(i-j) =rrflv(i-j).
(2·45)
Comparing it with Eq. (2 · 44), we conclude that if the coupling force K (R) is
42
S. Takeno
of the nearest neighbour type, G0 (o?) is symmmetric about w 2 = (1/2) wi, and
if the atom interacts with other atoms too, Go (w 2) becomes asymmetric : The
high frequency part of G0 (ol) is enhanced. For three-dimensional lattices, exact
calculations of Go (w 2 ) has been carried out by Montroll 23 J' 24 J only for a simple
cubic lattice with nearest neighbour interaction. In each expression for the
moment, the effect of isotopes appears as an addition term to the moment of the
regular lattice. As was shown in (I), here we meet with a random walk
among impurity sites. In general the squared-frequency distribution vanishes
except in limited regions of a? and then we are particularly interested in the
transition region between the allowed and forbidden bands. Moreover, as will
be shown later, the squared-frequency distribution near the top of the band is
seriously modified by the perturbation. Therefore we can say that the moment
method does not always provide a good description of the spectra in the high
frequency region, so the construction of the spectra with the use of the moment
method is omitted here.
.
§ 3.
Localized vibrational modes
The principal aim in this section is to solve the secular determinant (2 · 30)
for some simplified cases and then to obtain the frequencies of the localized
modes and the criterion for their appearance. When the spacial arrangement
of impurities has certain geometrical symmetricities, Eq. (2 · 30) is reduced to
the product of secular equations of lower dimensions. The solution of each
equation then yields the frequencies belonging to particular types of the vibrational modes. In order to study the type of the vibrational modes explicitly,
it is necessary to consider the amplitude of the vibration. For this purpose we
shall return to Eq. (2 ·1) and rewrite it in a somewhat different form:
..._,K(R~c-R 1-) u (R)
,:;._,--·-"
li.k
m
· u (R)
+ /.(U
j
=
2
(t)
2
u (R)
j •
(3·1)
Remembering Eq. (2 · 4), we can solve it symbolically as follows :
1
u= -w2 -
n
L0
Vu, V=i.w 2 L.L1(R-R,).
(3 ·2) *l
i=l
_Using the coordinate representation, we have
*l When w2 is within the band w2(k), we meet with the scattering problem.
concerned with an outgoing wave, then the solution is given by
u=u<Dl+---1---Vu
w2 -L0 -iE
'
where
u<OJ
satisfies the equation
If we are
43
Energy Spectrum of Lattices with Defects. II
"
(3·3)
Uz=Aw 2 l...:,ali(w2 )ui,
i=l
where we have used a simplified notation u 1 in place of u (R 1) . Equation (2 · 30)
thus results from the vanishing of the determinant formed by the coefficients
of the u's belonging to the impurity sites, which is well known in impurity
problem. 21 J First we shall obtain the criterion for the appearance of the localized
modes without specifying simple models.
(I) One-impurity problem
The secular equation
IS
given by
1- Aw2a 0 (w 2) = 0 .
(3·4)
Considering the fact that w 2 a, 1 (w 2) is a monotone decreasing function of (u 2 , we
can conclude that Eq. (3 · 4) yields a single solution if and only if J>J.o>O,
where A0 is given by
io= { lim w2ao((u 2) } -1 •
(3·5)
ru2-+roL2+0
Therefore we arrive at the conclusion that (1) the heavier isotope does not
cause the localized mode, (2) if }0 = 0, the localized mode can take place no
matter how small ). is, so long as it is positive. The amplitude of the vibration
of the atom around the impurity is given by the equation
(3·6)
where w0 2 is the solution of Eq. (3 · 4) and we have located the impurity atom
at the origin R = R 0 . When n isotopes are well separated from each other, the
nondiagonal elements of the determinant (2 · 30) can be neglected. Then the
frequencies of the localized modes are given by the solution of Eq. (3 · 4), which
are of course n-fold degenerate.
(2) Two-impurity problem
Without loss of generality, we may locate two isotopes at the sites R 0 and
Then Eq. (2 · 30) is separated into a pmr of equations:
R 1 respectively.
1-)w2 {a0 (w2) +a1 (w2)} =0,
(3·7)
=0 .
(3·8)
1- }(u 2 {a0 ((tl) -
a 1 (w 2 )}
It is easily seen from Eq. (3 · 3) that the former gives the eigen-frequency of
the symmetrical mode u 0 = u 1 , while the latter gives that of the antisymmetrical
mode u 0 = - u 1 . Since the functions w 2 {ao (ol) ± a 1 (w 2)} decrease monotonically
as w2 increases, Eq. (3 · 7) yields a single solution if and only if J.>J./ >O, where
)/ satisfies the equation
(3·9)
44
S. Takeno
while Eq. (3·8) yields a single solution if and only if J. 1>i.1->0, where J. 1is given by
J.J- = [ lim w 2 {a0 (tu 2 )
-
a1 (w 2) } ] - 1 •
(3 ·10)
ru2-7-ruL2
Remembering the fact that w 2a 1 (oi) is a monotone decreasing function of IR1 1,
we get the relation between the number of the localized modes and A as shown
in Table I, where R. denotes the impurity site closest to the origin. From this
Table I.
number of localized modes
.l<.l.+
0
at least 1
always 1, however, at most 2
always 2
.lc+<.l<.lo
.lo<.l<.l..l>.l.-
table we can conclude that in general the number of the localized modes does
not exceed the number of impurity atoms. Let the solutions of Eqs. (3 · 7) and
(3 •8). be
and lUj 2 respectively, then We have
wr
w;iz>wtz>wo2>wi2>w;;2,
w;
2 -
for all j (=/=c).
(3 ·11)
w;; 2 gives the maximum separation of two-impurity levels which are split
from a doubly degenerate level of pair of isolated impurities. As will be shown
later, this is important in considering the width of the impurity band in random
lattices.
(3) Th1·ee-impzn·ity pmblem
In studying a system of more than two impurities, one cannot in general
solve the secular equation (2 · 30). Here we confine ourselves to consider a
simplified case in which three isotopes are arranged on a straight line with
equal spacing. Let the location of three isotopes be R 1 , R 0 , and Rr, respectively, then Eqs. (3 · 3) become
(3 ·12)
By putting l = j, 0, and j', we get the secular equation determining w 2 •
easily verified that there exist two types of vibrational modes, for which
(3 ·1) symmetrical mode u 1 =;= ur
{1- J.w 2a0 (w 2)} [1- J.w 2 {a0 (w 2)
(3 · 2)
+ a11 , (w
antisymmetrical mode -u1 =
2) } ]
-2 {J.w 2a1 (w 2)} 2 = 0,
It is
(3 ·13)
- ur, u 0 = 0
1- i.lu 2 {a 0 (w 2)
-
aw (w 2)} = 0 .
(3 ·14)
These vibrational modes are shown schematically in Fig. 1 and Fig. 2. Equation
(3 ·13) yields a pair of single solutions if and only if i.>i. 1 ± (jj') >O, respective-
45
Energy Spectrum of Lattices with Defects. II
ly for the plus and minus sign, where /. 1 ± (jj') are given by the equations
(3 ·15)
Fig. 1.
Fig. 2.
Schematic representation of the symmetrical modes.
Schematic representation of the antisymmetrical mode.
while Eq. (3 ·14) yields a single solution if and only if 1.>1. 2 (jj') >O, where
(jj') satisfies the equation
/. 2
(3 ·16)
Next we shall apply the results obtained above to simple cubic lattices with
interaction between nearest neighbours only; although our principal aim is to
study the vibration of a three-dimensional lattice, in the case of one-impurity
prolem, both the one-dimensional and the square lattices will be studied in order
to elucidate the nature of the localized modes according to the dimensionality
of· the lattice. It is well known that the noncentral force must be included in
this model if it is to be stable against shear. With the use of this model, the
lattice vector Ri is represented by
(3 ·17)
where a is the lattice constant, i, j, and ·k are the unit vectors m the x, y, and
z directions respectively, and w 2 (k) is given by
r
2(k) = :Er. {1- COS (k.a)},
m
v=l
r
with
W£2= 2 :E /'.,
(3 ·18)
v=l
where r specifies the number of dimensions of the lattice, and
r.=K./m,
with
r1>r2=rs,
(3 ·19)
K 1 being the central force constant, K 2 and K 3 the noncentral force constants.
With the aid of Eqs. (3 ·18) and (3 ·19), the integral (2 · 26) can be reduced to
T=1,
(3· 20a)
r=2,
(3·20b)
r=3,
(3 · 20c)
where
l(w' 2 ; i 1 )
=
"'
\exp( -ol 2z)IiJz)dz,
vo
(3 · 21a)
46
S. Takeno
"'
I(wn; ih i2) = \
~xp( -w 12 z)I;
1
(r/z)I,.(r2 1z)dz,
(3·21b)
~
0
00
I(w 12 ; ih i2, i3) = \ exp (- w 12z) I;Jrr' z) I;. Cr2 1z) I; 3 Crs' z) dz,
(3 · 21c)
~
0
with
r
r = (I/r)
:E r.,
r
{U
12 = (w 2 -
v=l
:E r.) lr,
r.' = r./r,
(3. 22)
v=l
(3. 23)
In Eqs. (3 · 2I) the I's is the Bessel function of the imaginary argument. If
we denote the maximum value of the dimensionless frequency w 12 by w 1L2 , it IS
given by
r
w/ 2 = (wi- :E r.)/r.
(3·24)
v=l
In one- and two-dimensional lattices, the integrations of Eqs. (3 · 20) can be
carried out analytically, yielding the following results 25J :
(I) For the one-dimensional case
I (wl2; i1) = (wl4 -I) -1/2 (wl2 +
V wl4 -1) -I ill'
(3. 25)
which approaches infinity as w 12 -» I ;
(2) for the two-dimensional case
I(wl2; i1, i2) = Cr/) 1' 11 Cr/t·l
Vn(w12) 1' 1
:Ere Cli11 +I) /2) + p+q) rc (li21/2) + p+q+ I)
Cli1l+p)!(li2l+q)!p!q!
p,q
X(IL)2p
(_rl_)2q
wl2
wl2
'
(3 ·26)
which is divergent as a/ 2 --»r/ + r/. In a special case of i1= i2= 0 and r1 = r2,
it can be expresssed in a closed form as
I(wl2. 0 0) =1_ E(2/wl2)
' '
rr V w 14 - 4 '
(3. 27)
where E(x) is the complete elliptic integral of the second kind:
,.,2
E(x)
= fvi- x 2sin20 dO .
J
(3 ·28)
0
In the three-dimensional case, however, the analytical in.tegration of (3 · 2I) is
impossible, so that we must resort to the numerical calculations. For a simplified case r1 = r ~ = ra Crr' = r21 = rs' = 1) , we performed this program with the use of
47
Energy Spectrum of Lattices with Defects. II
Simpson's method, and the result is presented in Table II in which the I's is
expressed as a function of w' 2 for various values of i 1 , i 2 , and i 3 • Here we get
the relation
(3. 29)
Table II.
I(o/2; ih i 2, i 3) =
);-ID 12Z[; 1 (z)l; 2 (z)l; 3 (z)dz
for a/2=3.0, 3.5, 4.0, 4.5, 5.0, 5.5, and 6.0
as a function of il> i 2, and i 3 •
3.0
3.5
4.0
4.5
5.0
5.5
--------
6.0
----------
0.4990
0.3410
0.2824
0 0 1
0.1658
0.0645
0 0 2
0.0778
0.0220
0 0 3
0.0501
0.0035
0.00126
0.000611
0.000342
0.000210
0.000137
*
0.0242
0.0128
0.00791
0.00533
0.00381
0.00283
0 1 2
0.00765
0.00310
0.00159
0.000920
0.000577
0.000384
0 1 3
0.00236
0.000714
0.000297
0.000147
0.000081
0.000048
0 2 2
0.00318
0.00100
0.000427
0.000214
0.000118
0.0000705
0 2 3
0.00120
0.000286
0.000100
0.0000428
0.0000203
0.0000111
0 3 3
0.000527
0.0000972
0.0000281
0.0000103
0.00000440
0.00000210
1 1 1
0.0125
0.00540
0.00287
0.00170
0.00108
0.000729
1 1 2
0.00486
0.00166
0.000742
0.000382
0.000216
0.000131
1 1 3
0.00172
0.000451
0.000167
0.0000740
0.0000369
0.0000201
1 2 2
0.00228
0.000625
0.000236
0.000106
0.0000538
0.0000293
0 1 1
0.2437
0.2153
0.1936
0.0424
0.0307
0.0235
0.0187
0.0152
0.01260
0.00923
0.00737
0.00607
0.00516
---
0 0 0
0.1762
1 2 3
0.000935
0.000199
0.0000626
0.0000244
0.0000109
0.0000053
1 3 3
0.000435
0.0000726
0.0000198
0.00000647
0.00000256
0.0000011
2 2 2
0.00122
0.000271
0.0000875
0.0000346
0.0000156
0.0000077
2 2 3
0.000553
0.0000971
0.0000264
0.00000908
0.00000363
0.0000016
2 3 3
0.000277
0.0000389
0.0000897
i
3 3 3
0.000150
0.0000170
0.0000335
1
0.000000958 0.0000003
o.oooooo881 0.000000280 0.0000001
0.00000269
where the symbol "Per." means that one 1s to add the terms with zh z 2 , and i 3
cyclically permuted.
Now we shall actually calculate the frequencies of the localized modes inthe one-, two-, and three-dimensional problems. In considering the three-dimensional case, owing to the incompleteness of the preparation of the numerical
table of the l's, we confine ourselves to the case K1 = K 2 = Ka. By using Eqs.
(3 · 20), (3 · 22) and (3 · 23), let us reexpress Eqs. (3 · 4), (3 · 6), (3 · 7), (3 · 8),
(3 ·13), and (3 ·14) in terms of w' 2 and the l's.
(I) One-impurity problem
According to Eqs. (3·20)-(3·23), Eq. (3·4) 1s rewritten in the forms
J.(o/ 2 +l)J(w' 2 ; 0) =1,
(3· 30a)
/.(w' 2 +2)J(w' 2 ; 0, 0) =1,
(3. 30b)
48
S. Takeno
A(w' 2 +3)I(w' 2 ; 0,.0, 0) =1.
(3·30c)
By using Eqs. (3 · 5), (3 · 25), and (3 · 26), it is verified that for the one- and
two-dimensional lattices, ) 0 = 0, so that the localized vibration due to a single
impurity atom can take place so long as m' <m, no matter how small the mass
difference m- m' is. For the three~dimensional case, using Table II, we get
Ao=
1
~_!_
6 X 0.4990
3
(3·31)
Then from Eq. (2 · 2') we can conclude that the localized mode can take place
if and only if
(3. 31')
m'<C2/3)m.
More rigorous calculations which explicitly take into account the difference between the central and the noncentral force constants will be presented elsewhere.
At this stage it is interesting to investigate the effect of the change in the force
constant on the lattice vibration. The localized vibrations due to a single substitutional impurity atoms are studied in Appendix A. In order to see how the
localized modes appears according to the dimensionality of the lattice, it is convenient to consider a quantity x= (w 2/wi) rather than w' 2 • For simplicity, for
the two-dimensional case we shall assume K 1 = K 2 • Then by virtue of Eq s.
(3 · 25) and (3 · 27), Eqs. (3 · 30a) and (3 · 30b) become respectively
)=J~-
(3 ·32a)
1 '
X
)=; E{1/(2~-1)} j~- ~·.
The solutions of the above equations and
Eq. (3·30c) are shown graphically in
Fig. 3, in which x is plotted as a function of). From this figure we can conclude that the frequency of the localized
vibration decreases as the number of
dimensions of the lattice increases. The
localization of the vibration around the
impurity atom can most easily be seen
by considering Eq. (3 · 6). Confining
ourselves to the three-dimensional case,
we can write it as
2.0
1.5-
0
(3·32b)
1.0
Fig. 3. Localized mode due to a single
isotope in one· and two- and threedimensional lattices.
i.
Uz/Uo = ( -1) 1i. (m' 2 + 3)I(w' 2
;
lh l2, la).
(3 ·33)
Using Table·II, we have plotted in Fig. 4
49
Energy Spectrum of Lattices with Defects. II
CCJ91)
w'Z
w'2 =4
0.1
6.0
-1=1/2
(01_1) C0J2l
I I
I !
/ l i
: , I
f
:
:: ,I I f
:
,I ,I I ,
: I , !
: , I :
a/ / j
;a
I
(012)
I
0.01
5.0
(~3)
(022)
• (913)
:
/i i I
!
,/I i
// i /
(023)
0.001
// I
4.0
(2?2)
(133)
;;
(233)
0.0001 L__~---=--~--:-----;o-----===
1
2
3
4
5 vz~+z:+z:
Fig. 4. Amplitudes of vibration of atoms near
the impurity.
/b/ I
I
,,'' ,/ ' I 'b ,,/
,/ /
;'
/
;
I
// ,/ c
3.0
/
0.2
/
0.3
/
/'
'
0.4
/
'
0.5
0.6
0.7
Fig. 5. Frequencies of localized modes due to
a pair of isotopes. Curves a correspond to
the case i), curves b anc c to the cases ii)
and iii) respectively. Curve c represents
the frequency due to a single isotope.
the amplitude ratio u1/u 0 as a function of the distance V l 12 + l 22 + la 2 from the impurity atom for the case w' 2 = 4 and .i. = 1/2. It is seen that near the impurity
atom, the localized vibrational amplitude decays nearly exponentially.
(2) Two-impurity problem
According to Eqs. (3·20c), and (3·22), Eqs. (3·7) .and (3·8) become
1-.i. (w' 2 + 3) {I(w' 2 +3; 0, 0, 0) ± ( -1)1J(w' 2 ; jr,j2 ,j3)} =0,
(3·34)
which are reduced to the following pair of equations,
1-.i.(a/ 2 +3) {I(w' 2 ; 0, 0, 0) +I(w' 2 ;jr,j2 ,js)} =0
(3. 35)
1- .i. (w' 2 + 3) {I(w' 2 ; 0, 0, 0) - I(a/ 2 ; jr,j2 , js)} = 0.
(3 ·36)
The solution of the former gives the higher frequency, while the solution of
the latter the lower frequency. From the above equations we can prove the
fact that when J is even, the frequency of the symmetrical mode is higher than
that of the antisymmetrical one, and vice versa. The results of numerical calculations are shown in Fig. 5 for the cases i) j 1 = 1, j 2 = j 3 = 0, ii) j 1 = 2, j 2 = j 3 = 0,
and iii) j 1 = j 2 = j 3 = =. Here we may let Rc = i . Then from this figure .i.d and
(; are given respectively by
50
S. Takeno
(3. 37)
X; ""0.50.
(3) Three-impurity problem
Here we limit our discussion to a simple case in which three-isotopic impurity atoms are arranged on the x-axis with closest mutual separation. Then
we can let R 1 = (1, 0, 0) and RJ' = ( -1, 0, 0) respectively. According to Eq.
(3 · 20c), (2 · 22) and (3 · 23), aw (oi) and a 1 (oi) appearing in Eqs. (3 ·13) and
(3 ·14) can be· rewritten respectively as
ala1 (w 2 ) = (w' 2 +3)J(w' 2 ; 1, 0, 0),
(3. 37)
cv'2
6.0
Substituting these expressions m
Eqs. (3 ·13) and (3 ·14), we can
perform numerical calculations in a
straightforward way, and the results
are presented in Fig. 6. Let the
critical value of i. for the appearance
of the highest and the lowest frequencies in the n-impurity problem
be i.,:: and i.; respectively, then from
Fig. 6, we get
5.0
4.0
0.7
0.8
Fig. 6. Graphical solutions of Eqs. (3 13) and
(3·14).
;.
i.;- "'--'0.22,
i.; "'--'0.55 '
(3. 39)
Comparing with the two impurity
problem, we may infer the following
relations:
(3. 40)
§ 4.
Distribution of in-band frequencies and concentration expansions
of frequency spectrum and free energy
In this section we shall consider the second term of the right-hand side of
Eqs. (2 · 23) and (2 · 35), which can be considered as correction terms to g0 (s)
and G0 (w 2) respectively. As before, first we shall study one- and two-impurity
problems. Then one can write the correction terms to g 0 (s) and G0 (w 2) simply
as o-1 (s) and G1 (l,}) for the one-impurity problem, and as o-2 (s; R) and G2 (w 2 ; R)
for the two-impurity problem, respectively, R being the relative separation of
two impurities.
(1) One-impurity problem (n =I)
Remembering Eqs. (2 · 22) and (2 · 29), we have
51
Energy Spectrum of Lattices with Defects. II
(4·1)
(4·2)
where
Po(s) =sgo(s), Po'(s) = (d/ds)Po(s), PiJ(s) =sg,,J(s),
(4·3)
with.
PiJ (s) = Pli (s), PJJ (s) =Po (s), PoJ(s) = P1o (s) = P1 (s).
(2) Two-impurity problem (n = 2)
Let the relative separation of a pair of impurities be Rh then we have
a-2 (s.
'
R.) = _ _!__ {
J
N
Po' (s) + P/ (s)
1-i.{Po(s) +PJ(s)}
+
Po' (s)- P/ (,5)__}
1-i.{Po(s) -PJ(s)} '
(4·4)
G.(w•. R.) = __1_ _!]___[tan-1 i.w•{.TGo(lu2) +pJ(w2)}
'
J
Nrr dw 2
l+hu 2 {a0 (w 2) +a1 (w 2 )}
+tan-1 J.w2{11'Go(w2) -pJ(ai)} ·].
l+i.w 2{a0 (w 2) -a1 (w 2 )}
If R 1 =oo, p 1 (s), aJ(w 2) and
p1 (w
2)
(4·5)
can be neglected, then the a-2's and G2 's ap-
proach respectively
a-2 (s; R 1) = 2a-1 (s),
(4·6)
G2 (w 2 ; R 1) = 2G1 (w 2).
(4·7)
Thus the interaction between a pair of impurities can be described by the equations
(4·8)
G2 (w 2 ; R1) - 2G1 \w 2)
= _
_l__ _!]___[tan-1 {
Nrr dw 2
2Aw 2{ (1 + i.w 2a 0) -:rG_0~J.ai£!.1f11} ---}
(l+i.w 2a 0 ) 2 - (i.w 2) 2 (11'2G0 2 -{J/-a/)
l
_tan -1 {--']..i.w~7C}o (l±h~~£1'o) __ }
(1 + J.w 2a 0 ) 2- (i.w 2r:G0 ) 2
J.
(4 . 9)
Since tan-1x=x as X-40, in the low frequency region or in the limit of small
i. Eq. (4·9) approaches
(4·10)
(3) n-impurity problem
S. Takeno
52
Let O"n(s; Rr, R2, ···, Rn) and Gn(w2 ; Rh R2, ... , Rn) be the correction terms
to g0 (s) and Go (w 2) respectively when n impurity atoms are located at the sites
Rr, R 2 , · · ·, Rn-l and Rn, then we have
(4·11)
i=1,2, .. ·,n;
j=1,2, ... ,n.
If n impurities are well separated from each other, the nondiagonal elements
of the determinants can be neglected, and we obtain
IRi-:-Rjl~oo,
(4·13)
Gn(w 2 ; Rr, R2, ... , Rn) =nGl(w 2) as IR, -R1 1~=.
(4·14)
O"n(s; Rl, R2, ... , Rn) =n(Tl(s)
as
When they are distributed at random, the expressions for the O"n's and Gr.'s must
be averaged over all configurations of impurities. Here we can expand (1/N)Tr~T(s)
and G(<,i) in terms of the concentration of impurities. Then it is easily seen
that Eqs. ( 4 ·1) and ( 4 · 2) yield the terms proportional to c and Eqs. ( 4 · 8) and
( 4 · 9) the terms proportional to c2 , c = n/N being the concentration of impurities.
From Eqs. (4·8), (4·9), (4·13) and (4·14), we have
<Nl_ Tr IT(s)) =go (s) + <O"n(s; R1, R2, .. ·, Rn))
(4·15)
2
=Go (a?) + NcG1 ((li) + N_£_ 'Ew (R) {G2 (w 2 ; R) - 2G1(w2 )} + · · ·,
2 R(+O)
(4·16)
where w (R) /N is the probability that two impurities are located with relative
distance R, and the angular bracket denotes an average over all configurations
of impurities.
By using Eq. (2 · 38), we can obtain the interaction energy U ((3, R) between
a pair of impurities, which is defined as the difference between the free energy
of a system of two interacting impurities with relative separation R and that
of a pair of isolated impurities. Let the correction terms to Fa ((3). in ihe oneand two-impurity problems be F 1((3) and F 2((3 ; R) respectively, then it is given
by
(4·17)
Energy Spectrum of Lattices with Defects. II
53
F 2 (p ; R 1) and F 1 (p) can be obtained by using Eqs. (2 · 38), ( 4 · 2), and ( 4 · 9) as
follows:
(4 ·18)
x [tan-1 f-c 1 !i~:~:~ 2~·~~~-~(~~?;;;;9!a/).} Jdz
+ ~[ ~
(wj
+ wj) + ~log{ (1-ePI?mJ+)
(1-ePfiwr)}
J,
(4·19)
where w 0 , <u!, and wj are the solutions of Eqs. (3 · 4), (3 · 7) and (3 · 8) respectively. In terms of F 1 (p) and U (p ; R), F (p) can be expanded in powers of c
as follows:
2
F (p) = Fo ((1)
+ NcF1 ((1) + N_£_ I:
2
R(*Ol
w (R) U ({1; R)
+ ···.
(4· 20)
Next we shall study the actual form of G1 (w 2) by using a simple model.
Owing to the mathematical difficulty, the analytical form of Go (w 2) in the whole
frequency region has not been obtained. For this reason, we shall consider the
low and the high frequency regions separately.
(1) G1 (<i) in the low frequency region
Here the denominator in brackets in Eq. (4·2) can be replaced by 1, and
then G 1 (ai) can be expressed asymptotically as
(4·21)
If we adopt, as before, the model of simple cubic lattices of 1, 2, and 3 dimensions
with interactions between nearest neighbours only, a} (k) is given by Eq. (3 ·18)
and the corresponding squared-frequency distribution approaches 24 >
(4· 22) *>
*> The w2-dependence of G0(w2) is unaltered even when the lattice differs from the simple
cubic lattice and the coupling forces extend over more than the nearest neighbours.
54
S. Takeno
where r denotes the dimensionality of the lattice.
we obtain
Inserting it into Eq. ( 4 · 21),
Thus in the low frequency region the frequency spectra are increased or
decreased according as whether m'>m or m'<m, which is of course to be
expected from the fact that the perturbation due to isotopes is proportional to
...l. For studying the pair correlation effect, here we must consider Eq. (4·10).
However, it is of extremely higher orders as compared with G1 (oi), so that it
can be neglected. Thus we can conclude that in the low frequency region the
correlation among impurities such as pairs, triplets, etc., can be neglected and
the correction term is proportional to c. This is due to the fact that in each
expression for the correction tyrm the effect of the perturbation due to isotopes
enter into the form i.w 2 rather than i. itself, and here it has a minor effect.
(II) G 1 (oi) in the high frequency region
Here G0 (oi) depends on w 2 through the form (wL2 -oi) 112 • The correction
term contains the term (d/ dw 2 ) G 0 (w 2) , so that G 1 (w 2) diverges at w2 = oh2 except
for a two-dimensional lattice. The squared-frequency distribution of a square
lattice with nearest neighbour interaction was obtained by Montroll, 24 ) and it is
given by
(
2
, :rwV Oh2 -
K(
W2
41/~ )
wV wi- w2
Go(w 2) = .·
I
1
K(wVwL 2 -w2 )
2
\ 2;r vrlr2
4vrlr2
(4·24)
where the K is the complete ·elliptic integral of the the first kind. Of course,
Eq. (4·22) with r=2 is the special case of the above equation. By the use of
the power series expansion of the K's, Eq. ( 4 · 24) can be expanded asymptotically as
(4· 25)
(4·26)
In the one- and the three-dimensional lattices, the divergency of the correction
term at WL 2 results from the fact that Taylor's expansion of the spectra about
the point a}= wL2 cannot be used. In these cases it is not adequate to take the
55
Energy Spectrum of Lattices with Defects. II
limit N-HxJ before taking the limit E---70. If N is finite, a branch cut on the
imaginary axis of the complex s-plane becomes a series of poles and the spectra
of in-band frequency are of a-function type. Here we are concerned with the
shift of each in-band frequency. Now we must return to Eq. (2 · 39). For the
sake of simplicity, let us confine ourselves to the consideration of one-impurity
problem. In this case Eq. (2 · 39) becomes
O+ioo
exp(ts)~log{1-i.sg0 (s)}ds,
cp(t) =h0 (t) +__!_ _l_C ·
N 2rri J
ds
(4· 27)
0-ico
where m place of Eq. (2 ·17) g0 (s)
go(s) =
IS
given by the equation
[Go(~) dz---?N1.L;
J s-zz
~
• s-zz.
(o) '
(4· 28)
where z. (OJ is the squared-frequency of the unperturbed lattice. Noting that
(d/ ds) log(1- i.p 0 (s)) ---?0 as isi---?oo, we can reduce the Bromwitch integral in Eq.
( 4 · 27) to the contour integral on a closed path C which contains all zetosand
all poles of the function 1- i.p0 (s) . By the fundamental theorem of residue,
Eq. ( 4 · 27) yields the result19l
cp (t) =~I; exp (iz.t),
N•
where z. (v = 1, 2,- · ·, N)
IS
the solution of the equation
•
2
~"
..::.....
N
so that G(w 2)
IS
(4· 29)
•
1
(1}-z.
1'
(4· 30)
given by
(4· 31)
y
Fig. 7. Graphical solution of Eq. (4·30) for -l>O. The
arrow denot:os the shift of each in·band frequency.
Equation ( 4 · 30) represents the dispersion relation for i., and has been studied
by several authors in the case of one-dimensional lattices. Equation ( 4 · 30) is
solved graphically as shown in Fig. 7.
S. Takeno
56
§ 5.
A random lattice and perturbation calculations
Let us suppose that n isotopic impurity atoms are distributed at random in
a lattice . . Here the concentration c=n/N is assumed to be finite. Then for the
evaluation of G(w 2) , we must make use of the following procedures: (a) solving the simultaneous equations (2 ·16) for a fixed configuration of impurities,
and then taking the trace, and (b) averaging the resulting expression over all
configurations of impurities and then taking the limit s-?iw2 + E • Thus we have
(5·1)
where the angular bracket denotes the average mentioned above. In the previous
section we have expanded G((u 2) into the power series of c. However, the
formula (4·16) which separates G(w 2) into the unperturbed and perturbed terms
is not so useful for the present case. Moreover it is very difficult to apply the
average procedure directly to G~ (w 2 ; R 1 , R 2 , · • ·, Rn), since the dimension of the
determinant ( 4 · 12) is of the order N.
Here we shall present an alternative method for solving Eq. (2·16) / 9J which
is particularly of use when n-H>O. By the method of iteration, they can be
solved as
(5·2)
where the letters a, b, c, ··· denote the impurity sites, while l and m are the
lattice sites for which the coordinate representation of the o-'s is taken. Equation
(5 · 2) is equivalent to the following set of .equations,
O'lm = glm + i.''i:,fla!Ja.,. with !Jam= gam+ i.''i:,fab!Jbm·
a
(5·3)
b
In order to prohibit the repeated sums over the impurity sites, we shall rewrite
the equation for JJ's as
JJ
with
am
=-1_- gam +J._ '\;'IFb JJb
d
d •';;JJ a m,
1 + ,.{'
d(s) =1-i.sg0 (s) =1-..ip0 (s),
(5·4)
(5·5)
where the prime on the summation indicates that a=/=b, and the use has been
made of the relation (2 ·17). Inserting Eq. (5 · 4) into Eq. (5 · 3), and considering only the diagonal element of the a-'s, we have
+ ... ,
where, as before, the primes on the summation indicate that in the sums over
b, c, d, .. ·, each suffix on the g's must not equal its left direct neighbour. · By
Energy Spectrum of Lattices with Defects. II
57
making use of the relation
(5·7)
which can be easily obtained by using the integral representations (2 · 17) and
the periodic boundary condition:
:E exp(i(k-k') ·R
1)
=Ni3(k-h'),
(5·8)
l
we have the result
r
(~ Tr a-)=go- ~~<Paa')- ( ~ ~L;'<PabPba')- (~ r~L;'~'<PabPbcPca')- ···.
(5·9)
The second term of the right-hand side is obviously proportional to c and in
fact it is equal to Nca-1 (s).
Equation (5 · 9) is the basic equation with which we start to calculate
1
N- <Tra-(s)). It is a kind of perturbation expansion in terms of the exact solution of one-impurity problem, and considerably improves the approximation in comparisian with such a conventional perturbation method as expanding N-1 <Tra-(s))
in terms of the power series of A. In order to give approximate expressions for
the perturbation expansion to any desired order, a graphical method is presented.
For this purpose let us make bel d, - p' cal d, and - p' old ( = p' aal d) correspond
P.c(s) = ~,
d(s)
Fig. 8.
p;"(s) =~---~---':,
_ d(s)
_
P6(s) =
d(s)
O
Graphical representation of held, -P'cafd, and -P'ofd.
to the line, the dashed line, and the white point respectively as shown in Fig.
8. beld can be considered as a " walk " from the site Rb to the site Rc, while
- p' cal d can be considered as the last walk from the site Rc to the starting site
Ra. Thus we can represent each term in the right-hand side of Eq. (5·9)
graphically as shown in Fig. 9, in which for each impurity site denoted by a
vertex point, a ±actor A is associated, while for the impurity site Ra denoted by
the white point a factor - po' I d is associated. From Eq. (5 · 9) it is seen that in
each graph, the dashed line or the white point can apear only once. Therefore
N-1 <Tr a- (s)) is graphically represented by a sum over all the graphs which
represent various possible walks starting from and terminating at the site R,.
At this stage it is to be noted that the determinant ( 4 ·11) is equivalent to such
diagrammatic expansion as shown in Fig. 9.
When we consider the walk from the site Rb to the site Rc, it is more convenient to consider the following series in place of bel d :
ro
(Pbcl d.) :E (ipbcl d) 2" = i.pbclbbc,
JI=O
(5 ·10)
58
S. Takeno
~0
2 •
1
3
4
1
2
--------- ----------------------- ------------------ ------- ------1
3
4
I
a::::>
<)
---------
b
<:x:::>
b.
5
<J
Q
I
-----------------------~
-
-
----------------- ------ ------
:
00. If Llli
-c)>
4:,.
:~~~
I
--------- ------------------ ----i -- --------------- ------- ------
<>
b
oc::x::::>
~
:
~o1y
O
6
4:\
~
j[~
~-~j ~ ~
l
L Lt>
~ 1()<)<)(>
o·
------------------ ________________ _[ ------------------- ------- -------
06./{d:~~~
~~
4:::::Fig. 9.
Graphical representation of the typical terms in (5·9).
X is equal to the exponent of 0/d), and Y is the total number
of points (vertex points + a white point).
where
(5 ·11)
Similarly in place of - p' cal d, we get
"'
- (Pca I d)~ CJ.Pcald) 2" =
1
J-1=0
h,,.(s)
b1,.(s)
i.p' cal bca ·
(5·12)
~ ~ ~ ~ + ~- + ~- + ~ + --------
-j;/,,.(s) ~ ~- ~ ~------~
b1,.(s)
-
~
+ ~ + ~- + b~c + ________ _
----
--~-- ·
~
Fig. 10. Graphical representation of hc(s)jbbc(s) and
-h/(s)jbbc(s)
Graphical representation of Eqs. (5 ·10) and (5 ·12) are shown in Fig. 10. By
using these thick and double lines as a new frame, let us reconstruct all graphs
59
Energy Spectrum of Lattices with Defects. II
in Fig. 9 except for the isolated white point. This reconstruction can be achieved by replacing each line and dashed line by the thick and double lines respectively. For convenience's sake, let us call that part of Fig. 9, which is enclosed
by the broken line, and its remaining part the part (I) .and the part (II) respectively. Then it is easily seen that we have no need of applying the replacement to the graphs in the part (II). Thus we get "new graphs", some of
which are shown in Fig. 11. By means of this procedure, almost all graphs
1
0
2
0
2
3
4
<>
--------- -----------------------
l:l.
Fig. lL
Graphical representation of new graphs.
can be included in new graphs; here only the graph with asterisk has been left
out, which represents the correlation among triplets.
Now it is convenient to consider a quantity ¢a' defined by
(5·13)
Here, of course, we must exclude such new graphs that appear in the part (II)
in Fig. 9. For hc/bbc there corresponds a "new walk" from the site Rb to the
site Rc so that ¢a, represents the sum of all new walks starting from and terminating at the site Ra'. Then N-1 (Tr o-(s)) can be well approximated by the
series 20 J
(5 ·14)
(5 ·15)
where double primes on th~ summation indicate that all suffixes must be different. A graphical representation of Eq. (5 ·14) IS given in Fig. 12, where in
60
S. Takeno
Fig. 12.
Graphical representation of Eq. (5·14)
Fig. 13.
Graphical representation ·of
~.
~=~=o<::::D +~+ ~+··········
Fig. 14.
Graphical representation of
~-
each vertex and white point, a factor (A (1 +~))/d) is associated. *l An ap. proximate expression for f can be obtained by the same manner. Let us represent f graphically as shown in Fig. 13, then we can classify graphs for f
according to the number of returns to the starting site Ra' on the way (see
Fig. 14). Here we take only those graphs in which different walks include
no common intermediate sites, and make use of the following approximation,
f='1+'12+'13+ ... =-1--1,
1-'1
(5·16)
where '1 represents the sum of all "irreducible". new walks which never pass
the starting sites on the way. '1 itself can be calculated in the same way as Eq.
(5 ·14) (see Fig. 15). Considering Fig. 15, we can approximate '1 by the senes
(5 ·17)
Fig. 15.
An approximate expression for '11·
Equations (5 ·13)- (5 · 17) determine the expression for N-1(Tr a- (s)) in which
the correlation among impurities is exactly taken into account up to pairs.
Unfortunately, however, both of series (5 ·14) and (5 ·17) cannot be summed up as they stand. For this reason, here we shall replace each thick line
and double line in all new graphs in Figs. 12 and 15 by the line and the dashed
line respectively. This procedure corresponds to make use of the following
approximation for the b's
*l Here the site Rat is to be understood as one of the vertex point or the white point. Owing
to the random arrangement of impurities, ~a' is obviously independent of a 1 .
Energy Spectrum of Lattices with Defects. II
bbc(s) =d(s) independent of b and c.
Then Eqs. (5 ·14) and (5 ·17) become respectively
r
<~ Tr (} )= g -- 1-d(A~(Paa')- ( 1 ~~A
0
7J = 1 ~~[
e~ ~;r ~"
(PabPba) +
'[;/'<PabPba')-
e~fA) 'f"
3
X
(PabPbcPca) +
61
(5 ·18)
er
e~ r
:!_A
(5 ·19)
f;
:E" (PctbhcPcaPaa) + ···] ·
(5 ·20)
abed
As stated before, these are expansions in terms of the solution of the oneimpurity problem. If the spacial arrangement of impurities is completely random,
they can be summed up to yield the results
(5 ·21)
J
1 + f ·)} -go (.s) .
cis [ go {s (1 -dcA
7J=d
The derivation of the above equations are presented in Appendix B.
Eqs. (5 ·16) and (5 · 22), (1 +f)/d is reduced to
(1 +~) /d= (1-7])
-ld- =[
1
1- (1-c)i.sgo(s) -c)sgo{s(1- 1 ~fd)}
(5. 22)
By using
rl
(5·23)
Equations (5 · 21) and (5 · 23), although they constitute transcendental equations,
yield the expression for N-1(Tr CJ(s)) in a closed form.
By making use of Eq. (5 ·1), now let us obtain the expression for G(li).
Let
(5. 24)
then, using Eqs. (2 · 9') and (5 · 21), we have
G ( o}) = l_
rr
r[ {1 - ci. .J (
w 2)
-
id
r (w
2) }
____!!__ f
dk
(2rr) 3 Jai(1-ci..J(ai)) -w 2 (k) -ici.w 2r(ai)
z--c-G_o~(z~)_d=-z_ _--=-_
_ ciP ( w 2 ) \
"{w2 (1-ci.LI(ai)) -z} 2 + {ci.w2 r(oi)} 2
rr
(5·25)
'
(5·26)
where
]
S. Takeno
62
(5·27)
G" (w 2) =cJ.o?T(w2 )
f
J [oi(1-cJ.LI(w
Go(z)
-z] 2 + [cJ.w 2T(o?)]2
dz
2) )
'
(5. 28)
Here I (A) denotes the imaginary part of A. Considering Eqs. (5 · 23), (2 · 26),
and (2 · 28), we see that L1 (o?) and T(oi) take different forms according as
whether we are concerned with the case (i) or the case (ii). Then from Eq.
(5·26), we have
for the case (i)
Ll(w2) =- -~--:t:_=-Jl-:c)i:_(IJ_~_o_(w 2)_~c~o?G_'_-_C<_u2_,_)_ __
[1- (1-c)J.w 2ao(w 2 ) -cJ.w 2G' (w 2) ]2+ [cJ.w 2G" (w 2) ] 2 '
r (w2) =
dw2G" ( w2)
[1- (1-c)J.w 2a 0 (w 2) -cJ.w 2G'(o?)] 2+ [da?G"(w2)]2'
(5 ·29)
(5. 30)
for the case (ii)
L1(w2)
=
1+ (1-c)J.w 2a 0 (a?) -ci.<u 2G'(a?)
[1 + (1-c)J.w a 0 (w 2) -ci.tu 2G' (w 2 ) ]2+ [ (1-c)i.w 2{1 0 (w 2) +cJ.w2G" (w 2)] 2 '
(5·31)
2
r (w2) = ___ _
-c) i.t':ij9_o_( o?) + ci.tu 2G" ( w2)
·
2
[1+ (1-c)i.o?a 0 (w ) -ci.w 2 G'(w 2)] 2 + [(1-c)i.w 2{1 0 (w 2 ) +cJ.w2G"(o?)] 2
(5. 32)
_
__ -----~~
The series (5 ·14) and (5 ·17) can be summed up, if we replace the b- 1 's
in the angular brackets by its average (b- 1 ) :
(5. 34)
and if we make use of the following approximation for the second term of the
right-hand side of Eq. (5 ·14) :
(5. 35)
According to Eq. (5 ·11), (b- 1 ) is expressed explicitly as follows:
(b-1 )
= 1/2( {1- A(Po+ PJ)} - 1 + {1- A(Po- P1)} - 1 )1,
(5. 36)
where the angular bracket denotes an average over all relative separation R 1 .
Then, remembering Eqs. (5·21), (5·22), (B·8), and (B·9), we have the results
Energy Spectrum of Lattices with Defects. II
N-1(Tr o-(s)) = (1-cJ. (1 + ¢) (b-1)) g0 {s (1-cJ. (I+~) (b-1))},
-n=
.,
cAs
d2(b-1)
[g0 {s(1-cJ.(1+¢)(b-1))} -go(s)J.
63
(5. 37)
(5. 38)
Obviously the approximation (5 · 35) is not valid at low concentrations ; however, it may be used at high concentrations. By virtue of Eqs. (5 ·16) and
(5 · 38), (1 + ¢) (b- 1) becomes
(I+¢)(b-1)= (1-r;)-1(b-1)
= {d(b-1)} 2 [d2(b-1) + ci.s{g0 (s) - g0 {s(I-c (1 + ¢) (b-1 ) ) } } ] - 1 •
(5. 39)
Let
lim(l+¢(iw2 + E))(b-1 (i(li+ E))=Ll(w 2) +iT((Il),
(5· 40)
f-)-0+
then it is seen from Eqs. (5 · 37) and (5 · 39) that here G((i), G' ((11"), and G" (w 2)
are still given by Eqs. (5 · 25), (5 · 27), and (5 · 28), respectively. By using Eqs.
(5·39), (5·11), (4·3), (2·17) (2·26), and (2·28), we can express Ll(w 2) and
(w 2) explicitly as
r
for the case (i)
J(w2)=
:
(b~iw 2 )_?_ 1 +ci.w 2 A(iw 2 ){a0 (w 2 )-G~(w 2 )}.
"
,
[(b(uu 2) -1 ) - 1 +cl.w 2A(zw 2) {a0 (w 2) -G'(w 2)} ] 2 + [cl.w 2A(uu2)G"(w2 )]"
(5· 41)
for the case (ii)
J(w 2)=Re[ lim
Hiro 2 +'
(b (s)
- l ) -1
.
1
.
+ cl.sA(s) [g 0 (s)- g0 {s(I- cl. (1 +
~
~)
],
(b(s)-1 ) ) } ]
(5. 43)
where
(5. 45)
Equation (5 · 25) shows us that G(w 2) can be expressed as the Poisson integral of Go (w 2) , so that we can evaluate the former if the latter is known.
ciw 2 J ((ll) and c)w2 (w 2) represent the shift and the damping of G0 (w 2) respectively,
which characterize the spectrum of the random lattice. In the case (i) if we
r
64
S. Takeno
let ~=0, F(ol) vanishes, whereas J(w2) remains finite. We may therefore conclude that here the spectrum are mainly determined by the shift term. As will
be shown later, this conclusion is valid only when i. is so small that it does not
produce localized modes; the damping term F(o}) plays an essential role for
the determination of the impurity band. In the case (ii) the damping effect is
mainly determined by G0 (w2) . We can therefore expect that the spectrum in the
middle of the band is decreased due to the damping effect irrespective of the
sign of i.. From Eqs. (5 · 27) and (5 · 28) we see that G" (w 2) is intimat~ly
connected with G(w 2) , so that G(w 2 ) depends on F((i). This result is to be
expected from the fact that the transition probability is proportional to the
final density of states.
§ 6.
Squared-frequency spectrum of a random lattice
Using the results obtained in the preceding section, we shall actually construct the squared-frequency spectrum of a random lattice. First we shall obtain
the general property , of the spectrum which does not depend on a particular
model of the lattice. In order to obtain the actual form of G(w 2) which is given
by Eq. (5 · 25), we must solve the equations determining L1 (w 2 ) and
(w 2 ) by
the use of a suitable approximation.
In the low frequency region, the damping factor F(w2) can be neglected,
and the shift factor L1 (o}) is approximately given by
r
L1 (w2)
=
1
1 + i.w 2a 0 (w 2)
(6·1)
Then by virtue of the relation
o(x) =~ lim __
E_
rr ·~o+
x•+ E2
(6·2)
'
Eq. (5 · 25) becomes
G(oi)=(1-
d
1 + i.w 2a 0 (oi)
)G { ·(1
0
(I)
-
c)
1 + ..lala0 (w 2 )
)}
'
(6·3)
which approaches
G(w 2) = (1-ci.)G0 {w2 (1-d)} as w2 -40.
Since Go (w2) is proportional to (w 2)
2
G(aJ)
=
(
1' 2 ,
(6·4)
Eqs. (6 · 3) and (6 · 4) become respectively
3 2
2
1 - - -d- - -) / Go (w),
1+i.w2a 0 (0J) 2
(6·5)
(6·6)
At low concentrations, the above equations are reduced respectively to
Energy Spectrum of Lattices with Defects. II
65
2
G(w 2) =(1-1_
. cl.
2 )Go(w ) ,
2 1 + l.lu2 a 0 (w)
(6·7)
G(w 2) = (1-l_ci.)G0 (w 2) as w2 ---70.
2
(6·8)
The result (6·8) is coincident with Eq. (4·23).
Next we shall make use of the approximation ~ = 0 (r; = 0). This corresponds to summing up only such~graphs that lie in the first column in Fig. 9. First
we shall consider Eq. (5·21). Then L1(w 2 ) and F(lu 2 ) are given respectively by
for the case (i)
L1(w 2) =Re{d(iw 2+ E)}-1=
. ;
2 ,
1-l.w ao(w)
(6·9)
F(w2) =I{d(iw2+ E)}-1=0,
(6 ·10)
for the case (ii)
L1 (li) =Re {d(iw2+ w)} - 1 =-------__!:+:_J.tu 2 a0~0-)_ _ - - [1 + J.w2ao ( w2) J2+ [J.w2~o ( w2) J2
(6 ·11)
F(w2) =I{d(ilu2+ E)}-1=
(6 ·12)
rrJ.w2Go(w2)
-[1+J.w2ao(w2)J2+ [J.w2~o(w2)]2
In the case (i), from Eq. (6·2) we have
G(o})=(lcl.
·
1-l.w2a 0 (w 2)
In order that this approximation
relation,
IS
)co{w (1- 1-J.wcl.a (w
2
2
0
2)
)}·
(6 ·13)
valid, ). and c must satisfy the following
0<1- 1-J.w2ao(w2)
cl.
<~L<1.
w2
(6 ·14)
Remembering that 1-l.w2a 0 (w2 ) vanishes at l.=i. 0 , we see that Eq. (6·13) cannot be used in the cases i.~i. 0 and i.<O. In the other case it may be used as
an approximate expression for G(w 2) which represents a tailing off of the main
continuum into the forbidden region,*> the maximum frequency w~l being determined by the equation
(o)
lUm =wL
{1
-
cJ.
1- J.w 2 a 0 (w2)
} -1/2
•
(6 ·15)
In the case (ii) we must put Eqs. (6·11) and (6·12) into Eq. (5·25) for the
evaluation of G(w 2). We now wish to consider the function a 0 (w2 ) that appears
m the denominators of the equations determining L1(w 2 ) and F(w 2 ) . Consider*l For one- and two-dimensional lattices it cannot be used in any case, since if J.>O, the
localized vibrational mode always appears no matter how small ;. is.
66
S. Takeno
ing Eq. (2 · 32), we can see that this function
becomes - a 0 (w 2) for ol>wL2 , and within the
band its behavior is determined by Go (w 2 ) . If
G0 (w 2) is a fairly smooth function, it will have
the general form of a dispersion curve as illustrated in Fig. 16. It is easily verified that within
the band a 0 (w 2 ) and - a 0 (w 2) are symmetric with
each other about the point w2 = (1/2)wL2 if
Go (w 2) is symmetric about the same point.
While the damping is mainly determined by
Fig. 16. Typical plot of a 0 (co2)
Go ( w 2) , so that we can see that it is most effective
vs. co 2.
in the middle of the band, whence the spectrum
of the unperturbed lattice is diminished in that region irrespective of the sign
of i.. Equation (6 · 3) shows that when i.<O, G0 (w2) is pushed towards the low
frequency region ; if we neglect the damping term, the maximum frequency w, is
determined by the equation
w=w {1+
8
L
eli.!'
}-.112
2
1-li.lolao(w )
(6 ·16)
Then we see that the effect of the damping causes a smearing out of the spectrum into the region w2 >w/ and it vanishes at d = w\.
Next we consider the spectrum which is given by Eq. (5 · 37). Since the
perturbation due to isotopes plays an important role in the high frequency region,
it is supposed that in the case (i) the pair correlation effect should be included.
Letting ~ = 0 in Eq. (5 · 37), we have
G(ol) = (1-ci.<b(iw2) -1 ))G0 {w2 (1-ci.<b(iw 2) -1 ) ) } ,
(6·17)
which means that c and i. must satisfy the relation
(6·18)
From Table I we see that if i."" ;: , the approximation (6 ·17) is no longer valid.
When i.<i.c +, it yields an approximate expression for the spectrum, the maximum
frequency (ui::~ being determined by the equation
w;::~ =
(UL
{1- ci.<b (iw2)
- 1 )}.
(6·19)
In order to see how the pair correlation effect modifies the spectrum which is
given by Eq. (6 ·13), we shall consider a quantity b1 (iw 2) - 1 - (1- hu 2a 0 (w 2) ) - 1 •
By virtue of Eq. (5 ·11), it becomes
. 2)-1 - {1 - ;.·.w2ao ( w2)}-1_
{J.w2a 1} 2
l?1 ( zw
- -..
{1- i.w 2 (a0 + a1)} {1- i.w2 (a0 - a1)} (1- i.w2 ao)
(6. 20)
Considering the fact that Eq. (6·17) can be used only when i.<i.:, we can prove
Energy Spectrum of Lattices with Defects. II
67
Fig. 17. Squared-frequency spectrum near the top of the band. Curve
(1) is calculated from Eq. (6·13), Curve (2) from Eq. (6·17),
and Curve (3) from Eq. (5·25) by using ~=0 and Eqs. (6·11)
and (6·12).
>
the relation b1 (iw 2 ) - 1 (1- l.w 2a 0 (w 2) ) - 1 for any j, so that w~~>w~l. Therefore
the tailing off of the spectrum is expected to be enhanced when we take into
account the higher order correlation effects such as pairs, triplets, quartets, etc.
However, there should be a limit of this tailing. The -spectrum which is obtained by using the approximation ~=0 is shown schematically in Fig. 17.
Equations determining J (w2 ) and
(w 2 ) constititute a set of transcendental
equations,. so we cannot solve them as they stand. In the case (i), however,
we can determine the edge of G(w 2) . Equations (5 · 25), (5 · 28) and (5 · 30)
show that if (I} satisfies the relation G"(w 2 ) =0, F(w 2 ) vanishes, and then G(w 2 )
and G" (w 2) become .respectively
r
G((li) = {1- c/.J (w2 )} G0 {w2 (1- ci.J (w2 ) } ,
(6. 21)
G" (w 2 ) = G0 {a?(1-c/.J (w 2) ) } .
(6. 22)
Thus the zeros of G(w 2) and G" (w 2) are coincident with each other and given by
the equations
1- ci.J (w 2)
=
wL2/
w2
(6. 23)
,
1-ci.J(w2 ) =0.
(6 ·24)
It is easily seen that the former equation gives the frequency of the top of
G(oi), while the latter that of the bottom of the impurity band. Using the
above equations, we can give G' (w 2 ) as
(6 · 25a)
(6·25b)
where we have used the well-known identity
(6. 26)
Then J ( w2 ) becomes
1
J(w2)
=
1- (1- c) l.w 2a 0 (w2 ) - ci.w 2G' (w 2 )
for Eq. (5 · 29),
(6. 27)
{
<b(iw2)-1 )-1 +cl.w2A~iw2) {a (w
0
2)
-G'((Ii)}
for Eq. (5·41). (6·28)
S. Takeno
68
If we use Eq. (6 · 27), Eqs. (6 · 23) and (6 · 24), are rewritten respectively as
(6. 29)
(6. 30)
while if we use Eq. (6 · 28) and take into account Eqs. (5 · 36), ( 4 · 3), and (2 · 26),
they become respectively as
(6 ·31)
1
2
{I
1
\(1/..i) -oi{a0 (oi) +a1 (w 2)}
1
+(i/i.\~-c~ 2-{a0 (w~)
) }
-a1 (w2)}
= c [I- w 2A (iw 2) {a0 (w 2) + a 0 (0)}].
-l
1
(6 ·32)
From Fig. 16, we have
- ao (wL 2) =lim a0(cth 2+E), a 0(0) >O.
(6 ·33)
E-:)>0+
Then considering Eq. (2 · 26), we can easily verify the relations
- ao (cuL2) >ao (w 2) , w 2a0(w2)
>1 ,
(6. 34)
which show that the right-hand side of Eqs. (6·29)-(6·32) are positive. We
can therefore prove the fact that when i.<O, the spectrum does not appear in
the region w 2 >wL2 • Since in this case F(w 2) vanishes at w 2 =wL2 , G(w 2) also
vanishes at w 2 = wL2·
Next we shall consider the case i.>O. Since 1-i.cu2 {a0 (w2) +a0 (0)}<0,
Eqs. (6·30) and (6·32) have no solutions respectively for the cases i.<Jo and
i.<i.:. In these cases Eqs. (6 · 29) and (6 · 31) yield the frequencies of the top
of the spectra which are shifted and tailed off into the forbidden region. Let
w;, and w;,c be the solutions of Eqs. (6 · 29) and (6 · 31) respectively, from Eqs.
(6 ·15), (6 ·19), (6 · 29) and (6 · 31), we can easily obtain the following relations:
(6. 35)
Thus the tailing off of the spectrum is enhanced as we improve the perturbation
calculation to higher orders, that is, if we take into account the number of walks
as many as possible for given vertex points. In the case i.>i.o, both of Eqs.
(6 · 29) and (6 · 30) have solutions, which means that an impurity band takes
place resulting from the broadening of a single impurity level due to an isolated impurity atom. Similarly in the case ..i>i.c +, Eqs. (6 · 31) and (6 · 32) yield
an impurity band which results from the broadening of many impurity levels due
Energy Spectrum of Lattices with Defects. II
69
to various relative separations of i pair of impurities. We can see from Fig. 5 that
here the number of impurity levels are determined by i. . By noting that the function i.w 2a 0 (w 2) +c[1-(•l{a 0 (w 2) +a 0 (0)}] decreases monotonically as oi increases
it is seen that Eq. (6 · 30) yields a single root wb 2 . Thus we can prove the fact that
the approximation (5 · 21) gives a single impurity band. A similar result will be
obtained from Eq. (6 · 32). This result is in conflict with those of one-dimensional
studies made by Domb et al. and Hori by using the moment trace method. It
seems to the writer that this conflict is due to the ill convergency of the polynomial fit to the spectrum in the high frequency region. As will be shown later,
Eq. (6 · 29) yields a single solution for a given ..l. Then we can conclude that
when an impurity band takes place, the spectrum of the main continuum does
not enter into the forbidden region w2 >wL2 • Here w,.2 - (Vb 2 represents the width
of the impurity band. Similar states of affair will hold for Eqs. (6 · 31) and
(6 · 32). Let the solution of the latter equation be w~e, then w!.e- w~e represents
the width of the impurity band. Here we get the relation w!w- w~e>w~- wb 2, since
Wme>wm and (Vbe<wb.
Therefore we can say that the width of the impurity
band increases as we take into account the correlation effect to higher orders.
Now we can obtain the criterion for the occurrence of the gap between the
main continuum and the impurity band respectively for the approximations (5 · 21)
and (5·37):
i.>lim[uia0 (w 2) +c[1-w2 {a0 (w 2) +a0 (0)} J]-1
for Eq. (6 · 30),
(6. 36)
m2-tw2L
(6· 37)
>lim [c [1- w2 {a0 (w2) + a 0 (0)} JJ -1
for Eq. (6 · 32).
(l)2~w2L
Now let us apply the results obtained above to a simple cubic lattice with
the nearest neighbour interaction. In order to perform actual calculations of the
spectrum, first it is necessary to determine the edge of the band (the main continuum or the impm:ity band). For this purpose we shall consider Eqs. (6 · 29)
and (6 · 30). By noting that G0 (w 2) is symmetric about w2 = (1/2) a)£2 , then
we have a 0 (0) =a0(wL 2+0). Using Eqs. (3·20c), (3·22), and (3·23), we can
rewrite Eqs. (6 · 29) 'and (6 · 30) respectively as
~
=
(w' 2 + 3)1(w' 2 ; 0, 0, 0)
+c[ (1-~~~~-:3) -
1
+ ((V 12 +3) {1(3; 0, 0, 0) -1((V 12 ; 0, 0, 0)}
~=
A
J,
(6. 38)
(w' 2 +3)1(w' 2 ; 0,0,0) +c[1- (w' 2 +3) {1(3; 0,0,0) +1(w' 2 ; 0,0,0)}]. (6·39)
70
S. Takeno
U sin~ Table II, we can solve these
equations graphically as shown in Fig.
18, where the solution of Eq. (3·30c)
is also presented in order to illustrate
how the frequency of a single isolated
impurity level is broadened into a
band.
So Jar we have not specified the
shape of G0 (w 2) . The complete G0 (w 2 )
curve was obtained by Montroll. The
results are presented here for the cases :
(1) rl=r2=ra and (2) r1=8r2=8ra
(see Fig. 19a and 19b). Then the
outline of the G(w 2) curve in the
whole region can be obtained from
the above arguments and from Eq.
(5 · 25) without carrying out detailed
calculations. For a given value of the
concentration c, we are mainly interested in the four cases: (A) i.<O, (B)
O<..t<J 0 , (C) ..l$..lo, and (D) ..t>io,
and we have shown the typical G(w 2)
w'2
c=0.01
6.0
c=0.3
r=0.05;J c=0.05 c=0. 1
c=0.1
5.0
4.0
c=0.3
1.0
.lo
.l
Fig. 18. Frequencies at the top of the main
continuum 0<-1 0 ), and at the top and the
bottom of the impurity band CA>-lo) as
a function of ,l for c=0.01, 0.05, 0.1, and
0.3. The thick line denotes the solution
of Eq. (3·30c).
2.0
1.5
1.0
1.0
0.5
0.5
a
Fig. 19. Squared-frequency spectrum of regular simple cubic
lattices. Figures a and b correspond to the cases (1) r 1 =
r 2 =r3 and (2) r1 =8r2 =8ra, respectively.
curves in Fig. 20 for the case (2) at low concentrations. Here the parts of
the original G0 (w 2 ) curve which are abruptly flattened near x=w 2/ol£2 ~0.2
and x~0.8 are smoothed out.
In plotting the G(w 2) curve numerically as a function of c and ), we shall
confine ourselves to consider only those L1 (al) and
(w 2) which are given by
Eqs. (5 · 29)- (5 · 32), and to deal with the simple cubic lattice with i. 1 = A2 = i. 3 •
For this purpose we must determine the analytical form of a 0 (w 2) . According to Eqs. (2 · 29) and (2 · 32), it is given by
r
Energy Spectrum of Lattices with Defects. II
o.\------~bc------,-17..o---'---
71
x
o/fG(w2)
impunty band
0~------c---,l~.o~-l--x
Fig. 20. Typical plot of the G(w2) curve. Figures a, b, c, and d
correspond to the cases (A) ..!<O, (B) 0<..!<.-!0, (C) ..!:::;..! 0, and
(D) ..!>.10, respectively.
(6. 40)
As was shown in Fig. 19, Go (w2 ) cannot be expressed as an analytic function in the whole region
of w2, so that a 0 (w 2) can be evaluated only approximately. Near w2 = 0, Go (w 2) =Cv' w2 • Then by
putting it into Eq. (6 · 40), we have
;---·----\.
1.0
\,,
•,
~~~~--r!-r.·-1,
0.6 0.8 1.0
W
2/ 2
WL
(6. 41)
where 2CwL should equal lim a 0 ((ui +E).
E_,.O+
However,
<
this result is valid only in the region w2 (1/2)
wL2 • By virtue of the fact that Go (w 2 ) is symmetric
with respect to (1/2) wL2 , a 0 (w 2) has the form as
shown graphically in Fig. 21.
Fig. 21. Graphical .represenIn order to plot G((u2) curve explicitly as a
tation of a- 0 (w2).
function of A and c, it is more convenient to introduce the following dimensionless quantities:
x = w2/wL2, G (x) = wL2G(l•i), G 0 (x) = wL2G 0 (w 2) ,
ao(x) =(uL2a'o(w2 ) , iio(x) =(uL2 ao(x), G'(x) =wL2G'(w 2 ) , G"(x) =wL2 G"(w 2 ) .
(6. 42)
Then Eq. (5·25), (5·27), and (5·28) can be rewritten as
72
S. Takeno
r
1
= ci.r
G(x)
yGo(y)dy
(6. 43)
J [x(1-ci.LI)-y] 2 +[ci.Tx]2'
rr
0
1
G'(x)
= \ _[x(1-ci.LI) -y]Go(y)dy
" [x(1-cyLI)-i.]2+ [c..lTx] 2
(6. 44)
'
0
1
-"( X-CI.
)- ·rX \
G
Go(y)dy
"[x(1-ci.LI)- y] 2 + [ci.rxr
0
(6 ·45)
.
.
These integrals can be evaluated numerically by the aid of Fig. 19 and by
evaluating Ll(x) and T(x) numerically.
In what follows we shall present the results of numerical calculations respectively for the cases (A) and (B).
1) The case (A)
It has been shown in the previous arguments that .in this case the approximation ~ = 0 does not cause any essential difficulty in the perturbation calculation. For this reason we shall employ Eqs. (6·11) and (6·12). By using Eqs.
(6 · 42) and (2 · 29), we can rewrite them respectively as
Ll (x) =
T(x)
.. -1_+ i.xao (x)
[1+..lxa0(x)J2+ [n).xG0(x)]2
(6. 46)
_rr)xGo(x)_~----
=
(6. 47)
[1+i.xa0(x)]2+ [n..lxGo(x)]2
Inserting these into Eq. (6 · 43) and using the
numerical values of G0 (x) and a 0 (x) as shown in
Figs. 19 and 20 respectively, we have carried out
numerical calculations for the cases i. = - 0.3,
-0.5 and c=O.l. The results are presented in
Fig. 22.
G(x)
2.0
ok-----~L"'o
___ x
2) The case (B)
Fig. 22. G(w2) curve for the
cases ,\= -0.3, and -0.5
Here we are mainly interested in the region
In the case i.~i. 0 , the approximation (6·9)
and (6 ·10) may be used in the region
1. However, the damping effect
plays an important role for determining the shape of the G(ai) curve near
the top, so that here we shall make use of the following approximation : Substituting Eqs. (6 · 9) and (6 ·10) in Eq. (6 · 45), we have
when c=O.l.
x=l.
G"(x)=rrGo{x(1
x>
ci.
1-i.xao(x)
)}·
Here m Eqs. (5 · 29) and (5 · 30) we shall neglect ci.w 2 x {a0 (w 2)
(6·48)
-
G' (w 2) } , then
Energy Spectmm of Lattices with Defects. II
73
we have
il (x)
= _______ _ _ }_-:= J.xao(x)
[1-i.xa0 (x) ] 2
_
+[drrxG
{x(1- 1-lxa
. (x)
.
0
c)
(6. 49)
-) }]
2
'
0
ci.rrxG0 { x(1--~-=--)}
r(x) = --------------- -------·----- 1-lxa-0 (x) .
- [1-i.xa0 (x) )2+
0
~)}
1-lxa0 (x)
[ci.-.xG {x(1--.
2.
J
(6. 50)
By usmg Table II and Fig. 19, we
can evaluate Ll(x) and F(x) numerically, and then by putting those
values into Eq. (6 · 42) obtain the G(x)
curve. The results of numerical calculations are shown in Fig. 23. In the
region x<1, we have used those J(x)
c=0.5
and F(x) which are obtained by replacing a 0 (x) by -a0 (x) in Eqs. (6·49)
and (6 ·50), and fitted the curve thus
obtained with the above results.
In the cases (C) and (D), in
1.3
:l:
1.1
1.2
order to obtain the complete
Fig. 23. G(x) curve for the cases J.=0.25,
curve, we must solve transcendental
c=O.Ol, 0.05, 0.1 and 0.5.
equations (5 · 25), (5 · 27), (5 · 28),
(5 · 29)- (5 · 31), and (5 · 32). Owing to the mathematical difficulty we have been
unable to perform this program. If we use Fig. 19, we could produce a
plausible G(x) curve in the region x>1; however, it is omitted here.
G(x)
Acknowledgements
The author would like to express his appreciation to Prof. K. Tomita and
Prof. E. Teramoto for their continual encouragements and helpful suggestions
during the course of this investigation at Department of Physics, Kyoto University.
He is particularly indebted to Prof. T. Matsubara for his wise advice for a
graphical method in perturbation calculations. His thanks are also due to the
members of Tomita Laboratory f~r a number of discussions.
Appendix A
Localized vibration due to a substitutional impurity atom
Here we shall study the localized vibration of a simple cubic lattice with
the nearest neighbour interaction due to a single substitutional impurity atom.
74
S. Takeno
In this case, the equations of motion for its neighbours also differ from those
of the regular lattice. Considering Eq. (3 ·17), we shall rewrite the time-independent displacement vector u (R 1) explicitly as Uz1 z,z 3 • When the impurity
atom is present at the origin, the time-independent equation of motion for it is
given by
K/ (2Uooo ~ Uwo ~ U-1oo),
w 2Uooo = ~~
m'
K/ (2Uooo ~ Uo1o ~ Uo-10)
+ ~~
m'
Ka'
+ - - (2uooo~ztoo1 ~uoo-1),
m'
(A·1)
where m' is the mass of the impurity, and K{, K/, and Ks' are the central
and the noncentral force constants between the impurity and its neighbours respectively. By using Eqs. (2· 4) and (3 ·18), we can decompose the above
equation into the sum of the regular and the perturbed terms as follows :
+ (/C2 + i.' + 1C2i.') (2ztooo ~ Uow ~ Uo-w)
+ (!Ca + i.' + !Cai.') (2uooo ~ ltoo1 ~ lloo-1) ,
(A ·1')
where
3
L 0 =w 2 (~ip),
with w2 (k)
=:Er.{1~cos(k.a)},
(A·2)
Jl=l
and
(A·3)
By the same procedure, the equations of motion of neighbouring atoms at
( ± 1, 0, 0), (0, ± 1, 0), and (0, 0, ± 1) are given as
(A·4)
(A·5)
(A·6)
Let us write the coordinate representation of the operator V m Eq. (2 · 3) explicitly as
(A·7)
Then we have the same equation as Eq. (2 · 3), if the coordinate representations
of V have the forms
3
v co, o; o, o; o, O) = 2I; (!C.+ i.' + IC/) r.,
(A·8)
11=1
V(O, 1; 0, 0; 0, 0)
= V(0,~1;
0, 0; 0, 0)
= ~
(1C1 +i.' +/C1i.')r1,
(A·9)
75
Energy Spectrum of Lattices with Defects. 11
v
co, o; o, 1 ; o, O) ='V co, o; o,
-1 ;
o, o) =
- (x;2 + i.' + /C2i.') r2,
(A·IO)
v (0, 0 ; 0, 0; 0, I) = v (0, 0 ; 0, 0; 0, -I) = - (!Ca + i.' + !Cai.') ra' (A ·11)
v (I, 0; 0, 0; 0, 0) = v (-I, 0; 0, 0; 0, 0) = - v (1, 1 ; 0, 0; 0, 0) =
- V( -1, -I; 0, 0; 0, 0) =
v (0, 0; 1,0; 0, 0) = v (0, 0;
(A·12)
-IC1/' 1 ,
-1, 0; 0, 0) = -
v (0, 0; I, 1 ; 0, 0) =
- V(O, 0 ; -I, -1 ; 0, 0) = - IC2/'2 '
(A ·I3)
V (0, 0; 0, 0; 1, 0) = V (0, 0; 0, 0; -I, 0) = - V (0, 0; 0, 0; I, I) =
- V(O, 0; 0, 0; -1, -I)=
(A·l4)
-IC3 /' 3 ,
all other V(ir,j1; i2,j2; ia,ja) =0.
(A·I5)
Remembering Eq. (3 · 3), we have the secular equation for our impurity problem:
(A ·I6)
with
ali (a?) =a (w 2
;
l1- ir, l2- i2, la- ia).
Thus we obtain the secular determinant of the seventh order which determines the frequencies of the localized modes. Owing to the geometrical symmetricity of the lattice, however, it can be reduced to the product of several
secular determinants of lower order ; we can find the following three types of
modes:
(I) s-type mode: u 100 = U-1oo = u 01o = u 0_ 10 = Uoo1 = Uoo-1> (2) three p-type modes: (2 ·I):
U1oo = - U-1oo, Uooo = Umo = Uo-10 = Uoo1 = Uoo-1 = 0, (2 · 2) : Umo = - Uo-w, Uooo·= U1oo = U-100 =
Uoo1 = Uoo-1 = 0, (2 · 3): Uoo1 = - Uoo-1> Uooo = U1oo = u-100 = Umo = Uo-1o = 0, (3) three d-type
modes ; (3 ·1) U1oo = U-100 = - Uo10 = - Uo-10, Uooo = Uoo1 = Uoo-1 = 0, (3. 2) ; Uo10 = Uo-10 =
- Uoo1 = - Uoo-1, Uooo = Uwo = U-100 = 0,
(3 · 3) : Uoo1 = Uoo-1 = - U1oo = - U-wo, Uooo = Uo1o =
Uo-10 = 0.
By using Eqs. (A·8)-(A·I6) and (3·20c), after some manipulations, we have
the secular equations for each mode :
(I)
s-type mode:
I- (x;1 + i.' + IC1 i.') {I (w' 2 ; 0, 0, 0) +I (oP ; I, 0, 0)} - (x;2 + i.' + x;2i.')
x {I(w' 2 ; 0, 0, 0) +I (w' 2 ; 0, I, 0)}
- (x;3 + i.' + x;3 ..i'){I (w' 2 ; 0, 0, 0) +I (w' 2 ; 0, 0, I)} - (x;t/2) {I (w' 2 ; 0, 0, 0)
+ I(w' 2 ;
- x; 2 {I(w' 2 ; 1, 0, 1)
+ I(w' 2 ;
2, 0, 0)
+ 2I(oP;
0, 1, 0)}- x;a{I(w' 2 ; I, I, 0)
1, 0, 0)}
+ I(w' 2 ;
0, 0, I)} = 0,
(A·17)
(2)
P"type modes :
S. Takeno
76
(3)
(2·1) 1-_5_ {I(w' 2 ; 0, 0, 0) - I(w' 2 ; 2, 0, 0)} = 0,
2
(A · 18)
(2·2) 1-~{I(w' 2 ; 0, 0, 0)- I(a/ 2 ; 0, 2, 0)} =0,
2
(A· 18')
(2·3) 1-~{I(w' 2 ; 0, 0, 0)- I(w' 2 ; 0, 0, 2)} =0,
2
(A·18")
d-type modes:
(A·19)
(3 · 2) 1-~ {I(w' 2 ; 0, 0, 0) + J((u 12 ; 0, 2, 0)} +
2
(K2
+ ,{' +
A I(w' 2 ; 0, 1, 0)
K2 1)
(A·19')
(3. 3) 1-~{I(w' 2 ; 0, 0, 0) + I(w' 2
2
;
o; 0, 2)} +
(!Ca + ,{' + ICaA') I(w' 2 ; 0, 0, 1)
(A·19")
We see that the frequency of the p-type modes depends only on the difference
between the force constants. When r1 =r2 =ra, three p-type modes and three
d-type modes are three-fold degenerate respectively. By virtue of the relations
(3 · 29), Eqs. (A ·17), (A ·18) and (A ·19) are reduced respectively to
(1)
s-type mode
1-3(1G+A' +IGA') {I(w' 2 ; 0, 0, 0) +I(w' 2 ; 1, 0, 0)}
- !:_ {I(lu' 2
2
;
0, 0, 0) + 6I(w' 2 ; 1, 0, 0)
+ 4I(w
12 ;
1, 1, 0)
+ I(w'
2 ;
2, 0, 0)}
= 0,
(A·20)
(2)
p-type mode (triply degenerate)
1-!:_ {I(w' 2 ; 0, 0, 0)- I(w' 2 ; 2, 0, 0)} = 0,
2
(3)
(A·21)
d-type mode (triply degenerate)
1-!:_ {I(w' 2 ; 0, 0, 0)
2
+ I(w
12 ;
2, 0, 0) - I(w' 2 ; 1, 1, 0)}
= 0,
(A·22)
Here the frequency of the d-type mode also does not depend on the mass
difference m- m'. The results of numerical calculations of Eqs. (A· 20), (A· 21),
and (A· 22) are shown in Fig. A. 1, in which we have plotted w' 2 as a function
Energy Spectrum of Lattices with Defects. II
77
s: i.'=I/2
I
i
i
i
i
i
i
i
i
s:i.'=2
I
I
!d
I
5
6
7
8
9
10
Fig. A.l. Graphical representation of the frequencies of the
localized modes due to a substitutional impurity atom.
The frequencies of p-type and d-type modes are three-fold
degenerate.
of tc, taking i.' as a parameter. From this figure we can conclude that when
the force constants between the impurity atom and its neighbours are sufficiently strong, a localized mode can take place even in the case i.' <O.
Appendix B
Derivation of Eqs. (5 · 21) and (5 · 22)
Let us consider the sums { (1 + C::) i./ d} • I:" (PabPbc" · · PdePea) and - { (1 + C::) •
abC···de
i./d} • 'L:" (PabPbc · · · PdePea') .
Using Eqs. ( 4 · 3) and (2 ·17), and then interchang-
abe···de
ing the order of summation and integration, we can express them respectively
as follows:
{ (1 +f)) ) • I:" (p
.
d
rabc--de
P .. ·p P ) =
ab be
de
ea
{J!.__
(1 + ~) /.s} •
(2rr)3
d
.
(B·1)
(B·2)
where we have defined
78
S. Takeno
= L.;" <exp [i {k(IJ · (Ra- Rb) + k( 2J · (Rb- Rc) + ···
abC···dB
If the spacial arrangement of impurities is completely random, in the limit of
N -HXJ, Eq. (B · 3) is in effect equal to
S.(k<1 Jk<2J, · ·., k<•l)
=
N-•L.;" exp [i {k<1J · (Ra- Rb)
abc ... de
+ k< 2J · (Rb- Rc) + ···
where Ra, Rb, Rc,. · ·, Ra and R. are considered as :random variables.
the use of the random phase approximation, we have
S. (k<1 J, k< 2l, · · ·, k<•l)
=c·a (k< J -k< l) i3 (k< J -k< l) ··.a (k<•-lJ -k<•l) a(k<•J -k< l).
1
2
2
3
1
Then by
(B · 5).
Putting Eq. (B · 5) into Eqs. (B ·I) and (B · 2), we get
·{(I+~)A}"L.;"<Pabhc···PaePea)=__!}__[{
d
(2rr) 3 l
abc ..de
X
l{
I
J s-i(l}(k)
(I+~) c)s}" (
d
_I .
s-wl(k)
I
s-iw 2 (k)
(I+~)c)s}" dk,
d
Is ) dk.
(B·6)
(B·7)
By the aid of these results, the series (5 ·I9) and (5 · 20) can be summed up
respectively as follows :
=
(I - (I + ~)
d
cA) __!}____ r
dk
(2rr) 3 Js-iw 2 (k)- ((I +~)d/d)s
(B·8)
7J =_I_
__!}____ rdl.:
. I+¢ (2:-r) 3 J
t {s-i£tl(k)
I
•=2
(I + c;) ds} v
d
cis [ g
=d
0
{s (I --d-el.
I+,; ·)} -go (s)] .
(B·9)
Energy Spectrum of Lattices with Defects. II
79
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
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