Vibrational states in a semi-infinite crystal with local defects

ht. 1. Engng Sci.

cml-7225/91

Vol. 29, No. 3, pp. 323-326, 1991

VIBRATIONAL

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STATES IN A SEMI-INFINITE WITH LOCAL DEFECTS

A. M. KOSEVICH,

M. L. POLYAKOV

CRYSTAL

and E. S. SYRKIN

Institute for Low Temperature Physics and Engineering, Ukr. S.S.R. Academy of Sciences, 47 Avenue Lenin, Kharkov, U.S.S.R. 310164 Abstract-The lattice Green function is found in the model of a semi-infinite fee crystal with scalar displacement. All types of surface waves arising in this model are analyzed. The analytical expression for the formation threshold of a local vibration due to presence of an impurity in the subsurface region is found. The singularities of the density of states are studied (including the Van Hove singularities). Their relation to alteration of the topology of the two-dimensional projection of the &frequency surface (appearance of a concave region) is shown. The effect of the small interaction between second nearest neighbours on the characteristics under investigation is included. In some cases this effect can be considerable, e.g. the infinity in the density of states at the boundary of the continuous spectrum of an infinite crystal vanishes. If the variables are appropriately changed the results describe spin excitations in the Heisenberg ferromagnet in the one-magnon approximation.

We obtained an analytical expression of the lattice Green function in the K representation in terms of a simple model. The poles of this function, as is known, determine the dispersion laws of oscillations of a semi-infinite crystal, including that of surface waves. Using the above Green function, we studied the condition of formation of a local vibration and its dependence on the parameters of the problem. Analysis of the density of states showed that the Van Hove singularities result from alteration of the topology of the two-dimensional projection of the isofrequency surfaces (with concave regions appearing). The displacement u(n) of an atom at a site n in an unloaded crystal satisfies the equation of motion: d2u(n) m~+iu(n)=O where m is the mass of the atom and the form of the linear operation iu depends on the model assumed (for definiteness, we shall deal below with a scalar model of a one-atomic fee crystal with interaction between nearest and second nearest neighbours). Represent the action of the matrix L as: iu(n)

= C a(n - n’)u(n’) + i,u(n)

+

t,u(n)

(2)

II'

where a(n -

q‘) is the

matrix of the force constants of the infinite ideal fee crystal:

sdn - 04n’) = a g Mn) - 0 + WI + B 2 HO - 0 + A211

(3)

LY and p bein the co?stants of coupling with nearest and second neighbours, respectively, and the matrices L1 and L2 providing vanishing of the interaction along the crystal free surface: i,u(n)

= cr[u(nl + 1, n2, -1) + u(nl - 1, n2, -1) + u(n,, n2 + 1, -1) + u(n1,

~2An)=Bb(~l~ The

n2-

n2, -I)-uh

1, -1)

-

n2,

4u(n19

n2,O)

&,,I$

wk,I +Bbh,

n2, -a-u(n1,

(4) ~2,0)14@

(5)

solution of the equation mo’G(n

1n’) - iG(n

1n’) =f&,,,;

6,,; 6,,,j

(6)

vanishing at infinity, will be called the Green function of stationary vibrations. Assume the force f to be applied to the site @Oh). It can be shown [l] that the Green 323

324

A. M. KOSEVICH

function in the k-representation

et 01.

is:

f

G(L &, rc,) = - m{02(K) - w”} e iK&

1

f

_

m{w2(K)-w’}‘A+-A_

-(l+A+A_

+2cu/jQo

[2~-~(cosK,+wsK2)e-iK3+~-~em2iK3]gh+$sinK,

A+A_+l+# *

gh+l+&-1

[

11

B

-A,Ax

(&+I

where w’(K) is the dispersion law of volume vibrations co2(K) =I;y2-

(7)

P

--&z-d

cos K1 cos K2 - cos rir, ~0s KS - cos K3 cos K,)

2P + ; (3 - cos 2K1 - cos 2Kz - cos 2K3) -$

A*=

(cos K1 + cos K,) f { $

(cos Kt + cos K,)’ - ;

a2

In +~(3-cosK1~~~K2)+3-co~2K~-cos2K~ B 1

{A+(1 -t/m)}”

gh =zT=-zy

-i-&y2

,

1

{A-(1 - -1”

h 200,

g-1 =

g1

The first term in (7) is the Green function for the infinite crystal. The second term gives the effect of the boundary. Let us use the present method by constructing the Green functions in the two simple limiting cases: (1) /3 = 0, the model under study transforms into the fee lattice with nearest neighbour interaction and (2) (Y= 0, the case of a simple cubic lattice. For #I = 0, we have

f GWI,

X *2

K2,

kl,)

_

02:Kl 1s

=

-

m{ 02(K) K

2

)

- 02)

eiKh -fz

b:a

* to2(K;

_

@2}

l+VGf

f2 - (60s K1 + cos K2)eeiK3] 2 + (cos KI + cm I(,)

a1

2

where r&(K1,

cosK1+cosK2

2 - cos K1 cos K2 -

K,) = 4 ;

t

W)

2 >I

2

cos Kx + cos K2

I

bI=GZ-l

Q1= (m/4cx)w2 - 3 + cos K, cos K2 ’

@l

For tu=O, we have

i&y+ 4~~~

f G(f(l,

K2,

&I

= -

1 m2 o_?(K) - d

M (co2(K) - co’> e

1 w2

-

c&K,)

m K,)

((1 _

e-2iKg} (11)

*2

where &(K1,

K,) =;

28 (2

- cos 2K1-

cm 2K,)

-l/2

a2=

-3 48

o2 -t 3 - cos* K1 - cos2 K,)

,

J-Vi3

b 2

a2

Vibrational states in a semi-infinite crystal with local defects

The Green function (7) determines the behaviour of possible applications of the Green function to analysis Let an impurity with the mass m' be at the distance after substitution of -CO* Amu,,, Am = m' - m for f, for the local vibration frequency:

325

the system. We shah only present one of of local vibrations. h from the crystal surface Equation (7), leads straightforwardly to the equation

where the notation of (9) is used and Amw2uo

Yr=

48

1 *

A+-A_-

l+A+A_+2E A+A_+1+2E

X &I+1

g’

B

go B>

+

gh-1 -

B A+A_vl

_&j/l

_

A+(gh+l

-

gh-l)

(a)

’

(b)

I 0

l-

Fig. 1

1814a

1

w2/w2m

1

326

A. M. KOSEVICH et al.

Yo=

Amw2u, 33

gh -

A+-A_-

l+A+A_+2E

g, B>

Relation (12) determines the implicit dependence of w2 on the parameters Write the conditions for the threshold of formation of a local vibration: Am(O) = 0.4m;

4)

Am, h, a and /3.

h =O;

i= $ m In-‘(1/31/o);

Am(h)= :rn

In-’ h ln-‘(I/3l/a),

Am(m)-$m

ln-2(1/311cu),

h=l h >> 1 h=cc

(13)

Expressions (13) suggest the conclusion that, if the interaction with second nearest neighbours is taken into account, the threshold of formation of a local vibration due to an impurity in the infinite crystal is finite. Naturally, such a situation stems from the behaviour of the density of vibrations of an ideal crystal in the high-frequency range of the spectrum. The expression for the vibration density can be reduced to the following form: 1 g(w’) = (2n)3

I

dK1 d& (A+ A A_) ((1 - A:)-ln

+ (1 - A?)-“2}

(14)

in which integration is made over the two-dimensional projection (K,, K2) of the isofrequency surface. The figure shows a rough density of states in the cases of #l = 0 (a) and /!I< 0 (b). One of the singularities (for w2 = a wi) remains practically the same, while that at w2 = wk behaves quite differently: g( 0;) = m for /3 = 0 and g(wLax) = 0 for /3 #O. All these singularities can be analyzed by studying the topology of the two-dimensional projections of isofrequency surfaces (the Van Hove singularities are associated with appearance of convex regions).

REFERENCE [l] $9zj,KOSEVICH, M. L. POLYAKOVand E. S. SYRKIN, Fiz. Nizk. Temp. lS(ll),

1194-1203 (in Russian)

(Received 10 November 1989; received for publication 3 September 1990)