Solid State Communications, Vol. 47, No. 5, pp. 3 7 1 - 3 7 4 , 1983. Prittted in Great Britain.
0 0 3 8 - 1 0 9 8 / 8 3 $3.00 + .00 Pergamon Press Ltd.
SURFACE PARAMETER CHARACTERIZATION OF SURFACE VIBRATIONS IN LINEAR CHAINS* Norberto Majlis t and Silvia Selzer t International Centre for Theoretical Physics, Trieste, Italy Diep-The-Hung Laboratoire de Magnrtisme des Surfaces, Universit6 Paris VII, France and Henryk Puszkarski* International Centre for Theoretical Physics, Trieste, Italy (Received 30 March 1983 by F. Bassani)
We consider the vibrations of a linear monatomic chain with a comp.lex surface potential defined by the surface pinning parameter a = A e -'~. it is found that in the case of a semi-infinite chain a is connected with the surface vibration wavenumber k = s + it by the exact relations: s = ~o, t = In A. We also show that the solutions found can be regarded as approximate ones (in the limit L ~" 1) for surface vibrations of a finite chain consisting of L atoms.
the bulk band, instead, the amplitude changes sign from one atomic plane to the next towards the bulk of the body, i.e. s = n. The present work is aimed at characterizing the surface vibrations of a semi-infinite chain in terms of the surface parameter, taken to be a complex quantity. This will be done by relating the surface vibration numbers s and t, introduced above, with the real and imaginary parts of the surface parameter. As a convenient simple model for investigation we choose a chain cot~sisting of identical atoms of mass m, with nearestneighbour Hooke's law interactions. The end atom constitutes what may be called the "surface" of the chain and it will be assumed to interact only with its nearest neighbour towards the interior of the chain. We shall consider, moreover, an additional potential term arising from the surface tension, and we shall restrict ourselves to investigating only longitudinal vibrations of the chain.
1. INTRODUCTION TIlE CONCEPT OF the surface parameter, the quantity describing the surface defect of finite crystals, has turned out to be very useful when studying magnetic surface states [1 ]. Recently, this concept has been generalized [2] by admitting the surface parameter to be complex. A real surface parameter is known to give rise to localized surface states, while a complex one generates the so-called quasi-localized surface states. The wavenumber of a surface state is in general, complex: k = s + it, where both numbers s and t are real, with s E (0, lr) and t E (0, +oo). The peculiar property of the quasi-localized surface states resides in the fact that their real part s belongs to the &terior of the interval (0, n), whereas for the localized surface states the only accessible values for s are 0 and rr. As a consequence o f this fact the amplitude of localized surface states, with the energy below the bulk band, decreases strictly monotonically towards the bulk of the specimen (s = 0), while for quasi-localized states (0 < s < rr) the amplitude oscillates decaying into the bulk. For localized surface states with the energy above
2. EXPERIMENTAL The equations of motion of atoms for the semiinfinite chain can be written as follows: m leo = ~I¢~ - d Wo,
* To be submitted for publication. i" Permanent address: lnstituto de Fisica, Universidade Federal Fluminense, Caixa Postal 296, 24.210 Niteroi, ILl, Brazil. ~t Permanent address: Physics Institute, A. Mickiewicz University, Matejki 48/49, Poznan 60-679, Poland.
m~ =~(~_,+wt.,-2w3,
/=1.2 .....
where Wt is the displacement from the position of an atom located at the site 1 . . . . . and a is the nearest-neighbour constant. The quantity or' represents a 371
(1)
equilibrium labelled 1 = 0, Hooke's law force surface tension
372
SURFACE PARAMETER CHARACTERIZATION OF SURFACE VIBRATIONS
will turn out that only approximate formulas, replacing exact results [equations (8a)] obtained for the semiinfinite chan, can be found for this case. The characteristic equation of a finite monatomic linear chain can be written in matrix form as follows [3]:
contributing in the following form to the potential on the end atom: AU = ~(a' -- a ) l ~ . Assuming WI(T) to depend harmonically on the time T, thus 14,'t = Uz e i'~r, we have
(2 - a ) U o -
U, = eUo,
- - U t - l + 2 U t - - U t ÷ l = cUt,
1=1,2 .....
(2)
(9)
DL(x. a. b) = O,
where x = 2 -- e, and
where we have used the notation
x--a /7,l G.) 2
Of ~
e = --,
a=2----.
Ot
(3)
The quantity a, being dependent on ,~', can be referred to as the pinning surface parameter; we assume that this quantity can be complex: a = A e -i~.
-1 x
-1
--1
x
--1
--1
x
--1
O~
D,.(x, a. b) =
(4)
We now proceed to solve the set of equations (2) looking for surface vibrations only. It has already been shown [21 that in the case of a complex surface pinning parameter, the wavenumber k for surface vibrations must also be considered complex: k = s + it with s E (0, 7r). By introducing the fictitious site I = --1 (see [3 ]), the set of equations (2)can be transformed into the following set of equations: - u z _ l ( k ) + 2 u t ( k ) - Ut÷l(k) = eU~(k),
1 = 0 , 1,2 . . . . .
(5)
with the following boundary condition: (6)
aUo(k) = U_,(k).
One easily recognizes the solutions of equations (5) to be of the form
--1
(7)
;(lO) --1 x--
the subscript L denotes the size of the determinant (being equal to the number of atoms in the chain), a and b are surface pinning parameters, for free and substrate surface, respectively. It may be easily shown that equation (9) is equivalent to the following equation: D t . ( x ) - - ( a + b ) l ) L _ t ( x ) + a b D t ~ _ ~ ( x ) = 0,
(11)
where we have denoted by Dn(x) = Dn(x, O, 0). First, we shall show that surface solutions of equation (11) in tile limit of infinitely many atoms, i.e. for L --* '~, are the same as those obtained for the sentiinfinite case. We divide equation (11) by DL_I(x ) and ta take into account that the following limit exists for tile analytic continuation of D , ( x ) to tile complex x plane (x --, x + ie) (see [41): lira
U~(k) = C e w',
Vol. 47, No. 5
DL_I(x + ie)
c~o
DL(x + ie)
= ~j.
(12)
where C is independent of I, and, by inserting equations ( 7 ) a n d ( 4 ) into equation (6), we obtain s = ~p
and
t = lnA,
(8a)
indicating that in the case of a semi-infinite system, the surface vibration numbers s and t are straightforwardly related [by the exact relation of equation (8a)] to the modulus A and phase ~0of the surface pinning parameter. On the other hand, substituting equation (7) into equations (5) we find the dispersion relation: 2 - - E ( k ) = 2 cos k,
(8b)
We rewrite equation (11) (in the limit L ~ ~") in the for form ~-I+ab~--(a+b)
= O.
(13)
The new quantity/~ is related to the diagonal element x of the characteristic determinant DL(x, a, b) by the following equality: + / j - i = x,
(14)
which is obtained by assuming
ut = ~uz-,,
(15)
which for k = s + it becomes e(k) = 2(1 - cos s ch t + i sin s sinh t).
(8c)
3. RESULTS AND DISCUSSION Now we make an attempt to evaluate surface vibration numbers s and t for a finite linear chain o f atoms. It
and substituting into the bulk equations (5). Equation (13) has two solutions, namely 1
~' = a-
and
1
/~2 = ~,
which are, in fact, solutions corresponding to two
(16)
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SURFACE PARAMETER CHARACTERIZATION OF SURFACE VIBRATIONS
surface vibrations localized, if lal > I and Ibl > 1, at the free and substrate surface. On the other hand, these two solutions can be directly obtained from equation (13) by inserting therein b = 0 or a = O. This means that in the approximation L ~ o~ used to obtain solutions (16), both surfaces are perfectly independent and they do not "feel" each other. This is why one of the solutions (16) is exactly the same as we have obtained for a semi-infinite case: substituting equation (15) into the first of equations (2) leads immediately to ~t. The quantity/~ defined by equation (12) can be expressed by the surface vibration numbers as follows [4]: = e i' e -t.
1
e -t = --~'A-L(1 --A -2) A
(18)
where + (--) corresponds to the symmetric (antisymmetric) mode [5]. i l L --, oo this equation gives the solution found above for the asymmetrical chain with complex surface parameters. Therefore, it may be suggested that the formula (18) can be treated as an approximate solution for surface vibration numbers t o f a finite chain (with arbitra~ surface conditions) if L ~" 1. Numerical calculations show that by using the first term of the r.h.s, o f equation (18) one obtains e -t within less than I 1% i l L / > 5 and A t> 1.15. Moreover, by analogy, we may expect that the remaining surface vibration number s can be described conveniently by the following formula:
s = ~o+ As(A. L ) ,
A J,
(19)
[
(, lo~ : l
i 3-
,t
-
n,/6 /
2,"
I:
Io
O-
_A/..,,
3o
2ii _. /~ ?? /:',? 3i
(17)
Assuming a = A e -i* and by having recourse to equation (17) we finally connect solutions (16) with the results obtained for a semi-infinite case [equation (8a)], as one should expect. The introduction o f arbitrary complex surface parameters (a. b) in a film of finite width (L < ~ ) implies in general that the phonon frequencies [i.e solutions of equation (11)1 become also complex. It is shown in the appendix, however, that if the parameters at both ends (a and b) are complex conjugate, we find 2 complex conjugate frequencies, and L -- 2 rcat ones. All numerical examples described below belong therefore to this special family of Ilamiltonians, which can be considered as the generalization of the llermitian type. It has been shown previously [5) for the case of a finite monatomic chain with symmetrical boundary conditions imposed by real surface parameters a = b --- A, that the inverse localization length t = I m k of the surface vibrations is given by the following approximate formula (valid for L >> 1):
373
A-1.15
]""
4o
' _' ....
L ~-
A
1,
,5
= 11
I
Fig. I. The correction As, as defined in equation (19). Dashed curve: As(A), for L = 11. Continuous curve: As(L) for A = 1.15. In both cases ~0 = rr[6. with As small when L 1> 5, A t> 1.15 as for t. Since the exact formula for As(A, L) is not available, we may gain some insight into the dependence o f As on A and L by having recourse to Fig. 1, where the numerical values of As obtained for ~p = n/6 are presented. We see that As(A, L) decreases rapidly with growing L or A, i.e. the behaviour of this contribution is, in some sense, similar to the contribution to t resulting from equation (18).
Acknowledgements - The authors (NM, SS and lIP) would like to thank Professors Abdus Salam, Mario Tosi and Erio Tosatti, the International Atomic Energy Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste. APPENDIX Consider L atoms at points l = --(L -- 1 )/2, --(L - - 3 ) [ 2 . . . . , - - I , 0 , 1,2 . . . . . L - 1//2. As in equation (5) we add two extra atoms at l = --(L + 1 )/2, (L + 1/2) [3J, at the same time imposing
u-(L+l)/2
= au-~-t)/21.
IlL+l/2
bilL_l] 2
=
[
J
(AI )
Then, as in equation (15), we substitute ut = (~l + 13~-z,
(A2)
in equation (5) and in particular in equations (A1), leading to a system of two linear homogeneous equations
SURFACE PARAMETER CHARACTERIZATION OF SURFACE VIBRATIONS
374
sh 2 u sin ~b sin ¢ + sh ~ v sin ¢'a sin ¢ + sin ¢
for the coefficients at and ~, which have a non-trivial solution if:
F(~) =
1--a~ l--~b l-a~ -ll-~-~b
= ~z~.t).
Vol. 47, No. 5
(A3)
For real a and b the condition = e t'~,
(A4)
which is satisfied identically in ~ if and only if:
(I) ( G , G ) = 0 or rr: or
leads immediately to e 2iL~ = e i°ta.b.~),
(2) u = v, G = - ~ b -
(A5)
e:it,~ = e i°Cu,v,cj,¢~,
or
rrm O(a, b, ¢,n) Cm = "~-- + 2L '
(A6)
which can be solved iteratively for ~0m. in the particular case a = b = 1, one obtains L real roots ~p,,, of the form: ~m = n m / L , m = 0 . . . . . L -- 1 (0 = 0 ) . For a (and/or b) greater than 1, there are one (or two) solutions with I~1 < I (¢ = pure imaginary) and L -- 1 or L -- 2 real solutions {¢,n}. Consider the case (a, b) = complex, a
b
In case 2
=
(AI0)
where ff - ffa = --ff~, and 0 is a complicated function of its arguments. There are at most L real solution of ( A I 0 ) . In fact, as shown in equation (16), there are two complex solutions ~t = l/a, ~,. = 1/b, when L ~ ~ . The continuity of the solutions with L leads to expect that even for L < '~ one finds two complex solutions, approximately equal to ~t = l / a , ~Jz = ~ , and L - - . "~ real solutions o f ( A l 0 ) . This is in exact agreement with the numerical results.
e zu ei~ka~
e :v e i6t' }) '
with 0 < (tka, ~k-) ~< rr, u, v = real. The condition for finding a real root ~pof (A3) and (A4) is I/:(01 ~ =
1,
which can be written as
RFFERENCES
(A7)
(A8)
I. 2.
II. Puszkarski, Prog. SurJl Sci. 9, 191 (1979). II. Puszkarski, Solid State Commtttt. 3 3 , 7 5 7
(1980). 3. 4. 5.
il. Puszkarski, Acta t'hvs. Poh,n. 3 6 , 6 7 5 (1969). S. Seizer & N. Majlis, t;hys. Rev. B26,404 (1982). H. Puszkarski,Acta Phys. eolon. A 3 8 , 2 1 7 (1970).