Tamm states at a distorted surface

Physica 79 B (1975) 3 5 0 - 3 5 8 © North-Holland Publishing Company

TAMM STATES AT A DISTORTED SURFACE J. NEUBERGER and C. RUTHERFORD FISCHER

Department of Phystcs, Queens College of City University of New York, Flushing, New York 11367, USA Received 28 May 1974; revised 19 November 1974

The Kronig-Penney model has been used to study the effect of surface distortions on Tamm surface states. The dependence of surface states on 8barrier strength, interatomic spacing near the surface, and other crystal-potential properties is considered.

1. Introduction Since Tamm's derivation 1) of the existence of surface states for a Kronig-Penney (KP) model 2) terminated at a free surface, various investigations have been undertaken to study the influence of special boundary conditions on these surface states. The effects due to deformation near a crystal surface have been considered by Koutecky3), Steslicka4) and Henzler s) while Davison and Tan 6) considered the effect on Tamm states of electric fields at the surface. Other references may be found in a survey by Davison and LevineT). That the interatomic spacing changes near a surface is indicated by both experiment and theoryT). In particular, calculations on properties of ionic-crystal surfaces by Benson and Claxton s) indicate a change in the spacing of 10% for NaC1. For transition metal surfaces, a calculation by Allan and Lannoo 9) including Born-Mayer repulsion with d electrons treated in a tight-binding approximation, showed a 10-20% change in spacing. Philips 1°) has reviewed some data on the irregularities at surfaces. More exact methods have been given by Appelbaum and HamannH). They develop a scheme where the electronic structure near surface is calculated in a self-consistent manner. The surface states of Si are discussed 12) and it is observed that a relaxed surface potential compares favorably with experimental data. Uncertainties as to the exact nature of surfaces preclude however detailed comparison with experiment. In many cases impurity layers give rise to properties that are not intrinsic to the element 350

J. Neuberger and C. Rutherford Fischer/Tamm states at a distorted surface

351

under investigation. Thus a simple model which predicts the t y p e o f surface. energy state that can be expected to exist, has the merit that it can yield results w i t h o u t involving further approximation and may aid in the interpretation o f experimental results. While the KP model is relatively crude, the existence o f exact solutions for the surface-state condition makes a consideration o f varied b o u n d a r y conditions tractable. In this paper, the model is applied to study the effect on surface states o f varying the interatomic spacing, location o f crystal-potential termination at the surface, and strength o f the 8-function barriers. The calculations are somewhat similar to those o f Steslicka 4) and Steslicka and Woj ciechowski13), b u t different parameters are varied to take account o f the work o f refs. 8 and 9. 2. The model Consider 14) the one-dimensional lattice shown in fig. 1. F o r x > 0 we choose the KP model with 6-function periodic potentials at x = na, n = 1, 2, 3, etc. The area under the 8 function is measured b y P where \ m 111 t P = lim vt_,o o \ ¢/2 b t . • bl-~0

(1)

The last cell is assumed to be shifted and its corresponding ~ barrier located at x = - 6 with an area assumed to be P just as for the rest o f the lattice. This condition can be relaxed as was done b y Steslicka4), b u t was V(X) ~r- FUNGTION

~ v3

v,

I

x !i

I

~v ~ ~

POTENTIALS

I

lID,

X l=

d

r|

Fig. 1. One-dimensional potential investigated in this paper. The deformed cell may lie to the right of the origin as well. Both ~ and d are allowed to vary independently.

352 J. Neuberger and C. Rutherford Fischer/Tamm states at a distorted surface avoided here where the main emphasis is on studies o f changes in 8 and d. The lattice is terminated by the constant potential V3 in region 3, - d > x > _oo. In region I the wave function can be written according to KP as I111 = G(eip x +

?te-ipx),

(2)

where )t = (ei/aa - e i p a ) / ( e - i p a + ~aa)

(3)

p2 = 2 m E ~ 2 .

(4)

and

Here E is the energy and/a the wave number. In region II 42 = A e i p x + B e - i p x ,

(5)

where constants A and B are related to G and ~ o f eq. (3). This relationship can readily be obtained by knowing that ~ is continuous across a 6-function potential and that a finite jump in the derivative is obtained proportional to the strength o f the potential, i.e., P. The result is A = (1 - P/ip)G - (PGX/ip)e 2ip6

(6)

B = (PG/ip)e -2ip6 + G?t(1 + P]ip).

(7)

and

Assuming 6 < d, and expanding, the result is A / G = 1 - (P/ip) - X(P/ip) (1 + Zip/i)

(8)

BIG = (P/ip) (1 - 2ip/i) + ;k(1 + P/ip).

(9)

and

For region III ~Oa = Ce'rX,

with

(10)

J. Neuberger and C Rutherford Fischer/Tamm states at a distorted surface

3'2 = (2m/h 2) (V3 - E ) .

353

(11)

Continuity of ~k2 and ~b3 at x = - d and similar conditions for their derivatives give: ~, = ip (Ae -ipd - Beipd)/(ACipd + Beipd).

(12)

This yields

A / B = - [(7 + ip)/(7 - ip)] e 2ipd .

(13)

When (8) and (9) are used in conjunction with (4) defining ;~, after considerable algebraic simplification an expression for e itaa is obtained: e i/aa = c o s ~ + (sin

~/~)oM,

(14)

where ~ = ap and

o M = 2aP(1 - 2P/i) + ~F(1 - 45P).

(15)

Here

F=(Tcos71-psin~?)/(',/sinri+pcos~)

and

77 = pd.

(16)

To simplify for comparison with other work, the following alternate notation may be introduced:

op=aP,

q2= 2mV3[¢i2

and

d/a=a.

(17)

Hence

a M = 2Op (1 - 2ap ~/a) + ~F(1 - 4Ol, ~/a)

(18)

[(aq)2 _ g2]~ c o s ~ - ~ sin a t F = [(aq)2 _ ~2]~ sin ~ + ~ cosa~ "

(19)

and

F o r the KP model, the relation between # and g is given by the well-known relationship2) cos ga = cos g + (sin ~/~)Op.

(20)

354 J. Neuberger and C. Rutherford Fischer/Tamm states at a distorted surface Surface states exist when/a is complex such that #=mr/a+iz,

z>0.

(21)

Inserting (21 ) into (14) and (20) respectively yields ( - 1 ) n e -az = cos ~ + (sin ~/~)0 M

(22)

( - 1) n cosh az = cos ~ + (sin I~/~)Op.

(23)

and

Subtracting (22) from (23) gives ( - 1) n sinh az = (sin ~/~) (Op - OM).

(24)

Combining (23) and (24) yields 2Ol, ~ cot ~ = ~2 + o ~ - 2OMOP.

(25)

This equation must be obeyed if surface states are to exist. Furthermore since z > 0 and in the n t h gap, nrr < ~ < (n + 1)lr, and sign [(sin ~)/~] = ( - 1 ) n , thus from (24) we see that an additional condition for surface states is (26)

Op > o M.

The surface-state equation (25) can be written in a more explicit form when using (18) and (19):

~2 + f \ l~ + aZOpf m ~ I - 4~0~ (1 + - Op -f}~ ,

~cot~=2Oe

(27)

with f = [(aq)2 _ ~21½~ cot tx~ - ~2 [(aq)2 _ ~2]~ + ~ c o t a ~

(28)

The auxiliary condition (26) becomes 0 > (Op + f ) [ 1 - 4 (6/a)op].

(29)

J. Neuberger and C. Rutherford Fischer/Tamm states at a distorted surface 355

3. Special cases When 5 = 0 and ~ = 1, we have Tamm's model and the above should reduce to his results. Eq. (27) becomes, letting x = ~ cot ~ and y = [(ag)2 _ / j 2 ] ~

x

2oe

\y~x

~

2ot , y - + x

A bit o f algebra gives x - - ~ cot ~ = ( a 2 q 2 / 2 O p ) - (a2q 2 - ~2)~,

(31)

i.e., Tamm's result (he defines q as our aq). Also the condition (29) becomes -f>

Op

(32)

or (ll 2 - x y ) / f y + x ) > Op.

(33)

Insertion o f (31 ) yields (after some simplifying) Op> (a2q 2 4- ~2)~-,

(34)

in agreement with Tamm. An additional condition is o f course that the allowed surface states should be forbidden bulk states. This implies that those values which obey (20) have to be excluded. This means [cos ~ + (sin ~/~)Op[ > 1.

(35)

A simple algebraic transformation shows that this can be written as cot (g/2) > ~/Op,

for 0 < ~ < 7r.

(36)

For Op = 1, 2, 3, 4 the above condition corresponds to ~ < 1.30, 1.72, 1.98, and 2.14, respectively. A second special case occurs when 5 = 0. This corresponds to one o f the cases discussed by Steslicka4). Using (7) o f Steslicka's 13) result for 133,= 1, the latter equation becomes (converting to our notation)

356 J. Neuberger and C. Rutherford Fischer/Tamm states at a distorted surface cot ~ = (aq)2/2°P + y cos 2c~ + [(aq)2/2~ -- ~] sin 2 ~ [(Y/G) sin ~ + cos ~ ] 2

(37)

A bit o f algebra shows that this expression is in agreement with our (27) w h e n / i = 0.

4. Results Surface states were calculated for the following choices of parameter: op = 1, 2, 3;d/a = 0.9, 1.0, 1.1; and 5/a = - 0 . 1 , 0, 0.1. In all cases some surface states were found for q ranging from values 0 to 4. Further study o f the results shows that when o = 1 the nature o f the surface state is not altered in a significant way by the surface distortion from a T a m m state (arising from the presence of the surface). For o = 2 and 3 however the deformed lattice admits different and sometimes dual surface states in contrast to the single surface states in the Tamm model. Thus we conclude that even small distortions in the lattice constant near the surface play a significant role in surface states. Some o f the results are plotted in figs. 2 - 4 where we chose d/a = 1.1, while 8/a was allowed to vary over its three values, - 0 . 1 , 0 , and 0.1 for each o f the three choices o f o. The other choices o f d / a gave rise to similar results. For all the cases shown, the energy o f the surface state, which is proportional to increases with increasing 6/a ratios. For o = 2 the same general behavior

//y

13-

/%'

12 -

II -

.

(,' /

or" = I ;o--:, $/" =-.I

I

1.0

I

I

I2

1.4

, 0

,'1

I

I

1.6

I8

Fig. 2. Solutions of eq. (27) for o = 1, d/a = 1.1.

J. Neuberger and C. R u t h e r f o r d F i s c h e r / T a m m states at a dtstorted surface

20 I

.I .f ./

S

15

I#'

o-

I/

-2

d/~."-I

i

I

0 ,*1

,

,

I

EO

J

30

I 4.0

Fig, 3. S o l u t i o n s o f eq. (27) f o r a = 2, d/a = 1.1.

3.0

...... ,

....-

•

2.0

/

O"

~// t/

,

1.0 o

=3

d/~ = 61~•

I 2.0

i

II - I

I 3 o

, 0

,+.I

i

I 4.0

Fi& 4. S o l u t i o n s o f eq. ( 2 7 ) f o r o = 3, d/a = l. 1.

357

358 J. Neuberger and C. Rutherford Fischer/Tamm states at a distorted surface

continues. However, a new feature arises as well. For 5/a = +0.1 there are present t w o allowed surface states over a range o f the q's. This implies that t w o bulk states have been vacated and are replaced b y surface states of higher energies. F o r o = 3, fig. 4 shows that when 8/a = 0.1 the allowed surface states appear at much larger values o f q and correspond to much higher energies. When the same results are presented in a different way which emphasizes h o w the strength of the ~ potentials, i.e., o, affects surface states, the general pattern is that larger values o f o shift the surface states to higher levels and to greater values o f q. While only one case o f dual surface states is shown in figs. 2 - 4 , we also found that dual surface states arise over a limited range o f q's for the following cases: d/a = 0.9, ~/a = 0 . 1 , o = 2, 3;d/a = 1, 8/a = 0.1, o = 2 or 3. Further study o f the results shows that when a = 1 the nature o f the surface state is not altered in a significant way b y the surface distortion from a Tamm state (arising from the presence o f the surface). F o r o = 2 and 3 however the deformed lattice admits different and sometimes dual surface states in contrast to the single surface states in the Tamm model. Thus we conclude that even small distortions in the lattice constant near the surface play a significant role in surface states. Acknowledgements The authors wish t o thank Dr. S. G. Davison and Dr. P. W. Levy for stimulating discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)

Ig. Tamm, Z. Phys. 76 (1932) 849. Phys. Z. Sowjet 1 (1932) 733. R. de L. Kronig and W. G. Penney, Proc. Roy. Soc. A130 (1931)499. J. Koutecky, J. Phys. Chem. Solids 14 (1960) 233. M. Steslicka, Acta. Physiol. Polon 30 (1966) 883. M. Henzler, Surface Sci. 9 (1968) 31. S. G. Davison and K. P. Tan, Surface Science 27 (1971) 297. S. G. Davison and J. D. Levine in Solid State Physics, H. Ehrenreich, F. Seitz and D. Turnbull, eds., Vol. 25 (Academic Press, New York, 1970). G. C. Benson and T. A. Claxton, J. Chem. Phys. 48 (1968) 1356. G. Allan and M. Lanoo, Surface Science 40 (1973) 375. J. C. Philips, Phys. Rev. B9 (1974) 2775. J. Appelbaum and R. Hamann, Phys. Rev. B6 (1972) 2166. J. Appelbaum and R. D. Hamann, Phys. Rev. Letters 31 (1973) 106. M. Steslicka and K. F. Wojciechowski, Physica 32 (1966) 1274. A preliminary account of this work is given in: Bull. Am. Phys. Soc. II (1974) 19, 79.