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Comments on “the vibration frequency of an atom at the surface of a tight binding metal” by C.M. Sayers
Surface Science 97 (1980) L320-L324 0 North-Holland Publishing Company
SURFACE SCIENCE LETTERS COMMENTS ON “THE VIBRATION FREQUENCY OF AN ATOM AT THE SURFACE OF A TIGHT BINDING METAL” BY C.M. SAYERS
K. MASUDA Department of Materials Science and Engineering, Midori-ku, Yokohama 227, Japan
Tokyo Institute of Technology,
Nagatsuta,
Received 21 January 1980
Recently, it has been shown that the simple tight-binding scheme is successful for describing the vibrational property of transition metal surfaces [ 1,2]. Sayers [3], using the Einstein model for lattice vibration, showed that the value of (06 - of)/& depends strongly on the filing of the band and can be greater than 0.5 (0.64) near the end of the transition series. Here, w. and o1 represent the vibration frequency (VF) of an atom in the bulk and that normal to the surface, respectively. To calculate VF, w. and ~1, he considered the repulsive core-core interaction energy, arising from the interatomic d-d repulsion and the increase in sp kinetic energy upon compression, as well as the attractive electronic (d-band) energy. The electronic energy is evaluated by using a non-self-consistent moment technique (second order treatment for the change in the second moment 6pis), while the former repulsive energy is simulated by a phenomenological Born-Mayer potential.
Fig. 1. Surface tension 7 for the simple cubic (iO0) surface calculated by the SC (solid) and NSC (dotdashed and dashed curves) methods. L320
L321
K. Masuda f Vibration frequency of an atom at surface of metal
Sayers’ non-self-consistent (NSC) approach, however, does lead to the unphysical result that the surface tension becomes negative near the beginning or the end of the transition series (see fig. 1). This indicates that the NSC treatment underestimates the electronic contribution to the surface tension and is inappropriate for studying the vibrational property of transition metal surfaces, especially near the beginning or the end of the transition series. In the present communication, surface vibrational property will be discussed more generally by taking into account the variations of the diagonal potential q (hereafter referred to as selfconsistent potential) resulting from charge oscillations near the defect. This treatment is self-consistent (SC) in the sense that it is capable of satisfying the Friedel sum rule [4,5]. The importance of selfconsistency (SC surface potential) for the surface electronic states has already been/pointed out by Allan and Lenglart [6]. We start with the local density of states (DOS) on atom i, obtained by the second moment appro&nation (continued fraction technique) [7] : Pi(E)
2E 2= (4b -
E2)“2 [@ - bie) E* - (2b - bie) aieE + b& + b&l-’
9
where uio and bto are given in terms of the moments pi,, as ($0 = I_cil = (Yi ,
bio = Pi2
-
P?I
(2)
’
This approximation ensures that the perturbed and unperturbed DOS, p&T) and po(f?), have the same band edges +2b”*. Following Sayers [3,7], we assume that the transfer integral tij between sites i and i varies as tij= to eXp(-qRij)
(3)
,
and bio E 7
fz exp(-2qRfj)
.
(4)
The change in band energy due to the deviation oi and bio on site i is then given by [41
beIO a
2n(b - bio)
(4b - E;)“*
+ flsin-’ (Ef/2b “*) + in] + CA_ - A +)
In @D)1’2 (4b +g -f*)A--fit 2 [R_@_T+ - R+T_)] “* I @ 1))“’
- @Z-T+ - R+T_)“* + CR-T, - R+T-)“*
K. Masuda / Vibration frequency
L322
of an atom at surface of metal
, &R-T,
(4b ‘g-f2)A+ -f-g + [R+(R_T+ - R+T_)] “’
sin-
- R+T_)l12
[T+(R+ + @R_)]
•tU&i) 3
1’2 ’
for f2 - 4g < 0, b.IO =
2n(b - bi,,) i
-
K_(K?
(5b) - (K, t K_) [sin-’ (Ef/2b “‘) + i n]
(4b - E#‘2
- 4b)1’2
. _,
4b - K-Ef
( ‘ln
K--K+
2b”‘IEf
- K-l
fin
K+(K,2 - 4b)1’2
1 + C(CQ)7
K+-K-
PC)
forf’-4g>O*, where f = -Qi(2b - bio)/(b - bio) 3 A, = [-(4b R, =A:
+g)*
+fA,
g = (bfo + baf)/(b - bio) 2
[(4b +g)2 - 4bf2]“‘/f, T,=4b-A:,
+g,
K,=[-f*(f2-4g)1’2]/2,
Ef=@+-Ef)/(Ef-A_),
D=T_@+T+,
and C((Y,)=A
{$(4b - Et)“’
- $ai[Ef(4b - Ef2)“2 t 4b(sin-‘(Ef/2b1’2)
t ir)]}. (6)
In eq. (5b), the +$r becomes plus for negative f and minus for positive f, while in eq. (SC) the +$r becomes plus for K+ less than -2b”‘, and minus for Kk greater than 2b1’2. In deriving eqs. (5b) and (5c), we have assumed that the SC potential oi completely screens. the defect (perturbation) and each atom is neutral. In this case the SC potential oli can be determined by Ef
Ef
J Pi(oi, E) do = J PO(E) -m -2x/b
dE.
(7)
This approximation is exact in the limit of large phonon wavelengths, and seems to be reasonable near a surface where the charge oscillations must be very small [5,9,101* To calculate the change in the total energy due to the introduction of lattice defect or the atomic displacement GRii between atomic sites i and j, we add a * In the case where the “split*ff” states appear below the lower band edge, their contribution
should be added.
K. Masuda / Vibrationfrequency of an atom at surface of metal
L323
Born-Mayer repulsive term between nearest neighbours E, (Rji) = Co exp(-pRii)
.
(8)
For typical transition metals with qRo = 3 .O and pR,-, = 9.0 (R, is the equilibrium interatomic distance), the parameter value Cc can be chosen so as to satisfy the equilibrium condition [ 111. From eqs. (5) and (8), it is possible to obtain both the electronic and repulsive contributions to the surface tension. In fig. 1 we show the surface tension 7 (per atom), calculated from the NSC and SC methods with qRo = 3 and pRo = 9, for the (100) surface of a simple cubic tight-binding metal as a function of the band filing: The energy is given in units of the half-bandwidth W (=2b”‘), and the degeneracy of the d-band is taken into account. Dot-dashed and dashed curves represent the 7 calculated by the NSC. methods; they are obtained by using rigorous and second order treatments (hereafter referred. to as NSC( 1) and NSC(2) treatments) for 6&a, respectively. In NSC(l), the electronic constribution to the surface tension is calculated by Ef (E - Ef) AP(E) dE s_-oD (Ap(E) is the change in the DOS for the surface atom), while in NSC(2), it is calculated by using the power expansion for 6~~s (see Sayers’ paper [3]). One can see that the surface tension 7 calculated by the SC method (solid curve) is always positive, while the 7 calculated.by the NSC method becomes negative near the beginning and the end of the transition series. This indicates that the NSC(1) and NSC(2) treatments underestimate the electronic contribution to the surface tension. We now calculate the change in the total (electronic and repulsive) energy due to a small static displacement x of atom i along a [ 1001 direction of the simple cubic lattice. Performing this calculation both for the atom at the (100) surface, 6El(x), and for the bulk atom, &E,(x), one can calculate (0; - c&/c& (ZE’d as FL = (o; - c&/w;
= [6E,,(x) - 6E1(x)]/6Eo(x).
(9)
In the above eq. (9), 6EI(x) is defined as &El(x) = @El(x) + 6EI(-x)}
/2 .
Using eq. (9), we have calculated FL for the (100) surface of a simple cubic tightbinding metal and presented the results in table 1 as a function of the band ftiing. For comparison, we have also presented the results of FJ_calculated by using the NSC(1) and NSC(2) methods. One notices in table 1 that the Nd (number of delectrons) dependence of FL obtained by the SC method is completely different from that obtained by the NSC methods. Furthermore, one notices that NSC(2) method (employed by Sayers in ref. [3]) does not give the accurate values of FL both for the half-filed and almost full bands (compared to the results obtained by the NSC( 1) method).
L324
K. Masuda / Vibration frequency
of an atom at surface of metal
Table 1 (~‘0 - w$/w$ (‘FL) calculatedby NSC(l), NSC(2)and Nd FL (NW 1)) FL U’JSC(2N FL (SC)
5.0 0.32 0.4 1 0.32
6.0 0.37 0.43 0.32
7.0 0.47 0.47 0.31
SC methods a) 8.0 0.59 0.53 0.28
9.0 0.65 0.58 0.22
9.4 0.66 0.60 0.18
a) qRo = 3, pRo = 9.
From the results of surface tension 7 in fig. 1 and those of FL in table 1, one can observe that FL is in close relation with the surface tension y. The SC method which gives the “reasonable” (positive) values of surface tension even for the almost filled band leads to the result that FL is less than 0.5 for the whole band region. This is in agreement with the result of Matsubara’s calculation [12] using the interatomic potential method. On the other hand, the NSC methods (both NSC(l) and NSC(2)) lead to the result that FL is greater than 0.5 for a small number of electrons or holes in the band and smaller than 0.5 for a half-filed band. Unfortunately, however, these values of FL are unreliable since the attractive and repulsive contributions are not appropriately evaluated for Nd .zG1 and Nd Z 9, as mentioned before. In this respect, it must be noted that the NSC approach gives some spurious crystal instabilities (e.g., strange behavior of the surface relaxation) for a nearly empty or full d-band [S ,9]. Thus, it appears that the drastic softening observed for Ni(1 lo), Pt(100) and Pd(100) surfaces [13,14], FL= 0.68-0.78, is beyond the scope of the simple tight-binding theory.
References [l] G. Allan, Surface Sci. 85 (1979) 37. [2] C.M. Sayers, J. Phys. Cl2 (1979) 4333. [3] C.M. Sayers, Surface Sci. 82 (1979) 195. [4] G. Allan, M. Lannoo and L. Dobrzynski, Phil. Mag. 30 (1974) 33. [5] G. Allan and M. Lannoo, Phys. Status Solidi (b) 74 (1976) 409. [6] G. Allan and P. Lenglart, Surface Sci. 30 (1972) 641. [7] C.M. Sayers, J. Phys. F9 (1979) 123. [8] J.P. Gaspard and F. Cyrot-Lackmann, J. Phys. C6 (1973) 3077. [9] G. Allan and M. Lannoo, Surface Sci. 40 (1973) 375. [lo] K.P. Bohnen and J.P. Gaspard, Z. Physik B28 (1977) 43. [ll] F. DucasteRe, J. Physique 31(1970) 1055. [ 121 T. Matsubara, J. Phys. Sot. Japan 40 (1976) 603. [ 131 H.B. Lyon and G.A. Somorjai, J. Ched Phys. 44 (1966) 3707. [ 141 J.M. Morabito, R.F. Steiger and G.A. Somorjai, Phys. Rev. 179 (1969) 638.
SURFACE SCIENCE LETTERS COMMENTS ON “THE VIBRATION FREQUENCY OF AN ATOM AT THE SURFACE OF A TIGHT BINDING METAL” BY C.M. SAYERS
K. MASUDA Department of Materials Science and Engineering, Midori-ku, Yokohama 227, Japan
Tokyo Institute of Technology,
Nagatsuta,
Received 21 January 1980
Recently, it has been shown that the simple tight-binding scheme is successful for describing the vibrational property of transition metal surfaces [ 1,2]. Sayers [3], using the Einstein model for lattice vibration, showed that the value of (06 - of)/& depends strongly on the filing of the band and can be greater than 0.5 (0.64) near the end of the transition series. Here, w. and o1 represent the vibration frequency (VF) of an atom in the bulk and that normal to the surface, respectively. To calculate VF, w. and ~1, he considered the repulsive core-core interaction energy, arising from the interatomic d-d repulsion and the increase in sp kinetic energy upon compression, as well as the attractive electronic (d-band) energy. The electronic energy is evaluated by using a non-self-consistent moment technique (second order treatment for the change in the second moment 6pis), while the former repulsive energy is simulated by a phenomenological Born-Mayer potential.
Fig. 1. Surface tension 7 for the simple cubic (iO0) surface calculated by the SC (solid) and NSC (dotdashed and dashed curves) methods. L320
L321
K. Masuda f Vibration frequency of an atom at surface of metal
Sayers’ non-self-consistent (NSC) approach, however, does lead to the unphysical result that the surface tension becomes negative near the beginning or the end of the transition series (see fig. 1). This indicates that the NSC treatment underestimates the electronic contribution to the surface tension and is inappropriate for studying the vibrational property of transition metal surfaces, especially near the beginning or the end of the transition series. In the present communication, surface vibrational property will be discussed more generally by taking into account the variations of the diagonal potential q (hereafter referred to as selfconsistent potential) resulting from charge oscillations near the defect. This treatment is self-consistent (SC) in the sense that it is capable of satisfying the Friedel sum rule [4,5]. The importance of selfconsistency (SC surface potential) for the surface electronic states has already been/pointed out by Allan and Lenglart [6]. We start with the local density of states (DOS) on atom i, obtained by the second moment appro&nation (continued fraction technique) [7] : Pi(E)
2E 2= (4b -
E2)“2 [@ - bie) E* - (2b - bie) aieE + b& + b&l-’
9
where uio and bto are given in terms of the moments pi,, as ($0 = I_cil = (Yi ,
bio = Pi2
-
P?I
(2)
’
This approximation ensures that the perturbed and unperturbed DOS, p&T) and po(f?), have the same band edges +2b”*. Following Sayers [3,7], we assume that the transfer integral tij between sites i and i varies as tij= to eXp(-qRij)
(3)
,
and bio E 7
fz exp(-2qRfj)
.
(4)
The change in band energy due to the deviation oi and bio on site i is then given by [41
beIO a
2n(b - bio)
(4b - E;)“*
+ flsin-’ (Ef/2b “*) + in] + CA_ - A +)
In @D)1’2 (4b +g -f*)A--fit 2 [R_@_T+ - R+T_)] “* I @ 1))“’
- @Z-T+ - R+T_)“* + CR-T, - R+T-)“*
K. Masuda / Vibration frequency
L322
of an atom at surface of metal
, &R-T,
(4b ‘g-f2)A+ -f-g + [R+(R_T+ - R+T_)] “’
sin-
- R+T_)l12
[T+(R+ + @R_)]
•tU&i) 3
1’2 ’
for f2 - 4g < 0, b.IO =
2n(b - bi,,) i
-
K_(K?
(5b) - (K, t K_) [sin-’ (Ef/2b “‘) + i n]
(4b - E#‘2
- 4b)1’2
. _,
4b - K-Ef
( ‘ln
K--K+
2b”‘IEf
- K-l
fin
K+(K,2 - 4b)1’2
1 + C(CQ)7
K+-K-
PC)
forf’-4g>O*, where f = -Qi(2b - bio)/(b - bio) 3 A, = [-(4b R, =A:
+g)*
+fA,
g = (bfo + baf)/(b - bio) 2
[(4b +g)2 - 4bf2]“‘/f, T,=4b-A:,
+g,
K,=[-f*(f2-4g)1’2]/2,
Ef=@+-Ef)/(Ef-A_),
D=T_@+T+,
and C((Y,)=A
{$(4b - Et)“’
- $ai[Ef(4b - Ef2)“2 t 4b(sin-‘(Ef/2b1’2)
t ir)]}. (6)
In eq. (5b), the +$r becomes plus for negative f and minus for positive f, while in eq. (SC) the +$r becomes plus for K+ less than -2b”‘, and minus for Kk greater than 2b1’2. In deriving eqs. (5b) and (5c), we have assumed that the SC potential oi completely screens. the defect (perturbation) and each atom is neutral. In this case the SC potential oli can be determined by Ef
Ef
J Pi(oi, E) do = J PO(E) -m -2x/b
dE.
(7)
This approximation is exact in the limit of large phonon wavelengths, and seems to be reasonable near a surface where the charge oscillations must be very small [5,9,101* To calculate the change in the total energy due to the introduction of lattice defect or the atomic displacement GRii between atomic sites i and j, we add a * In the case where the “split*ff” states appear below the lower band edge, their contribution
should be added.
K. Masuda / Vibrationfrequency of an atom at surface of metal
L323
Born-Mayer repulsive term between nearest neighbours E, (Rji) = Co exp(-pRii)
.
(8)
For typical transition metals with qRo = 3 .O and pR,-, = 9.0 (R, is the equilibrium interatomic distance), the parameter value Cc can be chosen so as to satisfy the equilibrium condition [ 111. From eqs. (5) and (8), it is possible to obtain both the electronic and repulsive contributions to the surface tension. In fig. 1 we show the surface tension 7 (per atom), calculated from the NSC and SC methods with qRo = 3 and pRo = 9, for the (100) surface of a simple cubic tight-binding metal as a function of the band filing: The energy is given in units of the half-bandwidth W (=2b”‘), and the degeneracy of the d-band is taken into account. Dot-dashed and dashed curves represent the 7 calculated by the NSC. methods; they are obtained by using rigorous and second order treatments (hereafter referred. to as NSC( 1) and NSC(2) treatments) for 6&a, respectively. In NSC(l), the electronic constribution to the surface tension is calculated by Ef (E - Ef) AP(E) dE s_-oD (Ap(E) is the change in the DOS for the surface atom), while in NSC(2), it is calculated by using the power expansion for 6~~s (see Sayers’ paper [3]). One can see that the surface tension 7 calculated by the SC method (solid curve) is always positive, while the 7 calculated.by the NSC method becomes negative near the beginning and the end of the transition series. This indicates that the NSC(1) and NSC(2) treatments underestimate the electronic contribution to the surface tension. We now calculate the change in the total (electronic and repulsive) energy due to a small static displacement x of atom i along a [ 1001 direction of the simple cubic lattice. Performing this calculation both for the atom at the (100) surface, 6El(x), and for the bulk atom, &E,(x), one can calculate (0; - c&/c& (ZE’d as FL = (o; - c&/w;
= [6E,,(x) - 6E1(x)]/6Eo(x).
(9)
In the above eq. (9), 6EI(x) is defined as &El(x) = @El(x) + 6EI(-x)}
/2 .
Using eq. (9), we have calculated FL for the (100) surface of a simple cubic tightbinding metal and presented the results in table 1 as a function of the band ftiing. For comparison, we have also presented the results of FJ_calculated by using the NSC(1) and NSC(2) methods. One notices in table 1 that the Nd (number of delectrons) dependence of FL obtained by the SC method is completely different from that obtained by the NSC methods. Furthermore, one notices that NSC(2) method (employed by Sayers in ref. [3]) does not give the accurate values of FL both for the half-filed and almost full bands (compared to the results obtained by the NSC( 1) method).
L324
K. Masuda / Vibration frequency
of an atom at surface of metal
Table 1 (~‘0 - w$/w$ (‘FL) calculatedby NSC(l), NSC(2)and Nd FL (NW 1)) FL U’JSC(2N FL (SC)
5.0 0.32 0.4 1 0.32
6.0 0.37 0.43 0.32
7.0 0.47 0.47 0.31
SC methods a) 8.0 0.59 0.53 0.28
9.0 0.65 0.58 0.22
9.4 0.66 0.60 0.18
a) qRo = 3, pRo = 9.
From the results of surface tension 7 in fig. 1 and those of FL in table 1, one can observe that FL is in close relation with the surface tension y. The SC method which gives the “reasonable” (positive) values of surface tension even for the almost filled band leads to the result that FL is less than 0.5 for the whole band region. This is in agreement with the result of Matsubara’s calculation [12] using the interatomic potential method. On the other hand, the NSC methods (both NSC(l) and NSC(2)) lead to the result that FL is greater than 0.5 for a small number of electrons or holes in the band and smaller than 0.5 for a half-filed band. Unfortunately, however, these values of FL are unreliable since the attractive and repulsive contributions are not appropriately evaluated for Nd .zG1 and Nd Z 9, as mentioned before. In this respect, it must be noted that the NSC approach gives some spurious crystal instabilities (e.g., strange behavior of the surface relaxation) for a nearly empty or full d-band [S ,9]. Thus, it appears that the drastic softening observed for Ni(1 lo), Pt(100) and Pd(100) surfaces [13,14], FL= 0.68-0.78, is beyond the scope of the simple tight-binding theory.
References [l] G. Allan, Surface Sci. 85 (1979) 37. [2] C.M. Sayers, J. Phys. Cl2 (1979) 4333. [3] C.M. Sayers, Surface Sci. 82 (1979) 195. [4] G. Allan, M. Lannoo and L. Dobrzynski, Phil. Mag. 30 (1974) 33. [5] G. Allan and M. Lannoo, Phys. Status Solidi (b) 74 (1976) 409. [6] G. Allan and P. Lenglart, Surface Sci. 30 (1972) 641. [7] C.M. Sayers, J. Phys. F9 (1979) 123. [8] J.P. Gaspard and F. Cyrot-Lackmann, J. Phys. C6 (1973) 3077. [9] G. Allan and M. Lannoo, Surface Sci. 40 (1973) 375. [lo] K.P. Bohnen and J.P. Gaspard, Z. Physik B28 (1977) 43. [ll] F. DucasteRe, J. Physique 31(1970) 1055. [ 121 T. Matsubara, J. Phys. Sot. Japan 40 (1976) 603. [ 131 H.B. Lyon and G.A. Somorjai, J. Ched Phys. 44 (1966) 3707. [ 141 J.M. Morabito, R.F. Steiger and G.A. Somorjai, Phys. Rev. 179 (1969) 638.