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Microscopic theory of diffusion on lattices with local distortion
Surface Science Letters North-Holland
235 (1990) L341-L347
Surface Science Letters
theory of diffusion on lattices with local dist~~ti~~ +
Microscopic
T. Ata-Nissila
and SC. Ving
Received
1990; accepted
13 March
for publication
29 May 1990
We present results of microscopic caIcu~ations of singIe local distortions in the adiabatic potential. We demonstrate local changes. In particular, we compare our results at low The results of these ~aIcu~ations are applied to the diffusion
adatom diffusion on models of centered rectangular surfaces allowing how the diffusion anisotropy and the Arrhenius form depend on these temperatures with predictions based on simple random walk arguments. anisotropy of H adatoms on a W(ll0) surface.
Simple lattice gas models have often been used to model diffusion in real systems [l-3]. In particular, the experimentahy observed diffusion anisotropies of 0 and H adatoms on a W(110) surface by Tringides and Comer [4] have been explained by Kjoll et al. [2] using a two-step lattice gas model of diffusion” In this model, an adsorbed H atom causes a local distortion of the surface potential, which breaks the symmetry of the hourgtass shaped adsorption sites. In fig. lb we show the displacement pattern when a hydrogen adatom is at one end of the hourglass region. This displacement pattern is reversed when the H atom sits at the other end of the hourglass. The gain in binding energy for this distortion comes from a subtle balance between the interaction of the H atom with the first and second layers of the substrate [5]. As a consequence of this distortion, the adsorption potential in the hourglass develops a double well structure and the surface potential has two distinct saddle points with energy barriers Al and A,. This effect has been idealized in the two-step model, in which the diffusion process consists of two distinct steps, namely the intracell and intercell jumps which occur with rates a and b over energy barriers Ai and A,, respectively. In
* This work is supported
0039~6028/90/$03.50
by an ONR grant.
0 1990 - Eisevier Science Publishers
the limit of zero coverage, the diffusion anisotropy factor can be written as [2]
where r = a/b is the branc~ng ratio, and $,/x~ = l/2 for the W(110) surface. The undistorted case corresponds to the limit r + 00, for which the anisotropy ratio recovers the simple geometrical value D,,/Il,, = 2 as observed for diffusion of 0 adatoms [4]. For any finite value of r the ratio is always less than two, and extensive theoretical calculation of the two-step model (21 qualitatively explain the experimental ratio Dyy/Dx, = 1.2 for H diffusion on W(110) [4]. The essential parameter r of the two-step model is expected to be temperature dependent. Assuming a simple Arrhenius form in the low temperature regime, we can write r = (aO/bO) e-Bfbl-d*i,
(2)
where the prefactors a0 and b0 for internal and external rates, respectively, are usuahy different. Obviously, the random walk theory can make no predictions about the additional complicated temperature dependence of these prefactors. Recently, we have developed [6,7] a microscopic theory of classical adatom diffusion and
B.V. (North-Holland)
In(D) (ARBITRARY UNITS) -9.7647
o
1.0
-7.1038
1.2
-4.4429
1.4
-1.7821
1.6
1.8
0.
0
ANISD~D~Y RAW (bf c Fig. 1. (a) Geometry of the W(ll0) surface (from ref. [Z]). S denotes the location of the classical saddle point with a barrier A,, which controls diffusion for an undistorted surface. L indicates the hourglass shaped adsorption sites for adatoms. (b) Schematic picture of the local distortion field p(r) induced by an adsorbed H atom. The initially flat hourglass site develops an additional saddle point with an energy barrier Ai, which controls the intraceh jumps in the two-step model. (c) I)J,Y(upper curve) and Oxx (lower curve) versus /3 on a semilogarithmic scale for a model of adatom diffusion on an undistorted W(ll0) surface (ref. [7]). Dashed line denotes the anisotropy ratio D,,/Dx,,,. We note that fi- 1 and A are scaled to a common energy unit in this and all the other figures.
computed the diffusion tensor D for a variety of different lattices 171. In particular, we demonstrated [7] how in the limit of very low temperatures, the theory recovers both the Arrhenius form of activated diffusion and a geometric limit of random walk theory for anisotropic, rhomboidal lattices with a single physical saddle point. When the theory is applied to a model of diffusion on an undistorted W(110) surface, Dyy/Ilx, --+ 2 universally for BT -+ 0, However, the random walk limit is generally reached at temperatures much lower than the Arrhenius behavior, because the former is sensitive to the renormal~~o~ of the prefactors corresponding to different spatial directions of diffusion.
In this Letter, we have undertaken the microscopic theory to compute D and the corresponding anisotropy ratio Dyy/Dxx for a variety of models of W(ll0) surfaces allowing for local distortions in the adiabatic surface potential. This allows us to examine the validity of the two-step model and the fur1 t~~~er~tu~e dependence of the parameter Y on a general theoretical basis. We find that although it is no longer possible to describe diffusion simpty as consisting of two separate jumps over the saddle points, except at extremely low temperatures, the main prediction (I) of the two-step model for the diffusion anisotropy still holds qualitatively. The local distortion lowers the anisotropy compared to the undistorted ease.
The detailed behavior of the ~~o~~~~ is sensitive to the entire adiabatic surface potential and not just the barrier heights Ai and A,. The ticroscopic theory of difksion gives the elements of the diffusion tensor as D/,“= liziQ;,‘(w;
C=O,C’=O),
(3)
where the matrix Q is g&-enby Q&+:
G, G?)
In (41, *f is the frequency and G and F” denote reciprocal idtice vectors. The quantity x-‘(G G’) is the Fourier transform of the invexaeden&y
f
denotes an adiabatic potent2 seea by the diffusing particle, and the u’s denote int~~a~t~o~sbetween the adatom and substrate atoms at 11, averaged over the substrate vibrational degees of freedom. SXJw; G - G’) is a memory function, which is the Fourier transform of the product of the inverse density and a friction tensor q&w; P>_A ~~~~s~o~ic expression for qPp can be fmmd in ref. f6f. 1x3this work, we will set qxxx= qvv= f, 9 = qYxrsf0, which will not affect the ~~~tat~ve f2Lxes of our resuks f7]_ We note that the theory is vaM at @lb~~~~e~~~~~~~ WithiIxthe high friction a~pro~rn~t~o~,and no ~~u~~~i~~~ about thermal activation or individual diffusion jumps we made. To obtain I&, from (3) one has to invert the infinite matrix Q with a truncated, finite set of G-vectors md verify convergence. To model a distorted centered rectangular lattice3 we first write the total dean sm%ee potential as
first term in (5) describes the sum of pair potentials from the surface layer, while the second is the co~t~b~ti~~ from an underlying layer of atoms at depth H. For the centered r~~~~~~~ case, the second layer is displaced ho~zo~t~~~~by 8, which causes the. second layer atoms to lie directly beneath the clean surface adsorption sites. fn (S), y and kg are constants. fn the presence 0-l an I3 adatom, a focaI d~sto~~onin the first Iayer of surface atoms occurs. For shon range H-W interactions, we can ~p~ru~rna~ the locat distor~~~~ by a u&form ~sp~a~rne~t afr) of rhe first layer since the d~spia~~rn~nt of the W atoms beyond the range of interaction has no effect on the adsorption potential. CIIxedisplacement a(r), however, is a strong f~ot~~~ of the position of the adatorn, By symmetry, it should vanish when the H adatom is at the center of the hourglass region. The displacement also has to be periodic, being an odd function of the .P and an even function of t&e x coordinates of the adatom rne~~~ from the center of the hourglass. The general f~rrn of the distorted surfzxe potential is The
V(#->= &+-
R, - #> e-y”:
i
In the knit u -+ 0, this becomes
By chrsosing au ~~prop~ate S(F)* one can then obtak the ~~~~p~n~~~ distorted adiabatic ~t~t~~ V,. However~perhaps tire easiest my to construct a desired potential is to modify the Fourier cornpo~~~ts of T/C@ for various reciprom cal lattice sectors while respecting the symmetry constraints ~rn~~$~dby rc(r). In fig. 1~:we show results of caiculations from ref. [73 for ,m atom on an undistorted surface (i.4, II = O), with an &iabatic potential V,(r> = $[cos@& V) + cos(G2-t)j. This ~~~pond~
to A, = I and Ai = 0. In the high ternperaturrz &r& the &-m-y cxmdy z.-ecwers the case of diffusicynin a viscous, uniform medhm fix which D is isotropic. Also, both the Arrhenius
Fig. 2. (a) Adiabatic potential for a mofid of a distorted W(llU) surFxe, with d, = 20, di = 9. The unit cell shown has an W atom at each corner, and one in the middle. As D consequence of the local distortion, there are two minima within each hourglass adsorption site. (b) .I$,, (upper curve) and I&, (IOWW curve) versus inverse temperature /j for the pocentiai surface in (a). The anisotropy ratio values for an u~~~s~orc~ L&~&. f&shed Iinej approaches its UXZ~VW&limit 2 for fi * co, but remains Wow the corresponding surface. (c) Adiabatic potential with d, ,= 100, 6, - 99. (d) D,Vy (lower CU~W) and L7y, (upper curve) versus p for the potential surface in (c). DY,/Dxx (dashed lirxc) approaches its asymptotic limit 2 only well below the temperature range shown here.
ANIS’QTROPY RATIO
(b)
show results of calculations of eS for a potential of fig. 2a: F$(Fq= P&(P) - F&n(G,
-p) 6 sia&+)j2.
Obviously, Vi is proportional to the magnitude of the displacement of W atoms involved in the local distortion. We have chosen the value of V, in v&(r) to be (3.0 + 3m)/S. This leads to the barrier vaIues Ai = 9 and A, = 20. The diffusion. tensor D is controlled by the larger exterr~al barrier A,, which appears in the &ective Arrhenius form Q, - D,,,),- ePpde at low temperatures. The ~eomet~c ratio ~~.~/~~~ = 2 is obtained in the limit /CT+ 0, as predicted by (2) and (2) with r --+ no- At ~~te~~~iate temperatures the anisotropy ratio is considerably smra#er than in the undistorted case. In fig. 2d we display results for an extreme case of potential of fig. 2c with F; 111 (100 + 3m)J8, where the barriers are almost equal, with Ai = 99 and d, =s:ZOO.Although D varies &out 36 i3xh-s in rnag~it~~e~ the anisotropy ratio remains very close to one and 3s not approaching its Ii~ting value of 2 at the temperatures shown here. However, the Arrhenius form is nevertheless obeyed to a very good degree of accuracy.
In the opposite case of a strong re~o~str~c~ion where Ai > A,, we expect the diffusion in the y-direction to be sq?IXressed by the larger i.nrernaI barrier, In fig. 3 we show results for a ~oten~i~~ of the form VA(r) = &(v) - Vt[sin(G,
lr)
+ sin(G,*r)]2
-k ~~icosfG,.*r)~cos(G-*r)], fOF Wti&
Wi? hZiVi5 ChC3St?~ $5 = 5,
@) TX = s9 k%W&lg
4, = 19.55. In (S) G.,= G, + Gz and G _B G, - G,. B,, and DYYnow follow distinct eff~Gtive ~he~~iL~s forms BX, - e--me and BYF- c-“~;, which causes the ratio ~~~~~~~~~to to
di
xi&fi
= 20.01,
in &irelow kzqxzrature litit. This is gmsdjlc-
ted by frt) correctly. Other cafcnlations with similar potentials and larger ratios of AJA, showed a more rapid approach of the anisotropy ratio towards zero. an interesting special case arises if Ai = A,. Refeting to the. twestep mde, we then expecf the ratio &..~DXx to approach a nontrivial fixed value aa k T ---*0, which for r = 1 equals -$I In figs_ 4b and 4d we show results of Calc~~Iations for potentials of the type (81, chosen such that A;f4, = I and varying the absolute magnitudes of the
Fig. 4. (a) Adiabatic potential with equal barriers d, = di = I&Q. (b) t)yj, (jower curve) and DXXfuppe~ curve} versw 8 for tfic =: 0.82. (c) potential surface in (a). In the limit fi -+ w the ratio D(,,,/D,, (dashed line) approaches a nantrivial fixed value D,,/& Another adiabatic potential with equal, steep barriers d, = Ai = 19.51. (d) D,, (lower curve) and D,, (upper curve) versus fi for the potential surface in (c). In tfie limit /? -* co, I&/D,, = 0.53 (dashed line).
barriers. In all cases, the low temperature behavior follows an ,A~ben~~s form I&, - FYY- e-p’e, with an anisatro~y ratio ~~~,~~~~~attaming a potential dependent fixed vague in the limit kT + 0, Interestingly enough, for potentials of the type described here these values are refatively close to the
prediction of the two-step model with a branG~ng ratio ir = 1. Ex~~riment~l~~ the diffitsion anisotropy ratio, Davy/Bx,rfor the ~/W~~~~~ system was observed to be a~~r~~rnat~~y 1.2 [4f_ Moreover, bath Dyy and LIT, were found to obey Arrhenius behavior
with the same activation energy of about 5 kc& According to our theory, this would indicate that for the diffusion of W the barriers on W(110) surface have values such that A, < A,. Previously, we have estimated [Z] that the branching ratio r = 3 at room temperature. Referring back to eq, (2) this would correspond to the case A, = 9, A, = 10 discussed above if we assume that a, = 41,. To summarize, our microscopic theoreticaX calcufations of models of the W(~~~~ surface with local distortions support the use of the phenomenological two-step model for qualitative purposes. In all the cases studied involving different ratios of the two barriers A/A, the zero temperature limit for the diffusion anisotropy factor is always correct& predicted by the two-step model formutae (1) and (2). At finite temperatures, the actual diffusion path is far mare complicated than the naive picture of the two-step model, and eq. (2) is not qu~t~tat~~~e~y correct, Nevertheless, the diffusion ~~sotro~y is well described by the simple two-step model with an effective temperature dependent branching ratio r. Moreover, within the lists temperature range of the diffusion experiments, r often remains approximately constant. The main feature of the two-step model, namely the reduction of the diffusion anisotropy from the ideal surface value DYY/I& = 2, is clearly snpported by our calculations. Another advantage of the microscopic calculations is the natural emergence of the Arrhenius form at low temperatures, and the d~t~~~ation of the proper a~t~vat~o~
barrier for the diffusion constant, which is of course beyond the simple lattice gas model. We find that for A, 2 A i, the Arrhenius barrier is always determined by d,, indep~nd~ni of the internal barrier. This is exactly what we would expect from analysis based on simpte diffusion trajectories. Experimental data [4] seems to indicate that this is the case for the H/W(XlO) system as ~31. For the case where A, < A i, the larger internal barrier controls diffusion along the y direction, and in the low temperature limit D is dominated by Dxx - e -p’e. Finally, we like to point out that because of the sensitivity of diffusion anisotropy to local surface distortions, experimental m~~s~r~rnents of this. ratio can be used as an effective tool In probing lo& changes in the adsorbate environment.
References K. Einder and w. R&r, in: Applications of the Maim Carlo Method in Statistical Physics, Ed. K. Binder (Springw, Berlin, 1987). J. Kjoll, T. Afa-Nissila and S.C. Ying, Surf. Sci, 2X8 (1989) L476; J. Kjoll, T. Ala-Nissila, S.C. Ying and R. Tahir-Kheii, to be published. M. Trizqides and R. Comer, Surf. Sci. 166 (1986) 419,440. M. ‘Ttingides and R. Gamer, Surf. Sci. 155 (1985) 254. J.W. Chung, S.C. Ying and P.J. Estrup, Phys. Rev. Lett. 56 (1986) 749. SK. Ying, Phys. Rev. B 41 (1990) 7068. T. Ala-N&&la and S.C. Ying, to be pubiished.
235 (1990) L341-L347
Surface Science Letters
theory of diffusion on lattices with local dist~~ti~~ +
Microscopic
T. Ata-Nissila
and SC. Ving
Received
1990; accepted
13 March
for publication
29 May 1990
We present results of microscopic caIcu~ations of singIe local distortions in the adiabatic potential. We demonstrate local changes. In particular, we compare our results at low The results of these ~aIcu~ations are applied to the diffusion
adatom diffusion on models of centered rectangular surfaces allowing how the diffusion anisotropy and the Arrhenius form depend on these temperatures with predictions based on simple random walk arguments. anisotropy of H adatoms on a W(ll0) surface.
Simple lattice gas models have often been used to model diffusion in real systems [l-3]. In particular, the experimentahy observed diffusion anisotropies of 0 and H adatoms on a W(110) surface by Tringides and Comer [4] have been explained by Kjoll et al. [2] using a two-step lattice gas model of diffusion” In this model, an adsorbed H atom causes a local distortion of the surface potential, which breaks the symmetry of the hourgtass shaped adsorption sites. In fig. lb we show the displacement pattern when a hydrogen adatom is at one end of the hourglass region. This displacement pattern is reversed when the H atom sits at the other end of the hourglass. The gain in binding energy for this distortion comes from a subtle balance between the interaction of the H atom with the first and second layers of the substrate [5]. As a consequence of this distortion, the adsorption potential in the hourglass develops a double well structure and the surface potential has two distinct saddle points with energy barriers Al and A,. This effect has been idealized in the two-step model, in which the diffusion process consists of two distinct steps, namely the intracell and intercell jumps which occur with rates a and b over energy barriers Ai and A,, respectively. In
* This work is supported
0039~6028/90/$03.50
by an ONR grant.
0 1990 - Eisevier Science Publishers
the limit of zero coverage, the diffusion anisotropy factor can be written as [2]
where r = a/b is the branc~ng ratio, and $,/x~ = l/2 for the W(110) surface. The undistorted case corresponds to the limit r + 00, for which the anisotropy ratio recovers the simple geometrical value D,,/Il,, = 2 as observed for diffusion of 0 adatoms [4]. For any finite value of r the ratio is always less than two, and extensive theoretical calculation of the two-step model (21 qualitatively explain the experimental ratio Dyy/Dx, = 1.2 for H diffusion on W(110) [4]. The essential parameter r of the two-step model is expected to be temperature dependent. Assuming a simple Arrhenius form in the low temperature regime, we can write r = (aO/bO) e-Bfbl-d*i,
(2)
where the prefactors a0 and b0 for internal and external rates, respectively, are usuahy different. Obviously, the random walk theory can make no predictions about the additional complicated temperature dependence of these prefactors. Recently, we have developed [6,7] a microscopic theory of classical adatom diffusion and
B.V. (North-Holland)
In(D) (ARBITRARY UNITS) -9.7647
o
1.0
-7.1038
1.2
-4.4429
1.4
-1.7821
1.6
1.8
0.
0
ANISD~D~Y RAW (bf c Fig. 1. (a) Geometry of the W(ll0) surface (from ref. [Z]). S denotes the location of the classical saddle point with a barrier A,, which controls diffusion for an undistorted surface. L indicates the hourglass shaped adsorption sites for adatoms. (b) Schematic picture of the local distortion field p(r) induced by an adsorbed H atom. The initially flat hourglass site develops an additional saddle point with an energy barrier Ai, which controls the intraceh jumps in the two-step model. (c) I)J,Y(upper curve) and Oxx (lower curve) versus /3 on a semilogarithmic scale for a model of adatom diffusion on an undistorted W(ll0) surface (ref. [7]). Dashed line denotes the anisotropy ratio D,,/Dx,,,. We note that fi- 1 and A are scaled to a common energy unit in this and all the other figures.
computed the diffusion tensor D for a variety of different lattices 171. In particular, we demonstrated [7] how in the limit of very low temperatures, the theory recovers both the Arrhenius form of activated diffusion and a geometric limit of random walk theory for anisotropic, rhomboidal lattices with a single physical saddle point. When the theory is applied to a model of diffusion on an undistorted W(110) surface, Dyy/Ilx, --+ 2 universally for BT -+ 0, However, the random walk limit is generally reached at temperatures much lower than the Arrhenius behavior, because the former is sensitive to the renormal~~o~ of the prefactors corresponding to different spatial directions of diffusion.
In this Letter, we have undertaken the microscopic theory to compute D and the corresponding anisotropy ratio Dyy/Dxx for a variety of models of W(ll0) surfaces allowing for local distortions in the adiabatic surface potential. This allows us to examine the validity of the two-step model and the fur1 t~~~er~tu~e dependence of the parameter Y on a general theoretical basis. We find that although it is no longer possible to describe diffusion simpty as consisting of two separate jumps over the saddle points, except at extremely low temperatures, the main prediction (I) of the two-step model for the diffusion anisotropy still holds qualitatively. The local distortion lowers the anisotropy compared to the undistorted ease.
The detailed behavior of the ~~o~~~~ is sensitive to the entire adiabatic surface potential and not just the barrier heights Ai and A,. The ticroscopic theory of difksion gives the elements of the diffusion tensor as D/,“= liziQ;,‘(w;
C=O,C’=O),
(3)
where the matrix Q is g&-enby Q&+:
G, G?)
In (41, *f is the frequency and G and F” denote reciprocal idtice vectors. The quantity x-‘(G G’) is the Fourier transform of the invexaeden&y
f
denotes an adiabatic potent2 seea by the diffusing particle, and the u’s denote int~~a~t~o~sbetween the adatom and substrate atoms at 11, averaged over the substrate vibrational degees of freedom. SXJw; G - G’) is a memory function, which is the Fourier transform of the product of the inverse density and a friction tensor q&w; P>_A ~~~~s~o~ic expression for qPp can be fmmd in ref. f6f. 1x3this work, we will set qxxx= qvv= f, 9 = qYxrsf0, which will not affect the ~~~tat~ve f2Lxes of our resuks f7]_ We note that the theory is vaM at @lb~~~~e~~~~~~~ WithiIxthe high friction a~pro~rn~t~o~,and no ~~u~~~i~~~ about thermal activation or individual diffusion jumps we made. To obtain I&, from (3) one has to invert the infinite matrix Q with a truncated, finite set of G-vectors md verify convergence. To model a distorted centered rectangular lattice3 we first write the total dean sm%ee potential as
first term in (5) describes the sum of pair potentials from the surface layer, while the second is the co~t~b~ti~~ from an underlying layer of atoms at depth H. For the centered r~~~~~~~ case, the second layer is displaced ho~zo~t~~~~by 8, which causes the. second layer atoms to lie directly beneath the clean surface adsorption sites. fn (S), y and kg are constants. fn the presence 0-l an I3 adatom, a focaI d~sto~~onin the first Iayer of surface atoms occurs. For shon range H-W interactions, we can ~p~ru~rna~ the locat distor~~~~ by a u&form ~sp~a~rne~t afr) of rhe first layer since the d~spia~~rn~nt of the W atoms beyond the range of interaction has no effect on the adsorption potential. CIIxedisplacement a(r), however, is a strong f~ot~~~ of the position of the adatorn, By symmetry, it should vanish when the H adatom is at the center of the hourglass region. The displacement also has to be periodic, being an odd function of the .P and an even function of t&e x coordinates of the adatom rne~~~ from the center of the hourglass. The general f~rrn of the distorted surfzxe potential is The
V(#->= &+-
R, - #> e-y”:
i
In the knit u -+ 0, this becomes
By chrsosing au ~~prop~ate S(F)* one can then obtak the ~~~~p~n~~~ distorted adiabatic ~t~t~~ V,. However~perhaps tire easiest my to construct a desired potential is to modify the Fourier cornpo~~~ts of T/C@ for various reciprom cal lattice sectors while respecting the symmetry constraints ~rn~~$~dby rc(r). In fig. 1~:we show results of caiculations from ref. [73 for ,m atom on an undistorted surface (i.4, II = O), with an &iabatic potential V,(r> = $[cos@& V) + cos(G2-t)j. This ~~~pond~
to A, = I and Ai = 0. In the high ternperaturrz &r& the &-m-y cxmdy z.-ecwers the case of diffusicynin a viscous, uniform medhm fix which D is isotropic. Also, both the Arrhenius
Fig. 2. (a) Adiabatic potential for a mofid of a distorted W(llU) surFxe, with d, = 20, di = 9. The unit cell shown has an W atom at each corner, and one in the middle. As D consequence of the local distortion, there are two minima within each hourglass adsorption site. (b) .I$,, (upper curve) and I&, (IOWW curve) versus inverse temperature /j for the pocentiai surface in (a). The anisotropy ratio values for an u~~~s~orc~ L&~&. f&shed Iinej approaches its UXZ~VW&limit 2 for fi * co, but remains Wow the corresponding surface. (c) Adiabatic potential with d, ,= 100, 6, - 99. (d) D,Vy (lower CU~W) and L7y, (upper curve) versus p for the potential surface in (c). DY,/Dxx (dashed lirxc) approaches its asymptotic limit 2 only well below the temperature range shown here.
ANIS’QTROPY RATIO
(b)
show results of calculations of eS for a potential of fig. 2a: F$(Fq= P&(P) - F&n(G,
-p) 6 sia&+)j2.
Obviously, Vi is proportional to the magnitude of the displacement of W atoms involved in the local distortion. We have chosen the value of V, in v&(r) to be (3.0 + 3m)/S. This leads to the barrier vaIues Ai = 9 and A, = 20. The diffusion. tensor D is controlled by the larger exterr~al barrier A,, which appears in the &ective Arrhenius form Q, - D,,,),- ePpde at low temperatures. The ~eomet~c ratio ~~.~/~~~ = 2 is obtained in the limit /CT+ 0, as predicted by (2) and (2) with r --+ no- At ~~te~~~iate temperatures the anisotropy ratio is considerably smra#er than in the undistorted case. In fig. 2d we display results for an extreme case of potential of fig. 2c with F; 111 (100 + 3m)J8, where the barriers are almost equal, with Ai = 99 and d, =s:ZOO.Although D varies &out 36 i3xh-s in rnag~it~~e~ the anisotropy ratio remains very close to one and 3s not approaching its Ii~ting value of 2 at the temperatures shown here. However, the Arrhenius form is nevertheless obeyed to a very good degree of accuracy.
In the opposite case of a strong re~o~str~c~ion where Ai > A,, we expect the diffusion in the y-direction to be sq?IXressed by the larger i.nrernaI barrier, In fig. 3 we show results for a ~oten~i~~ of the form VA(r) = &(v) - Vt[sin(G,
lr)
+ sin(G,*r)]2
-k ~~icosfG,.*r)~cos(G-*r)], fOF Wti&
Wi? hZiVi5 ChC3St?~ $5 = 5,
@) TX = s9 k%W&lg
4, = 19.55. In (S) G.,= G, + Gz and G _B G, - G,. B,, and DYYnow follow distinct eff~Gtive ~he~~iL~s forms BX, - e--me and BYF- c-“~;, which causes the ratio ~~~~~~~~~to to
di
xi&fi
= 20.01,
in &irelow kzqxzrature litit. This is gmsdjlc-
ted by frt) correctly. Other cafcnlations with similar potentials and larger ratios of AJA, showed a more rapid approach of the anisotropy ratio towards zero. an interesting special case arises if Ai = A,. Refeting to the. twestep mde, we then expecf the ratio &..~DXx to approach a nontrivial fixed value aa k T ---*0, which for r = 1 equals -$I In figs_ 4b and 4d we show results of Calc~~Iations for potentials of the type (81, chosen such that A;f4, = I and varying the absolute magnitudes of the
Fig. 4. (a) Adiabatic potential with equal barriers d, = di = I&Q. (b) t)yj, (jower curve) and DXXfuppe~ curve} versw 8 for tfic =: 0.82. (c) potential surface in (a). In the limit fi -+ w the ratio D(,,,/D,, (dashed line) approaches a nantrivial fixed value D,,/& Another adiabatic potential with equal, steep barriers d, = Ai = 19.51. (d) D,, (lower curve) and D,, (upper curve) versus fi for the potential surface in (c). In tfie limit /? -* co, I&/D,, = 0.53 (dashed line).
barriers. In all cases, the low temperature behavior follows an ,A~ben~~s form I&, - FYY- e-p’e, with an anisatro~y ratio ~~~,~~~~~attaming a potential dependent fixed vague in the limit kT + 0, Interestingly enough, for potentials of the type described here these values are refatively close to the
prediction of the two-step model with a branG~ng ratio ir = 1. Ex~~riment~l~~ the diffitsion anisotropy ratio, Davy/Bx,rfor the ~/W~~~~~ system was observed to be a~~r~~rnat~~y 1.2 [4f_ Moreover, bath Dyy and LIT, were found to obey Arrhenius behavior
with the same activation energy of about 5 kc& According to our theory, this would indicate that for the diffusion of W the barriers on W(110) surface have values such that A, < A,. Previously, we have estimated [Z] that the branching ratio r = 3 at room temperature. Referring back to eq, (2) this would correspond to the case A, = 9, A, = 10 discussed above if we assume that a, = 41,. To summarize, our microscopic theoreticaX calcufations of models of the W(~~~~ surface with local distortions support the use of the phenomenological two-step model for qualitative purposes. In all the cases studied involving different ratios of the two barriers A/A, the zero temperature limit for the diffusion anisotropy factor is always correct& predicted by the two-step model formutae (1) and (2). At finite temperatures, the actual diffusion path is far mare complicated than the naive picture of the two-step model, and eq. (2) is not qu~t~tat~~~e~y correct, Nevertheless, the diffusion ~~sotro~y is well described by the simple two-step model with an effective temperature dependent branching ratio r. Moreover, within the lists temperature range of the diffusion experiments, r often remains approximately constant. The main feature of the two-step model, namely the reduction of the diffusion anisotropy from the ideal surface value DYY/I& = 2, is clearly snpported by our calculations. Another advantage of the microscopic calculations is the natural emergence of the Arrhenius form at low temperatures, and the d~t~~~ation of the proper a~t~vat~o~
barrier for the diffusion constant, which is of course beyond the simple lattice gas model. We find that for A, 2 A i, the Arrhenius barrier is always determined by d,, indep~nd~ni of the internal barrier. This is exactly what we would expect from analysis based on simpte diffusion trajectories. Experimental data [4] seems to indicate that this is the case for the H/W(XlO) system as ~31. For the case where A, < A i, the larger internal barrier controls diffusion along the y direction, and in the low temperature limit D is dominated by Dxx - e -p’e. Finally, we like to point out that because of the sensitivity of diffusion anisotropy to local surface distortions, experimental m~~s~r~rnents of this. ratio can be used as an effective tool In probing lo& changes in the adsorbate environment.
References K. Einder and w. R&r, in: Applications of the Maim Carlo Method in Statistical Physics, Ed. K. Binder (Springw, Berlin, 1987). J. Kjoll, T. Afa-Nissila and S.C. Ying, Surf. Sci, 2X8 (1989) L476; J. Kjoll, T. Ala-Nissila, S.C. Ying and R. Tahir-Kheii, to be published. M. Trizqides and R. Comer, Surf. Sci. 166 (1986) 419,440. M. ‘Ttingides and R. Gamer, Surf. Sci. 155 (1985) 254. J.W. Chung, S.C. Ying and P.J. Estrup, Phys. Rev. Lett. 56 (1986) 749. SK. Ying, Phys. Rev. B 41 (1990) 7068. T. Ala-N&&la and S.C. Ying, to be pubiished.