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A theoretical study of step bunching dynamics in the presence of an alternating heating current
B. Houchmandzadeh, C. Misbah, A. Pimpinelli
To cite this version:
B. Houchmandzadeh, C. Misbah, A. Pimpinelli. A theoretical study of step bunching dynamics in the presence of an alternating heating current. Journal de Physique I, EDP Sciences, 1994, 4 (12), pp.1843-1853. �10.1051/jp1:1994225�. �jpa-00247038�
Classification Pliysics Abstracts
61 50C 05.40 68.55
A theoretical study of step bunching dynanùcs in the presence of
an alternating heating current
B. Houchmandzadeh (~>*), C. Misbah (~>**) and A. Pimpinelli (~,***)
(~) Laboratoire de Spectrométrie Physique, Université Joseph Fourier (Grenoble I) & CNRS, B-P. 87, 38402 Saint-Martin d'Hères Cedex, France
(~) CEA/Départment de Recherche Fondamentale
sur la Matière Condensée, SPSMS/MDN,
Centre d'Etudes Nucléaires de Grenoble, 17 avenue des Martyrs, 38054 Grenoble, France
(Received 31 May 1994, accepted in final form 8 September 1994)
Rèsumé. Nous analysons théoriquement l'influence d'un courant alternatif sur la dynamique
des marches durant la sublimation de la surface vicinale d'un cristal. Nous expliquons l'absence d'instabilité de mise en paquet des marches dans des expériences récentes utilisant un courant alternatif Ceci est attribué
au fait que la période du courant est beaucoup plus petite que le temps nécessaire au developpement de l'instabilité- Notre analyse montre qu'un tel phenomène peut se manifester, dans le
cas du Si(111) à1250 °C, à des fréquences autour de 1 Hz. Il devrait
se présenter comme une alternance dans le temps entre une situation où les marches sont mises
en paquet et celle où elles sont régulièrement espacées. La détermination de la fréquence critique
au-delà de laquelle l'instabilité disparaît permet une mesure directe de la force électrique qui
est supposée s'excercer sur les adatomes. L'expérience proposée est une vérification cruciale de la validité de l'hypothèse d'électromigratiou. Les points actuellement controversés de
ce modèle
ainsi que des perspectives seront discutés.
Abstract. We analyze theoretically the influence of
an alternating electric current on step dyuamics during sublimation of
a vicinal surface. We explaiu quautitatively why in previous experiments the application of an alteruatiug current left the vicinal surface unaffected, 1e no
step-bunching. This is due to the fact that the excitation period used is much lower than the intrinsic time scale for the cooperative phenomenon that leads to step-bunching. Dur analysis suggests that step-bunching should be present, for Si(111) at 1250 °C, at a frequency of about 1 Hz. This should manifest itself by a bunching-debunching process m the course of time.
The determmation of the excitation frequency above which this process no longer exists should
give access to the strength of the electromigration force The proposed experiment should also
constitute a crucial test of the electromigration hypothesis. Discussions of sortie unresolved
puzzles together with outlooks are presented.
(* e-mail: [email protected].
(** e-mail: misbah©phase3.grenet.fr (*** e-mail: [email protected] fr.
@ Les Editions de Physique 1994
1. Introduction.
Many experiments [1-5j reported on step bunching during sublimation of vicinal Si(Ill) sur- faces, when trie heating source is a DC electric current. This phenomenon depends both on trie
temperature and on trie direction of trie heating current. In particular, reversing trie current direction transforms a stable temperature domain into an unstable one and mceuersa (see Fig.
1). It should be noted that experiments bave been performed with surface miscuts ranging
between 8' and 8° and that trie surface morphology diagram shown in figure 1 does not depend
on trie step spacing.
~
l
_ ~
1000 1200 1400 T(°C)
,
1200 1400 T(°C)
Fig. 1. Temperature range of step-bunching instability for ascending and descending current. Thick fines represent unstable regions.
Stoyanov [6, 7j proposed a simple but physically appealing theory to explain trie current- induced step bunching, based on trie addition of an electromigration force in trie step flow model of Burton, Cabrera and Franck (BCF)[8j: adatoms on trie surface are assumed to carry electric charges and therefore their Brownian motion is modified by trie electric field. More
precisely, this theory qualitatively accounts for trie appearance of step bunching at intermediate temperatures and its disappearance at higher temperatures when trie current points in trie
ascending direction. It also explains trie complementarity of trie step-up and step-down current
diagrarn (see Fig. l). As discussed by Stoyanov, trie cross-over region (when one goes from an unstable to a stable situation by increasing temperature) occurs basically when trie diffusion
length xs = @@ (Ds is trie surface diffusion coefficient, and T trie evaporation time) is of trie order of trie interstep distance, 1.
There are however several questions that remain to be clarified. First, trie cross-over tem-
perature T* is determined by x~ r~J1. Since xs is thermally activated, T* should depend on 1, contrary to what is observed (T* m 1250 °C for 20 À < 1 < 600 À). Second, trie diffusion
length bas been found from step velocity measurements [1, 9] to be at least 1 ~tm at T*, which is much larger than1 [Si.
In view of this unclear situation, it is important that one can propose new experimental protocols which may help to guide further developments. We propose here an experiment which con test the validity of the electromigration hypothesis, thereby giving the order of
magnitude of (the hitherto supposed) forces actmg on adatoms. It is based on the use of
an alternating current in the relevant range of frequencies. Experiments with an alternating
current have been already performed at a standard frequency v = 50 Hz. They concluded that
the alternating current leaves the vicinal surface unaffected (1.e. no step bunching). As we shall
see, a close inspection of the relevant time scale for the manifestation of step-bunching shows that such a frequency is likely to be too high to affect the vicinal surface. This is the time necessary for step bunching uJ~~ » v~~, so that before the step bunching instability (which is
potentially present) amplifies the current is reversed.
Dur conclusion is that if electromigration is the appropriate ingredient for step-bunching,
then we should observe the effect of the alternating current in the appropriate frequency range.
For example, for Si(Ill) at T*
r~J 1200 °C the appropriate range of frequency is in the range
1-10 Hz. The measure of the frequency threshold for the synchronization of step-bunching
and alternating current should give direct access to the amplitude of the electric force acting
on adatoms. Furthermore, the experimental observation of synchronization will reinforce our belief in the role of electromigration.
This paper is organized as follows. In section 1, we briefly recall Stoyanov's results for
a direct current and include the Schwoebel effect. In section 2 we analyze the effect of trie
alternating current. Finally section 3 is devoted to long time evolution of steps by integrating numerically trie nonhnear governing equations. Section 4 contains a discussion and conclusion.
2. BCF mortel with electromigration.
We consider an infinite set of straight steps on a crystal surface. During sublimation, adatoms diffuse on the surface with the diffusion coefficient Ds and evaporate with a lifetime T. A
constant electric force F acts on adatoms and induces an average velocity u
= DSF/kT. The adatom density c obeys
°~
= D~
°~~ ~S~ °~
~_ (i)
ôt ôx2 kT ôx T
Steps act hke adatom sources and sinks. Giving the attachement kinetic coefficients from the lower and upper terraces k+ (in the following, we use '+','-' to refer to quantities on the lower and upper sides respectively), and lengths associated with them d+
= Ds/k+ we zan write the boundary conditions at the steps
~ Î~~~
~dÎ~~' ~~~
°C F
~
C Ceq
(~)
ôx kT d-
where c~q is the equihbrium adatom concentration. The left hand side in (2, 3) (multiplied by Ds) represent the net flux of adatoms across the step, denoted by j+ and j-. The step velocity
is then given by
v " (A~)~~(J+ J-) ~ Q(J+ J-)> (4)
where zlc is the concentration difference between the solid and the gaseous phase adjacent to the step, and Q the area per atom in the crystal [loi.
Let us introduce different natural length scales. The first one is xs
= @@, the mean
free path of an adatom before it evaporates, usually called the diffusion length. This length
measures the scale over which the adatom concentration profile varies. We bave already defined
d+, which quantifies the adhesion of adatoms. Another natural length is the interstep distance 1 which characterizes the confinement of the adatom concentration field when xs > 1. Finally,
there is a length associated with the electromigration term, ( = 2kT/F.
To analyze step bunching, we begin with a set of equidistant steps, and perturb each of them
by a small amount (n. We use the quasi~static approximation which amounts to neglecting
c
c 1_____ ~
~~ ~
x
~i
j~
~ j+
ôc/ôt term in (1) [10]. This is warranted if steps move slowly enough so that the concentration field instantaneously adapts itself to the new terrace width. The concentration field on a terrace of width 1 is given by
c(x) =
e~/~ A cosh °~ + B sinh °~ c~q, (5)
xs xs
where a
= (1+ x)/(~)~/~ and A and B are integration constants. To obtain their expressions,
we make use of equations (2) and (3):
-ij d ai ad ai ad+ -ij
~~~~ " @ ~~ T~~~~~ i + ~ ~°~~ £ + ~ ~~P T ~~~
B(1) = (l ~~ cosh °~
+
°~~
sinh
°~ (l + ~~
exp (7)
D j z~ z~ z~ z~ j
where a
= (1+ x(/(~)l/~ and
D(1) = -(1+ ~~~~ +
~~ ~~
sinh °~ °
(d+ + d-) cosh °~. (8)
x~ xs xs xs
Let 1+ and 1- be the widths of lower and upper terraces (Fig. 2). The field in the region 0 < x < 1+ is given by expression (5) where we set 1=1+. Note that adatom concentration
at x = 0 (or 1) can be greater than c~q, depending on the value and sign of ( :
~~°~
= i (
+ j)1 (9)
Ceq Xs
where we take d+ = d- for simphcity. This phenomenon is well described by Stoyanov [7].
In order to evaluate the total diffusion current we need the expression of the field in the region -1- < x < 0 as well. It is obviously given by a similar expression by shifting x by an
amount 1-. Having determined the diffusion field on both sides of a step we are in a position
to write the diffusion currents:
a ~
e~ ~
aÎ+ °~~
Cosh ) ÎÎ ~
f~~ceq ~~ Î) siuh $ ~~
z~ ~
(ÎÙ)
(~~) " p(~~)d+Î
)+
_Dseq
J-(1-)
= ~lil~i 1((+ +1) sinh t~ + t+ cash t~ + °le~-/~)
+i~~~ ~~~)
Equations (10, II) can be used to compute the instantaneous velocity of the nth step
Vn = Q(J+(1+) J-(1-)). (12)
Equation (12) represents an infinite set of nonlinear equations for the evolution of terrace widths. We shall come back later to the full numerical study of this set. We now focus on the linear analysis of the instability. Consider a reference state characterized by a set of eqmdistant steps, separated by a distance 1, and let (n denote the fluctuation of the position of the nth step about its mean value. Thus, for the nth step, 1+ =1+ (n+i (n, 1- =1+ (n (n-i (see Fig. 2) and we can write its velocity as Vn = Vo + (n where
Vo " ii (J+(1) j- (1)) (13)
(n " 9+(n+1 (9+ + g-)(n + g-(n-1 (14)
and 9+"~ÎÎ~' 9-"~Îi~ ~~~~
Setting (n = (oexp(uJt + mçi) in (14), we obtain the dispersion relation
uJ = (g+ + g-)(cos çi 1) +1sin çi(g+ g-). (16)
The most unstable mode occurs for çi = ~r (that is for out-of-phase fluctuations [iii When
d+ = d-
= d, uJ« takes the simple form
xs ai 1
4QDsc~q à £ ~~~~ £ ~ ° ~~~~ [
~"
~ °~
(1
+ 1) sinh il + 2a cash il
,
~~~~
The term in brackets is what Stoyanov calls 16(1). For step up direction of the current, the slope of 4l is positive (step bunching appears) when xs » 1and negative when xs < 1 (see
Fig. 3). The cross-over region corresponds to xs r~J
1. When the current is step down, the
phenomenon is just opposite.
Stoyanov argued that at low temperatures xs »1and that therefore an ascending current
produces the instability. Increasing the temperature, xs decreases exponentially and may be-
come smaller than 1, so that the instability disappears. As 4l(1) is odd with respect to F (or(),
reversing the current induces again step bunching. This phenomenon was observed by various authors [1-5]. This has conferred at first sight some confidence to the mechanism put forward by Stoyanov. However, as stated in the introduction, several points remain questionable. The
determination of the diffusion length, on Si(Ill), which is based on the measure of the train
velocity, Vo er'~~ tanh(ilxs), the cross-over interstep distance above which Vo does not depend
on1is approximately xs), provided too high values (at least one order of magnitude greater
than 1) for temperatures r~J 1250 °C, le- in the experimental cross-over region. If such values
are correct, then according to the above theory, all observed vicinal surfaces should always be either unstable (step up current) or stable (step down current) with respect to step bunching.
0 40
0 30
/~
/ ~
Ù~
o zo
o io
ooo~~
~ ~~ ~~
ÎÎX
Fig 3. 4l versus é/~s for F > 0 Instability develops when 4l is an increasing function of é.
That is to say, there is at this point a contradiction between observation and the above theory.
Furthermore, some experiments [3] show no shift in the cross-over temperature for various mis- orientations (that is for various interstep distances). On the other hand, as the experimental evaluation of many physical parameters (e.g. F) is still challenging, it is at present not possible
to be conclusive. Proposing new tests to shed light on this unclear situation is highly desirable.
This will be the subject of the next section
Before concluding this section, we would hke to discuss the effect of the Schwoebel barrier
on step bunching and its comparison to the electromigration force. Although it is possible to
analyze our equation (14) in detail, we shall restrict ourselves to trie case where < xs,(.
This bas trie advantage to show trie relative effects clearly, and this will be largely sufficient for our purposes. For # = ~r, we bave
~" Î~ÎÎÎ~Î ~
~ )~~
~' ~~~~
One iees that trie Schwoebel barrier (r~J d- d+) dots as a renormalization of trie electromigra-
tion force (1If
r~J
F). Since d- > d+ (adatoms attach preferentially from trie lower terrace),
trie Schwoebel barrier reinforces trie step bunching instability. In principle even if F = 0 trie Schwoebel barrier may induce step-bunching. This case received much attention recently il Ii.
The current behef is that this effect is small as compared to the electromigration force -that
means that the Schwoebel effect would require a much longer time scale for the instability to
appear. We shall come back to this point later in this paper.
3. Trie alternating current case.
In this section we study the effect of an AC heating current where the force F cos(vt). For trie sake of simplicity we shall set d-
= d+. Now the diffusion equation (1) is non autonomous,
i-e- one coefficient depends on time. Therefore, in general one has to repeat the calculation
by means of a Floquet-Bloch analysis. However, this complication can legitimately be cir-
cumvented. Indeed, we can, in the expression of g+ (là), replace F by Fcos ut, or 1If by cos(vt) If. Consequently, the integration of this equation remains as simple as for the static
case. Let us outline the justification of this assumption. We can solve exactly the complete
time-dependent diffusion equation on a terrace, by means of Fourier transform. The evolution of trie kth component of trie concentration field, when F
= 0, reads
ck(t) = c~ + ék exp -(D~k~ + 1/T)t) (19)
where c[ is trie kth component of trie static solution. So, a fluctuation of trie static solution decays significantly after a time which is at most equal to T. When F # 0 the solution
is a little more comphcated, but, in the limit of a weak electric force (which seems to be
the case, see below), the above analysis can be used to leading order. At the cross-over
temperature (r~J 1250 °C) we have for Si(Ill),
T r~J ms and it decreases exponentially with
increasing temperature. Frequencies of interest being in trie range of 0-50 Hz, we con safely
continue to use trie quasi-static approximation; we neglect ôc/ôt in trie diffusion equation (1)
trie concentration field adapts itself instantaneottsiy to both the position of the step and the magnitude of the electric field.
Before going further, let us give an order magnitude of different length scales for Si(Ill).
The estimated value for ( is
r~J 100 ~tm I?i when xs < 1 ~m and 1,d+
r~J 100 À [iii. This permits us to hnearize the expression of ~(i) with respect to 1If. The evolution equation for the çi = ~r component of the fluctuation now reads
(« = uJ~ Cos(vt)(«, (20)
where uJ~ is the hnearized version of expression (17) with respect to 1If:
~~
2QDsc~q à l~S ~~~~ ~ l
~ ~~ Î ~~~~
~ Î ~~~~ s ~ ~° ~°~~ ÎÎ
The integration of this equation yields
(~ = (o exp ~~ sin ut) (22)
v
If v is small compared to uJ~, then for short times in comparison to 1/uJ~, sin ut r~J ut. That is, we obtain the same expression as in the static case; the instability con then take place (for
xs < 1) on a time scale t
r~J
1Iv. Since this time corresponds to the electric field period, one would be in a situation where step-bunching synchronizes with the field. For large enough frequencies (v » uJ~), however, the AC field would completely suppress the instability. Indeed, the instability requires a time
r~J 1/uJ~ to manifest itself, whereas the field direction oscillates
many times within this interval. That is to say, for before the adatom cooperatively responds
to the imposed field direction, this direction is reversed. In conclusion, in order to detect an
effect of the AC field one should select frequencies which are at most of the order of uJ~.
The estimated value of uJ~ for Si(1II), given below, shows that 50 Hz is too high so that it
suppresses step-bunching. For the following set of parameters at the cross-over temperature
1m d+ m 100 À,
xs m 10~ À, ( m 10~ À, Qc~q m 10~~, Ds m 10~~ À~/s we find uJ~/2~r m 1 Hz.
4. Long time evolution of trie vicinal surface.
The main fruit of the linear theory is the determination of the critical condition for the onset of the instability and the nature of those perturbations that are likely to grow first. If the long-
time behaviour of the instability is to be ascertained, then a nonhnear analysis is necessary.
