Kinematic Multiplication of Elementary Steps on the

ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2006, Vol. 103, No. 2, pp. 278–291. © Pleiades Publishing, Inc., 2006.Original Russian Text © A.Ya. Parshin, V.L. Tsymbalenko, 2006, published in Zhurnal Éksperimental’no

œ

i Teoretichesko

œ

Fiziki, 2006, Vol. 130, No. 2, pp. 319–334.

278

1. INTRODUCTION

As is well known, a faceted crystal grows due to thepropagation of elementary steps over faces, continuousgrowth being possible either in the presence of contin-uously operating sources of steps (such as screw dislo-cations and Frank–Read sources) or by means of newstep generation via two-dimensional nucleation. Thesteps can propagate either due to the surface diffusionor by the attachment of particles of another phase (liq-uid or vapor) occurring in contact with the growingcrystal. Under usual conditions, the velocity of step prop-agation is rather low, so that the kinetic energy of thismotion (being small compared to the rest energy) is fre-quently ignored in the analysis of step dynamics [1, 2].Evidently, in the other limiting case, the laws of conser-vation of the energy and momentum admit, for anappropriate law of dispersion, the decay of a propagat-ing step into two, three or more new steps. With allow-ance for the boundary conditions at infinity, we mayspeak of the generation of pairs of steps with oppositesigns, that is, of the generation of a new atomic layer bya rapidly propagating step.

According to the classical theory of crystal growth,the growing steps do not intersect. This condition

means that configurations possessing large excessenergy—such as overgrowth of one atomic layer overanother that corresponds to the intersection of steps ofthe same sign—are excluded from consideration. As forthe steps of opposite signs, this condition implies thatthe intersection (collision) of two such steps leads totheir annihilation in the region of contact, whichreduces the total length and, accordingly, decreases theenergy of steps. This property of steps must certainly beretained in the quasi-static case, whereby steps arepropagating at a sufficiently low velocity so that everypart of a step at any moment of time can be consideredas occurring in equilibrium, and the kinetic energy of astep can be neglected. The quasi-static approximationimplies that the relaxation time is small compared to the“collision time”

w

/

V

, where

V

is the relative velocity ofsteps and

w

is the characteristic step width (typically,on the order of interatomic distance). This condition isusually well satisfied, but it can be readily violated inthe case of a helium crystal surface at very low temper-atures, where the rate of energy dissipation tends tozero. Accordingly, the relaxation time exhibits infinitegrowth and, hence, the steps can propagate at very highvelocities, up to the velocity of sound [3]. It is naturalto suggest that, in this case, the colliding steps of oppo-

Kinematic Multiplication of Elementary Steps on the Surface of Helium Crystals

A. Ya. Parshin

a

and V. L. Tsymbalenko

b

a

Kapitza Institute for Physical Problems, Russian Academy of Sciences, Moscow, 119334 Russia

b

Institute of Superconductivity and Solid State Physics, Russian Research Centre Kurchatov Institute, Moscow, 123182 Russia

e-mail: [email protected]

Received March 14, 2006

Abstract

—The nonlinear dynamics of elementary steps on an atomically smooth crystal–liquid interface isconsidered within the framework of a weak coupling approximation. In fact, the proposed theory describesfaceting of the interface between solid and superfluid helium at low temperatures, when dissipative processesweakly influence the dynamics of growing steps. The main results are obtained by means of numerical calcu-lations. The law of dispersion of a step is determined and the dependence of its shape on the propagation veloc-ity is analyzed. It is established that a kinematic multiplication of steps is possible at a sufficiently high velocity,whereby a single step becomes unstable with respect to the formation of a pair of steps of opposite signs thatleads to the appearance of a new atomic layer. Under certain conditions, the collision of steps with oppositesigns may lead (in addition to the usual annihilation of steps) either to the step “transfer” to the adjacent rowwith the formation of a new atomic layer or to the reflection of steps from each other. Thus, a qualitatively newmechanism is proposed for the growth of crystals with atomically smooth faces in the absence of renewablesources such as screw dislocations. The effect of dissipation and external supersaturation on the dynamics ofsteps is considered. The possibility of experimental observation of the proposed mechanisms of step multipli-cation, as well as their probable relation to the unusual regimes of helium crystal growth at low temperatures,are discussed.

PACS numbers: 67.80.–s, 68.08.–p

DOI:

10.1134/S1063776106080115

ELECTRONIC PROPERTIES OF SOLIDS

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KINEMATIC MULTIPLICATION OF ELEMENTARY STEPS 279

site signs are not necessarily annihilated and can eitherpass “by inertia” one over another, thus forming a newatomic layer (see Fig. 1), or reflect from each other.

This paper theoretically demonstrates that, undercertain conditions, all the aforementioned nonclassicalprocesses can actually take place. As a result, thegrowth kinetics of crystals with atomically smoothfaces significantly deviates from predictions of the clas-sical theory. In the concluding section, we will discussthe experimental opportunities offered by this circum-stance. A brief outline of certain results was previouslypresented in [4].

2. MAIN EQUATIONS

In order to study dynamical growth processesinvolving elementary steps, it is necessary to develop aquite realistic model capable of describing displace-ments of a helium crystal surface on a microscopicscale. In formulating such a model, we must take intoaccount that the interface between superfluid and solidhelium features large quantum fluctuations [5], so thatthe correlation length and, hence, the effective stepwidth become large compared to the interatomic dis-tance [6]. Under such conditions, the most appropriatemodel for our purposes is that of the sine-Gordon typewith a continuous variable

ζ

(

r

) representing the fluctu-ation-average local displacement of the surface relativeto the equilibrium positions in an effective periodicpotential that describes the surface coupling to the crys-tal lattice [6–8]. The effective pinning potential provesto be small as compared to the surface energy, whichcorresponds to the weak coupling approximation that iswidely used (see, e.g., [9]) in the theory of phase tran-sitions related to the onset of crystal faceting (rough-ness transition). In addition, the presence of such a vari-able makes it possible to apply equations of macro-scopic hydrodynamics to the description of liquidmotions accompanying the crystallization and meltingprocesses.

The standard theory using the weak coupling approx-imation for the description of step dynamics [9, 10] doesnot take into account the kinetic energy of the system;that is, it restricts the consideration to quasi-stationaryprocesses. Making allowance for the kinetic energyexpands the field of applicability of the modified theoryup to the limits of validity of the concept of nondissipa-tive crystallization [5]. According to modern experi-mental data (see the recent review [11]), this domainextends at least up to velocities on the order of soundvelocity in liquid helium (

c

= 3.6

×

10

4

cm/s) and fre-quencies on the order of 10

11

Hz. As for the permissibletemperature range, this is limited only by the tempera-ture

T

R

of the roughening phase transition on the sur-face of a given orientation (e.g., for (0001) face of a

4

He crystal,

T

R

= 1.3 K). In fact, below we assumeeverywhere except Section 4.2 that the temperature issufficiently low to ignore the dissipation. In addition,

we restrict the consideration to step velocities below thevelocity of sound, so that the liquid (as well as the crys-tal) phase can be considered incompressible.

The Lagrange function for the system under consid-eration can be written as

(1)

where

a

is the interplanar distance (step height);

α

is thesurface tension (we neglect the anisotropy of

α

and,accordingly, do not differentiate between the surfaceenergy and stiffness);

ρ

l

and

v

l

are the density andvelocity of the liquid phase, respectively; and the factor

U

0

can be expressed in terms of the energy

β

per unitlength of the immobile step:

(2)

The last two terms in expression (1) describe the kineticenergy of the system. The first of these terms, repre-senting the surface contribution related to the rear-rangement of atoms in the surface layer on the passagefrom liquid to crystalline state [12], is relatively smalland does not qualitatively influence the results of anal-ysis. As for the second (volume) term, it can be trans-formed into a surface integral by assuming that theincompressible liquid fills the half-space

z

> 0 and by

Lα2--- ∇ζ( )2 U0 1 2πζ

a---------cos–⎝ ⎠

⎛ ⎞+ d2r∫–=

+ρ*2

------ ∂ζ∂t------⎝ ⎠

⎛ ⎞2

d2r∫ρl

2----v l

2 V ,d∫+

U0π2

16------ β2

αa2---------.=

R

Fig. 1.

Schematic diagrams illustrating the collisions ofsteps (a) with annihilation (a quasi-classical case) and(b) with the formation of a new layer (dynamic case). Adja-cent atomic layers are indicated by different cross-hatching.

(a)

(b)

280

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PARSHIN, TSYMBALENKO

taking into account the mass conservation in the courseof crystallization:

(3)

where

ρ

c

is the helium crystal density. Retaining onlythis term in the kinetic energy, we obtain the equationof motion for the crystal surface,

(4)

and the corresponding integrals of motion—the totalenergy

(5)

and the total momentum

(6)

The dimensionless coordinates

x

and

y

in Eqs. (4)–(6)are measured in units of

ξ

, and the dimensionless time,in units of

τ

:

(7)

For the (0001) face of the

4

He crystal, the values ofparameters are as follows:

α

= 0.25 erg/cm

2

[13–15],

β

/

a

= 0.014 erg/cm

2

[15], and

(8)

It should be noted that Eq. (4) was obtained using thecondition

|∇ζ|

�

1, or

|∇ϕ|

�

2

πξ

/

a

. Taking intoaccount the numerical values (8), we can conclude thatthis condition is well satisfied for the (0001) face, since

ρl

2----v l

2 Vd∫ρc ρl–( )2

4πρl----------------------=

× ∂ζ t r,( )∂t

------------------∂ζ t r ',( )∂t

--------------------d2rd2r 'r r '–

------------------,∫

12π------ ∂2ϕ t r ',( )

∂t2----------------------- d2r '

r r '–---------------∫ ∆ϕ– ϕsin+ 0,=

E1

4π------ ∂ϕ t r,( )

∂t-------------------∂ϕ t r ',( )

∂t--------------------d2rd2r '

r r '–------------------∫=

+12--- ∇ϕ( )2 1 ϕcos–+ d2r∫

P1

2π------ ∇ϕ t r,( )∂ϕ t r ',( )

∂t--------------------d2rd2r '

r r '–------------------.∫–=

ξ αa2

4π2U0

---------------,=

τρc ρl–( )2

ρl---------------------- a2

4π2U0

---------------ξ,=

ϕ 2πζa

---------.=

U0 1.5 10 3– α, ξ× 4a, τ 5 10 12– s,×≈ ≈ ≈

V0 ξ/τ 2.8 104 cm/s× c< 3.6 104 cm/s.×≈ ≈=

the characteristic values of |∇ϕ| in Eq. (4) are in thegeneral case on the order of unity. On the other hand,the condition of applicability of the weak couplingapproximation, which is expressed as a/ξ � 1, is satis-fied for this face with a small margin. Moreover, V0 isonly slightly below the velocity of sound in liquidhelium. For these reasons, the results presented belowoffer only a qualitatively correct description of theproperties of steps on the (0001) face. As for the otherfaces, the corresponding values of U0 and, hence, V0 aresignificantly lower [11], which ensures validity of allthe necessary conditions.

Real steps on the crystal surface are always slightlycurved, but the curvature is usually so small that thecurvature radius R is large compared to ξ. In this case,it is possible to ignore in Eq. (4) the dependence on oneof the coordinates (e.g., on y). The integral exhibits log-arithmic divergence and should be truncated either atthe face size (provided that it is smaller than R) or at R(expressed in ξ units, which eventually yields

(9)

3. METHODS OF NUMERICAL SOLUTION

The entire variety of numerical calculations can besubdivided into three types in terms of complexity.A simpler case was used to verify the correctness ofsolution obtained for a more complex variant. Thesetypes are as follows: (i) stationary motion of a singlestep (one-dimensional problem); (ii) evolution of anarbitrary profile (one-dimensional problem); and(iii) a symmetric two-dimensional (2D) problem.

3.1. Stationary Case

The general one-dimensional equation of motion (9)for the stationary propagation of a single step at a veloc-ity V can be reduced to an ordinary integro-differentialequation and written as

(10)

where ϕ = f(x – Vt), ϕ(–∞) = 0, ϕ(+∞) = ±2π. The pro-file f was determined by iterations using the targetmethod in the following sequence. The initial approxi-mation was chosen in the form of an immobile stepprofile

(11)

and used to calculate the convolution in the right-handpart of Eq. (10). Then, the equation was integrated

1π--- R

x ' x–---------------⎝ ⎠

⎛ ⎞ ∂2ϕ∂t2---------ln x 'd∫ ∂2ϕ

∂x2--------- ϕ.sin–=

d2 f

dx2-------- fsin–

V2

π------ R

x x '–---------------⎝ ⎠

⎛ ⎞ d2 f

x '2

d---------ln x 'd∫=

= V2

π------ fd

x 'd------- 1

x ' x–------------- x ',d∫

f 0 4 exarctan=

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 103 No. 2 2006


using the fourth-order Runge–Kutta method with anf '(0) slope at the initial point. The derivative was cho-sen so as to fit the solution at the end of the intervalx = 20 to the asymptotic behavior, which is described(according to Eq. (10)) by the following expression:

(12)

The obtained solution was substituted into the right-hand part of Eq. (10) and the procedure was repeated.For the velocities V < 1, the convergence is exponential,which implies that the slope at x = 0 depends on thenumber of iterations as is well approximated by theformula

(13)

where p ~ 1 for the velocities V ≤ 0.9. For somewhatgreater V, the rate of convergence decreases and theparameter p reaches ~5 for V ≈ 1. For still greater veloc-ities, we failed to find a stationary profile using thissimple method.

3.2. Evolution of a One-Dimensional Profile

Physically reasonable processes of collision, annihi-lation, and reflection of steps under arbitrary initialconditions and velocities lead to the surface profile of ageneral form, so that it is necessary to solve Eq. (9)directly. For solving problems of this type, we havedeveloped a finite difference scheme of integration. Inthe symmetric case, it is possible to implement a moreeffective Fourier transform and, in addition, to estimatethe accuracy of employed calculation algorithms.

In the general case, Eq. (9) can be converted usingthe Hilbert transform to the following form with anexplicit derivative with respect to time:

. (14)

Integrating by parts, we obtain a cumbersome expres-sion that is nevertheless more convenient for construct-ing an effective calculation scheme:

(15)

where a singular kernel in the integrand converges as1/x2. Since we assume that the profile is described by a

f x( ) π 1 x( )sgn+( )∼ 2V2

x---------.+

f n' f ∞' const np---–⎝ ⎠

⎛ ⎞ ,exp–=

∂2ϕ∂t2---------

1π--- ∂

∂x '------- ∂2ϕ

∂x '2--------- ϕsin–⎝ ⎠

⎛ ⎞ x 'dx ' x–-------------∫–=

∂2ϕ∂t2---------

1π--- ∂2ϕ

∂x2--------- ϕsin–⎝ ⎠

⎛ ⎞x '

∫–=

– ∂2ϕ∂x2--------- ϕsin–⎝ ⎠

⎛ ⎞x

dx '

x ' x–( )2--------------------,

continuous function possessing continuous derivativesat least up to the fourth order, the expression in theright-hand part of Eq. (15) (as well as in Eqs. (14) and(10)) is principal-value integrable. Replacing deriva-tives by second-order finite differences, we constructedan explicit scheme of solution with a temporal step ∆tand a spatial step ∆x. The general integration formulas(trapezoidal, etc.) without allowance for the kernel sin-gularity do not provide for the necessary accuracy ofconvolution calculations and lead to unstable solutions.For this reason, the integration was performed using analgorithm described in Appendix. The solution is stable(robust) provided that

(16)

which is analogous to the Courant condition for thehyperbolic equations. In most cases, the coordinate stepwas ∆x = 0.1 and the temporal step was ∆t = 0.02, whilethe number of nodes in the finite difference scheme gridwas varied between 1000 and 10000 depending on thesystem evolution scale. The solution accuracy was eval-uated on a grid with a spatial step of 0.05 and a tempo-ral step of 0.005. The differences did not exceed 1%.

In the case of collision of symmetric steps withequal and opposite velocities, we used effective Fouriertransform techniques. Equation (14) for the Fouriercomponents takes the following form:

(17)

where ϕk is the profile harmonic with a wavevector kand Fk(sinϕ) is a Fourier transform for the sine of theprofile. In these terms, the problem reduces to solving asystem of ordinary differential equations coupled by afunction in the right-hand part. At each time step, inte-gration of the system of equation was followed by aninverse Fourier transform that restored the step profile.Then, the Fourier transform of the sine of this functionwas used to construct the next iteration. The even char-acter of ϕ allowed using fast Fourier transform withrespect to cosine only.

As was noted above, the correctness and conver-gence of calculation methods was checked on test prob-lems. For the method described in this section, the testconsisted in calculating step profiles for a series ofvelocities and comparing the established profile withthe results obtained in a stationary case as described inSection 3.1 for the initial profile corresponding to animmobile step (11). Another criterion of accuracy wasprovided by a comparison of the scenarios of symmet-ric collisions of steps obtained by both methods. Theresults of such verification showed that a disadvantageof the proposed algorithm in application to an arbitraryprofile was an insufficiently high accuracy of convolu-

∆t2

∆x3--------- 1,<

d2ϕk

dt2----------- k 3ϕk+ k Fk ϕsin( ),–=

282



tion calculations, which led to smoothening of the stepprofile and a related increase in the critical values ofvelocities (see below). A comparison of the results ofcalculations using this scheme to those provided by themore accurate calculations of step profile, energy, andmomentum for a stationary step (Section 3.1) and thecalculations of symmetric step collisions gave an esti-mate of the solution accuracy. The maximum differencebetween the obtained values of parameters did notexceed 10%, which can be considered as the accuracyprovided by the given computational scheme.

An additional criterion of correctness for theobtained solution consisted in the verification of thelaw of energy conservation as

(18)

This check was made for several velocities and showedthat the energy remained constant with time to within5–10%, which provided an additional estimate for thesolution accuracy.

3.3. Two-Dimensional Case

Calculations of the propagation of steps in the 2Dcase require significantly greater computational facili-ties as compared to the one-dimensional case. This cir-cumstance restricted our possibilities to solving a sym-metric problem for colliding steps. The high symmetryof the problem made possible the use of an effectivemethod based on the Fourier transform analogous to

E1

2π------ R

x ' x–---------------⎝ ⎠

⎛ ⎞ ∂ϕ x( )∂t

--------------∂ϕ x '( )∂t

----------------ln x x 'dd∫=

+12--- ∂ϕ

∂x------⎝ ⎠

⎛ ⎞2

1 ϕcos–+ x.d∫

that described in the preceding section. In the generalcase, the equation of motion was as follows:

(19)

Applying the Fourier transform, we obtain

(20)

where fkm is the amplitude of a harmonic with thewavevector k. We obtained a system of M × N ordinarydifferential equations. Owing to the symmetry of theproblem, the expansion into series was performed forcosines only, which reduced the number of equationsby half. Specific features of the Fourier transform addi-tionally reduced this number by a factor of 4.

The solution algorithm was verified by solving aone-dimensional problem (one-dimensional step mov-ing in the 2D case, profile determination, symmetriccollisions). The obtained solutions showed good coin-cidence (to within the accuracy indicated above) bothwith respect to the evolution of step profiles and the val-ues of critical velocity.

4. MODELING THE MOTION OF A SINGLE STEP

4.1. Profile, Energy, and Momentum of a Single Step

A solution to Eq. (1) for V = 0 is a monotonic func-tion of x. As the velocity increases, the profile acquirestwo symmetric humps (Fig. 2) and decays at the infinityaccording to a power law (instead of the exponent)according to Eq. (12). Substituting this profile intoexpressions for the energy and momentum,

(21)

with the parameter M expressed as

(22)

we can determine the law of dispersion. The results ofthese calculations are presented in Fig. 3. At smallvelocities, the energy is a quadratic function of themomentum. As the velocity approaches unity, thisdependence becomes virtually linear (“relativistic”case). It should be noted that the parameter M is notequivalent to a “mass” per unit step length, since thisquantity also depends on the profile shape, that is, onthe velocity. We have separately considered the ques-tion as to how the energy and momentum values depend

12π------ ∂2ϕ

∂t2--------- d2r '

r r '–---------------∫ ∆ϕ ϕ.sin–=

d2 f km

dt2------------- k 3 f km+ k Fkm ϕsin( ),–=

EMV2

2-----------

12--- ∂ϕ

∂x------⎝ ⎠

⎛ ⎞2

1 ϕcos–+ x,d∫+=

P MV ,=

M1π--- R

x ' x–---------------⎝ ⎠

⎛ ⎞ ∂ϕ x '( )∂x '

----------------∂ϕ x( )∂x

--------------ln x x ',dd∫=

8

6

4

2

0

0.20.4

0.60.8

1.0

10V

x

0

86

42

Fig. 2. Profiles of single steps uniformly propagating at var-ious velocities. Vertical scale corresponds to an interplanarspacing of 2π.



on the radius of integration in Eqs. (21) and (22). It wasfound that the values of parameters exhibit a ratherinsignificant (logarithmic) dependence on the size ofthe integration domain. This can be illustrated by theratios of parameters obtained using the integrationdomains of radii 100 (a) and 1000 (b) for two veloci-ties: V = 0.2. Eb/Ea = 1.06, Pb/Pa = 1.44; V = 0.95.Eb/Ea = 1.24, Pb/Pa = 1.32.

4.2. Step Propagation under Conditionsof Supersaturation and Dissipation

In the general case, the surface motion is accompa-nied by the energy dissipation. At low temperatures, thedissipation is related predominantly to the interactionwith phonons. This interaction has an essentially non-local character and its description goes far beyond theframework of the model used in our study. For a quali-tative description of the dissipation phenomena, let usintroduce a term proportional to ∂ϕ/∂t into the generalequation of motion (4). In the presence of dissipation,the stationary propagation of a step must be supportedby a supersaturation in the system, which is equivalent toadding a constant term into the equation of motion. As aresult, we obtain the following equation instead of (9):

(23)

Generally speaking, friction modifies the stepshape. Since the parameters δ and γ introduced in thisway are small at low temperatures, the shape distortionswill be small. The results of numerical calculationsusing the scheme outlined in Section 3.1 showed that,for γ = 0.1, the maximum velocity, and the correspond-ing supersaturation, the change in the profile caused bythe dissipation did not exceed several percent.

The rate of variation of the total energy of the stepin the presence of supersaturation and dissipation is

(24)

In the stationary case, the friction losses are compen-sated by the supersaturation and we obtain the follow-ing relation for the velocity of step propagation:

(25)

As can be seen from this formula, the velocity is pro-portional to the supersaturation only in the region ofsmall values. As the velocity grows, a change in the stepshape leads to deviations from the linear dependence.The results of numerical calculations of the dependence

1π--- R

x x '–---------------⎝ ⎠

⎛ ⎞ ∂2ϕ∂t2---------ln x 'd∫

= ∂2ϕ∂x2--------- ϕsin– δ γ ∂ϕ

∂t------.–+

dEdt------- δ d

dt----- ϕ xd∫( ) γ ∂ϕ

∂t------⎝ ⎠

⎛ ⎞2

x.d∫–=

V2πδ

γ ∂ϕ∂x------⎝ ⎠

⎛ ⎞2

xd∫---------------------------.=

of the velocity on the supersaturation are presented inFig. 4.

The presence of supersaturation leads to accelera-tion of the step. Let us estimate the time required forattaining a critical velocity sufficient for the appearanceof a new layer. As can be seen from Fig. 3, for V ≤ Vcthe energy is approximately a quadratic function of thevelocity and we can use the relation

(26)

dEdt------- d

dt----- MV2

2----------- const+⎝ ⎠

⎛ ⎞ MVdVdt------- 2πδV ,=≈≈

V2πδM

---------t.=

20

0 20

E

P40 60 80 100

40

60

1.00

0.95

0.70

0.40

Fig. 3. A plot of the energy E versus momentum P of a singlestep P. Figures at the points indicate the step velocities.

0.2

0 0.02

V

δ0.04

0.4

0.6

0.8

1.0

0.06 0.08 0.10

Fig. 4. Plots of the velocity of stationary motion of a singlestep in the presence of dissipation and supersaturation.

284



According to this expression, the characteristic timeof acceleration for the experimentally reasonablesupersaturations of δ ≤ 0.1 is on the order of M/δ ~103−104.

4.3. Small Oscillations

The spectrum of small surface oscillations (rip-plons) in the absence of steps can be readily determinedby linearizing Eq. (4), which yields

(27)

The minimum phase velocity for this law of dispersion

is . This result implies that the motion of steps with

velocities not exceeding is not accompanied by theemission of ripplons.

The spectrum of small oscillations of the stepsdepends on the velocity of propagation. Within theframework of the model adopted, each oscillation ischaracterized by a wavevector q directed along the step(y axis) and by the number of the corresponding mode.Since the step profile is described by a function of thecontinuous variable x, there is, generally speaking, aninfinite number of modes. Among these, there is onegapless mode corresponding to bending oscillations ofthe step. The law of dispersion on this mode for small qand arbitrary V can be determined as described below.

Let us seek for a solution to Eq. (4) in the followingform:

(28)

where f is a small deviation from the stationary profile ϕ0.The linearized Eq. (4) can be written in terms of theFourier components as

(29)

For fk = ikϕ0k + ψk , where ψk 0 at q 0 (andω 0), we obtain (upon multiplying by kϕ0k andintegrating) the following relation:

(30)

ω2 k2 1+( ) k .=

2

2

ϕ ϕ0 x Vt–( ) f x Vt–( )e iωt– iqy+ ,+=

k2 q2 1 ω kV+( )2

k2 q2+------------------------–+ +

⎝ ⎠⎜ ⎟⎛ ⎞

f k Ik ' k– f k ' k ',d∫=

Ik1

2π------ 1 ϕ0 x( )cos–( )eikx x.d∫=

ω kV+( )2

k2 q2+------------------------ k V2– k2V2

k2 q2+--------------------–

⎝ ⎠⎜ ⎟⎛ ⎞

ϕ0k2 k2 kd

∞–

∞

∫ 0,=

which can be rewritten as

(31)

For a small velocity, we have

which yields (in a logarithmic approximation withrespect to q)

(32)

The question concerning a gap in the spectrum of “opti-cal” oscillations is briefly considered in the nextsection.

4.4. Stability of Steps

As can be seen from Eq. (31), the frequency of thebending oscillations remains substantially positive forall velocities. This implies that a moving step, as wellas a propagating one, is stable with respect to the onsetof bending oscillations (at least log-wavelength ones).On the other hand, a step can be unstable with respectto the oscillations of some other type involving achange in the step shape (including the case of q = 0).

The possibility of such instabilities is related to thefact that the law of dispersion presented in Fig. 3 has adecay character. It can be shown that a minimum veloc-ity at which the decay is possible corresponds to theseparation of a step into three components with equalmomenta and energies (the decay into two steps isimpossible for the given boundary conditions at theinfinity). This phenomenon provides a qualitativelynew mechanism of step multiplication either in thesame atomic layer or with the formation of a new layer,depending on which one of the three possible scenariosis realized (Fig. 5). A numerical estimate based on thespectrum of Fig. 3 gives a critical velocity of about0.77. The fact of attainment of this critical velocitylevel does not imply that the process would necessarilybegin. This condition only indicates that the state of apropagating isolated step becomes metastable. Decayof this state can be induced by any inhomogeneity, inparticular, by a local curvature of the step. A threshold-less multiplication of steps begins upon reaching a limitof absolute instability.

ω2

q2 kV2 k2V2

k2 q2+--------------------–+

⎝ ⎠⎜ ⎟⎛ ⎞

k2ϕ0k2 kd

0

∞

∫

k2

k2 q2+--------------------ϕ0k

2 kd

0

∞

∫---------------------------------------------------------------------------.=

ϕ0k1

ik πk/2( )cosh---------------------------------,=

ω2 2q2

π 1/q( )ln----------------------

q2V2

2-----------.+=



Let us consider the propagation of an isolated step ata constant velocity V in the case of a deviation from theequilibrium shape ϕ0:

(33)

Assuming the deviation to be small and neglecting theterms of higher orders, we obtain the following equa-tion of motion for this deviation:

(34)

Accomplishing the Fourier transform, we arrive at thefollowing dispersion equation:

(35)

The onset of absolute instability corresponds to the casein which the frequency is zero.

However, instead of solving this equation, it was asimpler approach to model the decay of a step bydirectly solving the two-dimensional equation ofmotion (4). The small initial inhomogeneity was intro-duced in the form of a slight bending of the step. Theresults of calculations for V = 0.95 are presented inFig. 6. It should be noted that a control calculation per-formed for a straight step propagating with the samevelocity did show the appearance of a new layer withinthe same period of time, which agrees with the abovestatement concerning the metastability of such steps.As can be seen from Fig. 6, the surface begins to rise inthe middle of the step (where the curvature radius ismaximum). As a result, the nucleus of a new layer isformed by the time t = 50 and keeps growing (t = 60).

Numerical modeling showed only one of the threepossible scenarios of step decay depicted in Fig. 5(namely, the scenario presented in Fig. 5b). Accordingto this variant, a new atomic layer is formed with thesame sign of the leading front as that of the initial step.The critical velocity for this process at a small initialperturbation is close to unity. As for the two other vari-ants of step decay, the question concerning the condi-tions for their realization has remained unstudied.

As was noted above, in solving the one-dimensionalproblem we failed to find a stationary step profile forV > 1. This velocity level probably corresponds to thepoint of absolute instability. An example of the evolu-tion of a one-dimensional profile for a step propagatingat a velocity of V = 1.2 is presented in Fig. 7, where thestep moving leftward generates a nucleus of the next

ϕ ϕ0 x Vt–( ) f x( )e iωt– .+=

1π--- R

x x '–---------------⎝ ⎠

⎛ ⎞ ω2 f– 2iωVf ' V2 f ''+ +( )ln x 'd∫–

– f '' ϕ0( ) fcos+ 0.=

f k 1 k2 ω2 2Vωk V2k2+ +k

--------------------------------------------–+⎝ ⎠⎛ ⎞ f k 'Ik ' k– k ',d∫=

Iq1

2π------ 1 ϕ0cos–( )eiqx x.d∫=

layer, which subsequently expands with time. It is inter-esting to note that a step with a double height keepsmoving in the initial direction. As can be seen fromFig. 6, this peculiarity is retained in the 2D case, at leastover a period of time used in the calculations. Elucida-tion of the question concerning stability of the doublestep requires additional investigation.

5. COLLIDING STEPS

In calculating the collisions of steps, we assumedthat the supersaturation and dissipation did not signifi-cantly influence the scenarios of collisions (i.e., δ and γvalues were assumed to be zero). This assumption wasjustified by the estimates presented in Section 4.2. Aswill be shown below, the collision time falls within10−100 units. During this period, the energy of the sys-tem cannot change more than by 1%, which is withinthe accuracy of our calculations. Therefore, we canconsider the process as the collision of steps moving byinertia.

5.1. One-Dimensional Case

In order to avoid time losses for establishing the sta-tionary shape of steps, we used the profiles calculatedfor a given velocity as described in Section 3.1 as theinitial approximation for the problems with collidingsteps.

The collision of steps propagating with equal andopposite velocities was played using two scenarios. Inthe region of small velocities (until reaching a criticalvalue), the steps always exhibited annihilation accom-panied by the emission of a packet of ripplons (Fig. 8).For the velocities above the critical level (Vc), wealways observed the formation of a new layer asdepicted in Fig. 9. This was also accompanied by the

(b)(c)

Fig. 5. Possible scenarios of step decay at a velocity abovethe critical value (V = 0.77): (a) initial step (arrow indicatesthe propagation direction, dashed line shows the level ofgrowing plane); the appearance of a nucleus in the newlayer behind the step front; (b) the formation of a nucleusahead of the step front; (c) decay of a single step into threesteps.

(a)

286



formation of ripplons, which propagated at a velocityexceeding the velocity of steps. As can be seen fromFig. 9, the velocity of steps changed rather insignifi-cantly, so that the energy of emitted ripplons was small

as compared to the energy of steps. The critical velocitydetermined using the Fourier method in a symmetriccase was about 0.4, which was somewhat lower than thevalue (≈0.45) given by the general method for an arbi-

0 40 80 120 160 200

Fig. 7. The decay of a step propagating at a velocity ofV = 1.2.

–60 –40 –20 0 20 6040x

Fig. 8. Symmetric collision of steps with annihilation.

0 20

4030

6050

Fog. 6. Diagrams illustrating the development of instability at a velocity (V = 0.95) admitting the decay of a step. The initial step isslightly bent. Figures indicate the current time.



trary step profile. To summarize these results, even thecollisions of steps propagating at velocities significantlylower than the sound velocity lead to the formation of anew atomic layer and the growth of a crystal face.

As was noted above, the collisions of steps witharbitrary initial velocities were calculated using thegeneral method. In this case, the scenarios of collision

processes become more diverse. In addition to the pro-cesses of annihilation and new layer formationdescribed in the preceding section, we observed themutual reflection of steps and the formation of a newlayer with retention of the pair of steps in the old layer(i.e., the formation of a new pair of steps). The formercase is illustrated in Fig. 10. In this process, the emis-sion of ripplons was insignificant, and the steps

–80 –40 –20 0 20 8040x

–60 60

Fig. 9. Symmetric collision of steps with formation of a newlayer and the emergence of ripplons. A small change in thestep velocity indicates that ripplons carry away a ratherinsignificant part of the energy.

0 40 80 12020 60 100x

Fig. 10. Mutual reflection of steps having different veloci-ties. As a result, the steps exchange their velocities andpropagate in opposite directions.

0 40 80 16020 60 100x

140120

Fig. 11. Collision of steps having different velocities. As aresult, a new layer is formed, the parent steps are retained inthe old layer, and the newborn steps move away from theparent couple.

–0.5

–1.0

1.0

1.0–0.5–1.0 0

0.5

0.5

Transmission

Reflection

Annihilation

Reflection

Transmission

V1

V2

Fig. 12. Schematic diagram summarizing the results of cal-culations for the step collisions in the one-dimensional case.The inner region correspond to the annihilation of collidingsteps. The regions of high velocities feature step transmis-sion with the formation of a new layer; outside theseregions, the steps exhibit mutual reflection.

288



exchanged energies without practically changing theirvelocities. The second scenario is realized at highervelocities and the pattern is more complicated, which isillustrated in Fig. 11. Here, the initial energy is redis-tributed between four steps. It is interesting to note that

the pair of steps in the new layer propagates in thedirection determined by the left initial step.

Figure 12 gives a summary of the scenarios of colli-sions for arbitrary velocity relations. Regions corre-sponding to different regimes are apparently symmetric

Fig. 13. Diagrams illustrating the collision of symmetric round steps propagating at V = 0.6. Right-hand images show the step pro-files in two mutually perpendicular planes passing through the center of symmetry.

40

12

200–20–40

6

0



relative to the bisectors of quadrants. This pattern pro-vides the possibility of an additional evaluation of theaccuracy of numerical calculations.

5.2. Two-Dimensional Case

Under real conditions, purely one-dimensional col-lisions of straight parallel steps considered above arelow probable. Steps generated by 2D nucleation,Frank–Read sources, and spiral growth are curved. Pre-viously, it was pointed out [4] that the collisions ofsteps with large radii of curvature must proceed accord-ing to a scenario close to that for the one-dimensionalcase. This conclusion is based on the fact that a pertur-bation arising at the intersection site does not accumu-late with time, since the point of intersection is movingat a velocity exceeding the velocity of perturbationtransfer along the steps (as long as the size of the inter-section region is small compared to the curvatureradius).

Verification of this statement and investigation ofthe collision of curved steps required the solution of a2D problem. We used a rectangular calculation regionwith a side length ratio of 1 : 2 for monitoring a singlecollision of two ring steps with the initial velocities setalong their radii. The spatial and temporal steps (0.20and 0.02, respectively) were chosen so as to ensure sta-bility of the algorithm (see relation (16)). In the firststage, we simulated the collisions of steps with smallradii (~5). It was established that the critical velocity inthis case increases to reach a level of about 0.85. Theinfluence of the step curvature on the critical velocityvalue was not specially studied. In the second stage, thecalculation region had dimensions 512 × 1024, the stepradius was set to about half of the smaller size (~50),and the calculations were performed for the velocities0.3, 0.4, and 0.6.

Figure 13 (left column) shows the evolution of thesurface upon the collision of steps propagating at avelocity of 0.6; the right column presents the mutuallyperpendicular cross sections by planes passing throughthe center of symmetry of the system (solid and dashedcurves refer to the profiles parallel to the short and longsides of the calculation region; the scales of all curvesare identical, as indicated in the bottom right plot). Thesolid profile, which is perpendicular to the steps at thepoint of contact, is analogous to the profile obtained ina one-dimensional case. As can be seen, the temporalevolution of this profile is actually analogous to theresults obtained on solving the one-dimensional prob-lem (see Fig. 9), in agreement with what was suggestedin [4]. The results of numerical calculations revealed afine structure of the surface upon collision, which couldnot be determined proceeding from qualitative consid-erations. Indeed, the velocity of perturbation transferalong the step in the initial stage was smaller (as indi-cated in [4]) than the velocity of the geometric point ofintersection. According to the results of calculations,this leads to a retardation of this region and the forma-

tion of the next layer, as manifested by two symmetricpeaks near the nucleus of the new layer. The angle pointon the intersection line is rectified rather slowly, andthese peaks are retained until the end of the calculation.A detailed plot shows that waves are spreading from theangle point both along the step and over the entire sur-face. Once formed, the nucleus of the new layer keepsincreasing in size and changes in shape from oval tospheroidal.

The results of calculations performed for a velocityof 0.4 showed that the central nucleus in this case is notformed, but two symmetric long-lived peaks that reachthe level of the next layer appear at the point of inter-section. During the time of calculations, these peaksmoved away from the center of the system, neitherchanging in size nor decaying, showing evidence for arather long lifetime. At a velocity of 0.3, these peaks didnot reach the next layer and decayed to vanish withtime.

Thus, we can ascertain that the collision of curvedsteps leads to the nucleation of a new layer, in agree-ment with what was declared previously [4]. New qual-itative evidence is that the steps of smaller curvatureradius must possess a higher velocity for the formationof a new crystalline layer. The appearance of peaks onthe surface at the point of intersection is probably alsoa general property of collisions. The results of simula-tions revealed a large variety of scenarios of the colli-sions of steps propagating over a crystal surface. Forexample, calculations performed for the collisions of

1

0 0.1

δp, mbar

T, K0.2

2

Fig. 14. A comparison of the experimental data for the“burstlike” growth [3] to the results of calculations usingformula (26). Circles correspond to supersaturations atwhich this growth takes place. The curve shows δp values atwhich the stationary velocity of steps reaches a critical levelof about 0.4. As can be seen, small supersaturations at tem-peratures below 0.15 are sufficient to accelerate steps up tovelocities admitting their kinematic multiplication.

290



steps having sinusoidal shapes showed that a collapseof the isolated region gives rise to a peak even at arather small velocity. Therefore, it is not excluded thatsome geometric features of the steps can both increaseand decrease in the critical velocity.

6. CONCLUSIONS

The results of numerical simulation of the propaga-tion of elementary steps on the surface of helium crys-tals and the collisions of such steps showed that, at asufficiently high velocity, nuclei of a new atomic layercan be generated both by a single propagating step anddue to the collision of steps with opposite signs involv-ing a “transfer” to the adjacent layer; alternatively, thecolliding steps can exhibit mutual reflection from eachother. Thus, a qualitatively new mechanism is proposedfor the growth of crystals, which is based on the phe-nomenon of kinematic multiplication of elementarysteps.

As was pointed out in the Introduction, the assump-tion that the growing steps do not intersect underlies theclassical theory of growth of the atomically smoothcrystal faces, in particular, the theory of spiral growth.One of the main features of this growth mechanism—anoncumulative character whereby the growth rate isindependent of the number of growth dislocations—isa direct consequence of the absence of such intersec-tions [2]. With allowance for the phenomena of kine-matic multiplication of steps, the dependence of thespiral growth rate on the number of dislocations can beexpected. Naturally, the dependence of the growth rateon the degree of supersaturation must change as well.Thus, the existing theoretical interpretation of theexperimental data on the spiral growth of helium crys-tals [3] may need serious correction.

As was pointed out above, the conditions of applica-bility of the theory proposed in this study are well sat-isfied for all faces of 4He crystals except for (0001),where they are valid with a small margin. Therefore,predictions of the proposed theory in this case may notbe valid on a strict quantitative level. However, virtuallyall the available experimental data with a few excep-tions refer to this very face. From this standpoint, ofmost interest are the experimental investigations intoqualitatively new phenomena in which the predictedeffects can be manifested.

The kinematic multiplication of steps probablyplays an important role in the phenomenon of “burst-like” growth on a dislocation-free (0001) face of 4Hecrystals. Figure 14 shows experimental data [3] on theaverage values of supersaturation for which the (0001)face exhibited a transition to the fast growth. The solidcurve presents the results of calculations of a criticalvelocity for the step mobility (friction) taken fromexperimental results of the same study. As can be seen,at temperatures up to 0.15 K, the supersaturations aresufficient for accelerating the steps to a critical velocity

at which their kinematic multiplication begins. Athigher temperatures, the situation is reversed. However,one should not expect the exact coincidence of thisapproximate estimate with experiment: the process ofburstlike growth has a very complicated character,which can hardly be explained by a single factor suchas the appearance of a qualitatively growth mechanismon a dislocation-free crystal face. This is clear if wetake into account that, for example, a transition to thefast growth takes place after a rather long action ofsupersaturation on the crystal face, that is, depends onthe sample prehistory. However, since the observedsupersaturations accelerate the steps to nearly criticalvelocities, the kinematic multiplication of steps is evi-dently a significant factor of the “burstlike” growth.

Another interesting phenomenon is the observationof anomalously high growth rates for crystal faces attemperatures within 0.2–0.8 K on a freely growing 4Hecrystal (see short review [16]). This effect can also berelated to the kinematic multiplication of steps. Thewhole body of experimental data available at presentsuggests with a high probability that the anomalousgrowth as well as the burstlike growth have a commonphysical origin. Therefore, the above considerationsmay equally refer to the anomalous growth. Low valuesof the supersaturation, which are definitely insufficientfor the breakage of faceting boundaries and for the 2Dnucleation, can nevertheless lead to the transformationof faces and to an increase in their growth rate by twoto three orders of magnitude. At least no mechanismother that the kinematic multiplication of steps isknown that can so substantially change the surfacekinetics at such small deviations from the equilibriumstate.

Finally, it should be also noted that recently Abeet al. [17] showed that the kinematic multiplication ofsteps during their collisions well explains the observedfast growth of the (0001) faces on helium crystals underthe action of a high-intensity acoustic wave.

ACKNOWLEDGMENTS

The authors are grateful to A.F. Andreev, G.E. Volo-vik, R. Jochemsen, and V.I. Marchenko for fruitful dis-cussions and useful advice.

This study was supported in part by the Russian Foun-dation for Basic Research, project no. 05-02-16806-a,and Russian President’s program supporting the leadingresearch schools (no. NSh-5763.2006.2).

APPENDIX

An integral of the type

(A.1)

exhibits principal-value convergence, provided that the

If x( ) f 0( )–

x2---------------------------- xd∫=



function f is doubly continuously differentiable. Weseparate the singularity in the integrand as

(A.2)

and formulate a numerical scheme assuming that thefunction f is well approximated by a second-order poly-nomial on a segment of sufficiently small length 2h.This approximation allows the singularity (second termin (A.2) to be integrated. The x axis is divided into seg-ments of 2h length and the f function on each segmentis approximated by the polynomial

(A.3)

where the coefficients ai on each segment are deter-mined from the condition of equality of the values of fand the polynomial at the ends and in the middle ofeach segment. Substituting polynomials (A.3) intoexpression (A.2), explicitly calculating the integrals,and collecting terms, we obtain the desired numericalscheme:

(A.4)

With increasing the number n, the coefficients decreaseas 1/n.

REFERENCES

1. W. K. Burton, N. Cabrera, and F. C. Frank, Philos. Trans.R. Soc. London, Ser. A 243, 299 (1951).

2. A. A. Chernov, in Modern Crystallography, Vol. 3: Crys-tal Growth, Ed. by B. K. Vaœnshtein, A. A. Chernov, andL. A. Shuvalov (Nauka, Moscow, 1980; Springer, Berlin,1984).

3. J. P. Ruutu, P. J. Hakonen, A. V. Babkin, et al., J. LowTemp. Phys. 112, 117 (1998).

4. A. Ya. Parshin and V. L. Tsymbalenko, Pis’ma Zh. Éksp.Teor. Fiz. 77, 372 (2003) [JETP Lett. 77, 321 (2003)].

5. A. F. Andreev and A. Ya. Parshin, Zh. Éksp. Teor. Fiz. 75,1511 (1978) [Sov. Phys. JETP 48, 763 (1978)].

6. S. V. Iordanskiœ and S. E. Korshunov, Zh. Éksp. Teor. Fiz.87, 927 (1984) [Sov. Phys. JETP 60, 528 (1984)].

7. E. Fradkin, Phys. Rev. B 28, 5338 (1983).

8. D. S. Fisher and J. D. Weeks, Phys. Rev. Lett. 50, 1077(1983).

9. P. Nozières, Shape and Growth of Crystals, in Solids Farfrom Equilibrium, Ed. by C. Godrèche (Cambridge Univ.Press, Cambridge, 1992).

10. P. Nozières and F. Gallet, J. Phys. (Paris) 48, 353 (1987).

11. S. Balibar, H. Alles, and A. Ya. Parshin, Rev. Mod. Phys.77, 317 (2005).

12. L. Puech and B. Castaing, J. Phys. Lett. 43, L601 (1982).

13. A. V. Babkin, D. B. Kopeliovich, and P. Ya. Parshin, Zh.Éksp. Teor. Fiz. 89, 2288 (1985) [Sov. Phys. JETP 62,1322 (1985)].

14. O. A. Andreeva and K. O. Keshishev, Pis’ma Zh. Éksp.Teor. Fiz. 46, 160 (1987) [JETP Lett. 46, 200 (1987)].

15. E. Rolley, C. Guthmann, E. Chevalier, and S. Balibar,J. Low Temp. Phys. 99, 851 (1995).

16. V. L. Tsymbalenko, J. Low Temp. Phys. 138, 795 (2005).

17. H. Abe, Y. Saitoh, T. Ueda, et al., J. Phys. Soc. Jpn. 75,023 601 (2006).

Translated by P. Pozdeev

If x( ) f 0( )–

x2---------------------------- xd

h

∞

∫ f x( ) f 0( )–

x2---------------------------- xd

h–

h

∫+=

+f x( ) f 0( )–

x2---------------------------- xd

∞–

h

∫

p x( ) a0 a1x a2x2,+ +=

f n f nh( ),=

I1h--- c j f j, c j–

j ∞–=

∞

∑≈ c j,=

c0 4, c1– 454--- 3,ln–= =

c2k 4 k2k 1+2k 1–---------------⎝ ⎠

⎛ ⎞ln 1– ,=

c2k 1+ 412--- 4k 1–( ) 2k 1+

2k 1–---------------⎝ ⎠

⎛ ⎞ln–=

+ 4k 5+( ) 2k 3+2k 1+---------------⎝ ⎠

⎛ ⎞ , k 1 …,= .ln

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Kinematic Multiplication of Elementary Steps on the