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Incommensurate Crystals at T = 0
Chapter 6
Incommensurate Crystals at Γ = 0
The subject of this chapter is the ground state of incommensurate crystals in the classical approximation. It is convenient to introduce the problem with one-dimensional models. The presentation will be organized as follows: the first three sections are concerned with the ground state and the excitation spectrum in such models; the effects of discreteness of the incommensurate crystal lattice on its translational properties (pinning of the soliton lattice) are considered in the subsequent two sections; then in Sections 6.6-6.9 we proceed to describe the ground state of the two-dimensional incommensurate crystal—orientational epitaxy and the soliton lattice.
6.1. Resonance Approximation The simplest one-dimensional model of a crystal in a periodic potential was proposed by Frenkel and Kontorova (1938) to describe the structure of dislocations. In this model one considers a one-dimensional chain of springcoupled particles placed in a periodic potential relief with period b (Fig. 6.1). 193
194
INCOMMENSURATE CRYSTALS AT
Γ=0
a
b
FIGURE 6.1.
A chain of particles coupled by springs in a periodic potential.
The potential energy U of such a system is
U = Σ [|(*n+1 " *n - af + K(x„)],
(6.1)
where a is the period of the free chain, xn is the coordinate of the nth particle, and λ is the stiffness of the spring. Note that due to the different dimension ality, the definitions of λ and V are different from those of Chapters 1 and 4. Let the periods a and b be close: a = b + δ,
6«b.
(6.2)
We shall assume that V is small: 2
|K(x)|«Afc .
(6.3)
Let us introduce the variable un = x„-
na.
(6.4)
Then minimization of (6.1) with respect to un results in the equation V\dn + un) un+x
K
- 2un + «„_! = — —-—"A
(6.5)
In view of the smallness of δ, the quantity ν\δη + un) is a slowly varying function of n. As follows from the weak-potential condition (6.3) and
195
RESONANCE APPROXIMATION
Equation (6.5), un is a slowly varying function of η as well. Therefore, the difference equation (6.5) can be approximated by a differential equation. This approximation. approximation will be called the resonance It is convenient to introduce a new variable φ(η) = δη + uw
(6.6)
which is the displacement of the nth particle from its position in the free chain in the absence of the potential. In terms of this variable the potential energy (6.1) takes the form
υ
=
Μ ί '
)
δ
+
η
φ
λ
ά
η
·
( 6
·
7 )
The resonance approximation can be also used at a = (M/N)b + <5, where Μ and Ν are integers and δ « b. Consider the particular case of a = (b/2) + δ. For simplicity, assume that Υ(φ) = — V cos qφ9 where q = 2n/b. Then the energy (6.1) can be written as
- Kcos qx„ + Kcos # „ >,
(6.8)
where χ„ = u2n + 2n<5, ψη = M 2 „ + I 4- (2n + 1)<5. First consider a commensurate phase in which φ and χ are n-independent. Then the energy (6.8) takes the form 2
2
U = ?j- Ιλ(φ - χ) + λδ - Vcos qx + Kcos # ] .
(6.9)
Minimization of (6.9) gives equations V X
" ^
=
~ 2λ
Q S in qX
'
m Sί Ζ
=
Sm
The solutions of these equations are of the form V
π X
=
2^-u °
qC Sm7t
mn +
T'
^
β Λ )ϋ
196
INCOMMENSURATE CRYSTALS AT Γ = 0
where m is an integer. This shows that if the period of the potential relief is double the period of the crystal, then atoms are trapped at places where the absolute value of the potential derivative V\x) is maximal, rather than at bottoms of wells. In the incommensurate phase φ and χ will be slowly varying functions of n. It is convenient to introduce new variables φη = (χ„ + φη)/2 and 2 ω„ = χ„ — φη> Then the formula (6.8) for I/, up to the K -order terms, in the new variables takes the form υ
= J
2δ
)
+
Ω
+ (
-
δ
+
vqVsmq^dn.
+
(6.12)
Minimization of ( 6 . 1 2 ) in ω yields I do ω
=
V
- 2 Λ Γ - 2 ΐ
9
8
ί
η
·
#
)
Substituting ( 6 . 1 3 ) in (6.12), we obtain after some transformations 0
-Μϊ- ) Μ
,+
Τ3Ρ"" ·*]*· 1
· Ι , 46)
In the more general case of a = (M/N)b + δ, similar calculations result in
U=
A
-
hj + ν^^Νφ^άη,
(6.15)
2
l
where A is a certain constant, hoc δ, and VN oc V(Vlkb f~ . Thus, for any degree of incommensurability the potential energy can be transformed to the universal form (6.7). The presented results were obtained in papers by Pokrovsky and Talapov ( 1 9 7 8 ) and Teodorou and Rice ( 1 9 7 8 ) .
6.2. Soliton Lattice A qualitative picture of the incommensurate phase and the simplest es timations of the soliton parameters were given in Section 1.2. Let us consider this problem in more detail. Different terms in the energy (6.7) correspond to opposite tendencies in the arrangement of the particles. On the one hand, particles tend to occupy the minima of the potential relief ν(φ). On the other hand, they tend to form a lattice with period a different from the period b of
( 6 1 3
197
SOLITON L A T T I C E
the potential. As a result of the tradeoff, at a certain critical value <5C the purely commensurate phase becomes unstable, and domains with broken in commensurability, or solitons, will form. The value of <5C can be found from the equilibrium equation for φη, which, in its turn, can be easily derived from (6.7): 2
, ά φ _ dV(4>) « -2 ^ · dn " άφ
(6.16)
This equation has the first integral
ffl-
λ 2
\ J - I
<*+ν(φ).
(6.17)
The equations (6.16) and (6.17) have a simple mechanical analogy. The equation (6.16) describes the deflection φ of a pendulum in the field ν(φ), η being the time. The equation (6.17) is the energy conservation law. The case of S > — Vmin, when the pendulum energy exceeds the potential energy at the top point of the trajectory, corresponds to the incommensurate phase. When δ = Sc = — Vmin, the pendulum stops at the top point. It leaves this point after an infinite time, then passes through the bottom point after a finite time and climbs again to the top after an infinite time. This trajectory corresponds to the formation of one soliton in the particle chain. Namely, at a very large distance from the soliton core all the particles occupy the minima of the potential Υ(φ). Let 0 0 as η — oo. Then at large distances from the soliton core φ is exponentially small. In the core φ = b/2 and φ' = άφ/dn « y/V/λ. On the right of the soliton core, as η - > + oo, the deviation of φ from b is exponentially small too (see Figs. 1.5, and 6.2). The soliton energy Es is the difference between the total energy (6.7) and the energy Uc of the com mensurate phase: t>c
=N |^+K
Ea =
0
j^2[V(4>)
m i n
J,
(6.18)
+ / J - δλ^άη
0)
= £< - Xdb,
(6.19) 0)
Here J V 0 is the number of particles in the chain, Sc = — VmiB, and £ < is the soliton energy at δ = 0. As follows from (6.19), the critical value δΒ, at which soliton creation becomes energetically favorable, equals <5C =
E^/Xb.
(6.20)
198
INCOMMENSURATE C R Y S T A L S AT 7 * = 0
Rewriting (6.19) in terms of the variable φ, using (6.17), we obtain
^
0)
= £ V 2 A [ ^ ) - K m i n] # .
(6.21)
In the case of Υ(φ) = - Kcos(27r/i# we find £ < = (4b/n)y/Xv. Solving Eq. (6.17) with S =
ν(φ):
do
, oc e x p ( - \ n - η0\^/φ)9
where κ =
ά ν(φ)/άφ 2
2
at the minimum of
Υ(φ).
(6.22)
199
SOLITON L A T T I C E
The formation of solitons implies a transition from the commensurate phase to the incommensurate one ( C — I transition). Such a transition should occur in the vicinity of any rational value of a/b = M/N. Hence, with change of <5, the commensurate structures must replace each other with arbitrarily high frequency. Hence, a situation of the "devil's staircase" type (Section 5.1) occurs. As shown in the previous section, the amplitude of the effective N potential is proportional to V , so that intervals of dc correspond to a given NI2 commensurate phase, and <5C oc V . Obviously, thermal fluctuations will smear out the fine structure of transitions. Thus, the formation of solitons becomes energetically favorable at δ > <5C. number of solitons produced at δ > <$c will be controlled by the The interaction between them. This results in the formation of a periodic lattice with the period determined by
da
(6.23)
y2[
The
energy of the soliton lattice per unit length can be easily found from
and (6.17):
U ΤΓ~
^2^+ν(φ)2άφ-δ^
=
λδ
Τι
Ν0
-
(6.24)
2
© + —τ—.
άφ
(6.7)
2
(Ο.
.24)
Minimizing with respect to the parameter S, we obtain δ as a function of $ and consequently, by virtue of φ as a function of δ. For instance,
(6.17),
δ
= Γ y/2[*
+ V(4>)]d.
(6.25)
Near the C - I transition δ — <5C has the following dependence on S:
where α is a certain numerical constant. The energy of the incommensurate and phase can be derived from
(6.24)
(6.25):
l/ = No|^--^+^<5 J. 2
(6.27)
200
INCOMMENSURATE CRYSTALS AT Γ = 0
At S = SG = - Vmin, the energy (6.27) coincides with the energy (6.18) of the commensurate phase. In the incommensurate phase 8 > &C9 and con sequently, at the same value of δ the energy of the incommensurate phase is always lower than that of the commensurate phase. For the potential of the form V() = - Κα>8(2π/ί>)φ the solution describ ing the incommensurate phase was first obtained by Frank and van der Merwe (1949a, b). Substituting this potential into (6.17), we find
where amn denotes the Jacobi amplitude, and k = y/2V/(V+ S) is the modulus of the elliptic function (Akhiezer, 1970). If Vmin= — V, then k = 1 at S = Sc = Κ the degenerate elliptic function coincides with the hyperbolic function, and the solution (6.28) describes an isolated soliton and coincides with (1.4). The relationship between δ and k is given by equation
(6.29)
where E(k) is the complete elliptic integral of the second kind. A plot of φ(η) for the soliton lattice is shown in Fig. 6.2. In the vicinity of the C - I transition the spacing / of solitons increases as
(6.30)
The corresponding formula for the energy, U
—
=
- 6bX + ^ q ^ e x p t - l / y i d ) j
0)
λδ
2
4 +^-,
(6.31)
has a simple physical meaning. The term 2s£ — 6bX is the energy gain for the formation of one soliton, whereas the exponential term describes the energy increase due to the exponentially weak repulsion of solitons. Minimization of (6.31) with respect to I leads readily to the formula (6.30) for the spacing of the soliton lattice.
201
SPECTRUM OF S M A L L OSCILLATIONS
6 3 . Spectrum of Small Oscillations As mentioned in Chapter 1, a principal feature of the incommensurate phase is a group of continuous transformations—translations of the soliton lattice leaving the energy unchanged. Indeed, the exact solution of the equation (6.17) can be written in the form (6.32) This solution contains an arbitrary parameter n 0, on which the energy of the incommensurate phase does not depend. Invariance under the group of continuous translations implies the existence of an acoustical branch (the Goldstone mode) in the spectrum of small oscillations in the incommensurate phase. In order to find a spectrum of the small oscillations it is necessary to add the kinetic term to the energy (6.7). The total energy is (6.33)
Η =
where m is the particle mass. Let φ(η, ί) be represented as φ0(Η) + Ψ(η, T), where φ0(Η) satisfies Eq. (6.16) and Ψ(Η, T) is small. In the linear approximation Ψ(π, T) satisfies the equation Δ
2
Ψ
m — 2^ — DT
Δ
2
Λ——*-2 DN
ά ν(φ ) Ψ—ΤΡΓάφ% 2
Ψ
+
0
= 0·
(6.34)
Evidently, the dispersion law in the commensurate phase will be of the form (6.35) where Ω is the frequency, Q is the wave vector, and Κ is defined in (6.22). In this case the spectrum has a gap Ω 0 = Y/Κ/ΙΗ. In the incommensurate phase the equation for oscillations of frequency Ω reads as (6.36)
202
INCOMMENSURATE CRYSTALS AT Γ = 0
2
Here we have replaced ά ν/άφο with λφ%/φ'0 using (6.16) (a prime denotes differentiation with respect to n). The equation (6.36) has the form of the Schrodinger equation with a periodic potential. The existence of a zero-frequency oscillation mode follows from the equation (6.36) at once. The corresponding solution is Ψ(Ω = 0, η) = φ'0, describing the translation of the soliton lattice. To simplify the notation we introduce the function 1,2
ι 2
τ = ΦΌ = {2[
(6.37)
The zero-frequency mode has zero quasimomentum, q = 0. For small q we search for a solution of the form Ψ = τ β χ ρ ^ Jjx^,
(6.38)
where χ is a periodic function of η with period / satisfying the condition (6.39) This constraint follows from the Flouqet-Bloch theorem on the properties of the solutions of equations with periodic coefficients: Ψ(η + Ζ) = βχρ(^Ζ)Ψ(η),
(6.40)
and from the periodicity properties of τ. Assume that at small q the dispersion law is of the form Ω = cq. Then the equation for χ resulting from (6.36) is 2
iq(x'z + 2χτ') + q (^j-
2
- χ ) τ = 0.
(6.41)
Now, multiplying (6.41) by τ, integrating between zero and /, and using the periodicity of χ and τ, we obtain
ioW"* )' ^ ' 2
2
0
(642)
This equation permits determining the sound velocity in the small-q limit. As
203
SPECTRUM OF S M A L L OSCILLATIONS
2
follows from (6.41), in the small-g limit χ = η/τ , where η is given by the equation
C'dn
(6.43)
Jo τ
Employing dn = άφ/τ, which follows from the definition τ(φ) = φ', from (6.42) and (6.43) we obtain
J^\
fil^
U2
C
(
6
4
4
)
In the incommensurate phase far away from the C - I transition the value of c is close to y/X/m. On the other hand, near the C - I transition,
c oc Near the C - I transition the spacing of solitons is large and the soliton lattice can be treated as a chain of interacting particles, the solitons themselves being regarded as particles. The frequency of oscillations in such a chain is determined by the standard formula Ω = Asm j .
(6.46)
The constant A can be easily determined from the small-*? limit when Ω = cq. Therefore, ^
2c . ql
T
s l n
T
( 6
·
4 7 )
The equation (6.47) describes the entire acoustic branch of the spectrum. Evidently, it is valid only near the C - I transition. The maximum frequency of the acoustic excitations, Ω ^ = 2c//, vanishes for δ - • <5C as
Ω™*(<5-<υ
1 / 2
m
[ΐη^-] •
These results were obtained by Pokrovsky and Talapov (1978).
(6.48)
204
INCOMMENSURATE CRYSTALS AT 1 = 0
6.4. Lattice Discreteness Effects The following two sections are mostly of a mathematical character. Readers not interested in the mathematical formalism can skip them and proceed to the end of Section 6.5, where a qualitative picture of pinning is presented. The discreteness of the lattice has a strong influence on the properties of the incommensurate lattice. One of them has already been mentioned: the infinite number of commensurate and incommensurate phases (the "devil's staircase"). Another such phenomenon—the pinning of solitons—is the subject of this section. We shall use the Frenkel-Kontorova model (see Section 6.1). Taking 2 λφ/2π) ν(\ — cos φ) for the potential, with fybjln instead of φ in (6.6), we rewrite the second-order finite-difference equation (6.5) as a system of two first-order equations: ω η+ ι =con
+ vsm((un + φη\
Φη+l π + Φη· =
ω
(6.49)
The second of these equations can be regarded as a definition of ωη. The system (6.49) can be treated as a mapping of the two-dimensional vector space (ωη9φ„) onto another two-dimensional vector space ( ω π + ,1 φ „ + 1) . Suppressing the subscripts, we rewrite (6.49) once more: ω' = ω + ν ύη(ω + φ\ φ' = ω + φ.
(6.50)
This formulation of the problem was suggested by Aubry (1979). In our presentation we shall follow the method developed by Pokrovsky (1981). Mappings of the type (6.50) were studied by mathematicians long ago and are known as canonical mappings. The above formulation of the problem makes it similar to those of the nonlinear dynamics. The basic theorems were derived by Poincare, Birkhoff, Kolmogorov, Arnold, and Moser (for a review see the book by Arnold and Avez (1968)). The principal ideas will be described here in a somewhat heuristic form without rigorous proof, to make the mathematical formalism more accessible to nonexperts. The reader can find more details, e.g., in the book by Lichtenberg and Liberman (1983). The mapping (6.50) is continuous and analytic, and its Jacobian equals unity, so that it is an area-preserving mapping having an inverse trans formation with the same properties. We shall call a curve Γ an invariant curve of the mapping Τ if any point of this curve (ω, φ) is mapped onto
205
L A T T I C E DISCRETENESS E F F E C T S
another point (ω', φ') belonging to the same curve Γ. If ω = (ο(φ) is an equation of the invariant curve, then by the definition (6.50) of the mapping Τ we can write ω'(ω(φ%
φ) = ω(φ'(ω(Φ),
Φ)1
(6.51)
Substituting the functional form of ω ' , φ' from (6.50) into (6.51), we obtain the equation for the function ω(φ): ω(φ
+ ω(φ))
= ω(φ)
+ νύη\_φ
+ α#)].
(6.52)
It is worthy to note that the invariant curve is not uniquely defined. Indeed, suppose a point (ω 0, φ0) is mapped into the point ( ω ΐ 9 φ ι ) . If we draw an arbitrary continuous curve joining these two points, it will be mapped under Τ into a continuous curve joining (ωϊ9φ1) and its map (ω 2,φ 2)· By virtue of the continuity of the mapping, the second curve can be continued in both directions. According to the method of its construction this curve is an invariant one. With some effort one can provide for the continuity of the derivatives at the junction points. Let us search for the infinitely differentiable solutions of Eq. (6.52) in the form of power series (v is supposed to be small): ω
= Σ
k
(6.53)
cokv .
k= 0 For the zero-order function ω 0 we obtain the equation ω 0 ( φ + ω 0) - ω 0 ( φ ) = 0,
(6.54)
with the obvious solution ω 0 = constant. The first-order term satisfies the equation ω χ ( φ + ω 0) - ω ^ φ ) = sin(<£ + ω 0) ,
(6.55)
which has the solution
ο>ι(Φ)
=
\
^sin y - ^
cos( + £ω 0).
(6.56)
The t?-linear term in the expansion (6.53) is small compared with the zeroorder term ω 0, if ω 0 is outside the narrow strips with width y f v along the lines
206
INCOMMENSURATE CRYSTALS AT Γ = 0
ω 0 = 2nm9 where m is an integer. We shall call these strips the first-order danger zones. The solution for the second-order term in (6.53) is
°>2(Φ) =
2
TS(1 — 2 cos ω 0) [sin (a> 0/2) sin ω 0 ] "
1
eos(2$ + §ω 0).
(6.57)
The second-order term in (6.53) is small compared with the first-order term if ω0 lies outside the first- and second-order danger zones. The latter are strips of width ν along the lines ω = n(2m + 1). It can be easily checked that in the nth approximation the denominator acquires a new factor sin(n
(6.58)
which has the solution 2
co = 2i?(C-cos<£),
(6.59)
where C is a certain constant. This approximation is equivalent to the continuous (resonance) approximation described above in Section 6.1. The values of C > 1 correspond to the periodic curves. The interval — 1 < C < 1 corresponds to the closed curves which at C = — 1 contract to the points at φ = (2m + 1)π, ω = 0. The points φ = mn, ω = 0, are the fixed points of the mapping T. At small v9 even values of m correspond to hyperbolic points, and odd ones to degenerate foci, the invariant curves being closed around them (see Fig. 6.3). Of particular interest is the case of C = 1, which corresponds to a separatrix joining the adjacent hyperbolic fixed points. This separatrix corresponds to the isolated soliton carrying one extra particle or hole. As long as the separatrix joins two fixed points, the corresponding soliton possesses a continuous degree of freedom. Indeed, starting with an arbitrary
207
L A T T I C E DISCRETENESS E F F E C T S
ω
FIGURE 6.3.
The pattern of the first-order invariant curves.
1
point on the separatrix and applying successively transformations Τ or f " , one can move gradually to the right or left, approaching φ = 2π or φ = 0 as w —• οο or η - • — οο, respectively. Geometrically this degree of freedom describes precisely the soliton center of gravity. We recall that if there is a continuous degree of freedom the soliton energy is independent of position. To proceed to the higher approximations we represent ω as ο; = ω where ω
( 1)
( 1)
+ ω
( 2)
+ .··,
satisfies the equation (6.59). Then ω (2)
(1)
-^-(ω ω ) + ^ ( ω αφ 2
( 1 ) 2
) τ^(ω αφ
( 1 )
( 2)
(6.60) will satisfy the equation (ί)
) = νω οοϊφ.
(6.61)
The solution should be chosen so as to avoid singularities at C < 1. Such a solution proves to be independent of new parameters: (2)
ω
= ^ιηφ
(6.62)
At small ν the second-order corrections do not change the qualitative pattern (η) n/2 of the invariant curves. The nth approximation term ω is of the order of v and is a linear combination of sin kφ and cos kφ with k running from 1 to n/2 for even η and from 1 to (n + l)/2 for odd n. As already mentioned, this structure of the higher-order terms can be most easily verified by a straightforward analysis of the equation (6.5). Thus, the qualitative pattern of
208
INCOMMENSURATE CRYSTALS AT Γ = 0
the invariant curves is unaffected by any higher-order corrections to the continuous approximation (6.59). By considering the second-order danger zone along the line ω = π, we obtain the equation for the function ω2(φ) = οο(φ) — π — (v/2) sin φ similar to 2 (6.58). The difference between (6.58) and co2 is in the substitutions ν ν and sin(0 + ω) -> sin2( + ω). The above method can be also used to obtain the zeroth-approximation invariant curve (ο{φ) = π + (υ/2) sin φ. A secondary structure of closed and open invariant curves separated by separatrices develops around this curve. The period of this superstructure is half the modulation period and half the period of the structure in the firstorder danger zone. The emerging pattern of the invariant curves is shown in Fig. 6.4. It consists of danger zones bounded by separatrices and layers of open curves in between them. 2 The separatrix intersection points are fixed points of the mappings 7^ T , 3 f , etc. From the physical point of view they correspond to the com mensurate phases. The separatrices correspond to solitons near a given commensurate phase. Open curves describe the periodic soliton lattices. The
FIGURE 6.4.
The global pattern of the invariant curves.
P I N N I N G OF THE SOLITON L A T T I C E
209
A
Β FIGURE 6.5.
Intersections of separatrices.
existence of layers of open curves is guaranteed by the Kolmogorov-Arnold Mozer theory. We have already explained how these layers are formed. Such trajectories can be shown to correspond to the acoustical branch in the excitation spectrum.
6.5. Pinning of the Soliton Lattice The power series in υ cannot converge at all v. Indeed, an arbitrarily small change of ν can transform a closed trajectory into an open one and vice versa. We can estimate the accuracy of the power expansions. The nth-order term contains sin[( + ω)η]. Let us expand it in powers of ω. The corrections will be small only if ηω « π/2. For the first-order danger zone ω ~ y/v, that yields the bound η ~ n/(2y/v). Up to terms of this order the expansion represents the invariant curve more and more accurately; then it diverges. This situation is typical of asymptotic expansions. A characteristic feature of such expan sions is that one may add exponentially small terms of the form exp(—const/^/ϋ) without changing expansion coefficients. One can prove to polynomial accuracy that some separatrix starting at one fixed point A passes through another fixed point B. However, in the general case with exponential accuracy, that cannot be guaranteed. This means that separatrices in general can intersect (Fig. 6.5). The splitting of separatrices takes place in an exponentially narrow region with width of the order of exp( —const/y/v) near the separatrices of the continuous approxi mation. The integral curves squeeze through this region from one fixed point
210
INCOMMENSURATE CRYSTALS AT 7 = 0
to another in a manner reminiscent of a random walk. This phenomenon is called Arnold diffusion. The intersection of separatrices is closely connected with the pinning of the solitons. Let O 0 be a certain separatrix-intersection point. Then the points 1 0 ± 1? 0 ± 2, . . . generated from O 0 by successive mappings Τ and f " will be the separatrix-intersection points as well. Thus, the generated sequences of intersection points tend asymptotically to the points A and S, respectively. Hence, the only way to leave A at η — οο for Β at η + οο is to follow precisely this sequence of intersection points. The only degree of freedom for the soliton is a choice of the initial point in this sequence. A typical solution in the stochasticity interval has the form of randomly distributed solitons and antisolitons. Outside the stochasticity interval there exist quasiperiodic invariant curves that never intersect. Starting with any point of this curve, we can obtain a periodic soliton structure with a continuous group of trans lations. The connection between the pinning of solitons and the stochasticity mode was first noted by Aubry (1979, 1980). There are special potentials in which solitons always possess a continuous degree of freedom. One potential of this class is the potential Vx defined by its derivative 1
2
1
νί(φ) = 4 t a n - { ( t a n h ^ ) s i n ^ [ l - ( t a n h ^ c o s ^ ] " } .
(6.63)
The finite-difference version of the equilibrium equation (6.5) has the solution <£w = 4tan
1
{expert
+ n 0) ] } ,
(6.64)
where n0 is an arbitrary constant that is a soliton center of gravity. In the limit of ξ = y/ϋ « 1 the potential νλ(φ) tends to v(l — cos φ). But the small difference between these potentials is of paramount importance, as it is a source of the pinning of solitons. It is easy to check that the soliton energy is n0-independent. To estimate the soliton pinning energy it is convenient to use the Poisson summation method. The total energy of the chain, U9 can be written as (see (6.1), (6.7)) (6.65) where
The m = 0 term in (6.65) corresponds to the continuous approximation. To
211
P I N N I N G OF THE S O L I T O N L A T T I C E
estimate the discreteness effects the m = 1 term suffices, since at small ν the higher harmonic contributions will decrease rapidly. In the special potential νχ(φ) contributions of two components of U cancel each other in (6.65). Then, for estimation purposes, ρ(φη) should be replaced by the difference ν(φ„) — ν^φη). The integration in (6.65) makes it possible to estimate the pinning energy: 1
2
Ep oc lb exp( - π / Jx>\
(6.66)
This asymptotic estimation is valid only at ν - • 0. 6.5.1. Qualitative P a t t e r n o f Pinning The above pattern of discreteness effects on the incommensurate phase can be presented in a more transparent, though cruder, way. For the isolated soliton the discreteness of the lattice gives rise to periodic potential relief, similar to the Peierls relief for dislocations (see, e.g., Hirth and Lothe, 1967), with amplitude Ep (see Pokrovsky, 1981). In the continuous approximation solitons form a lattice with spacing Z. The discreteness effect gives rise to the Peierls relief. Since the soliton-soliton interaction decreases exponentially with increasing / (see Section 6.2), at sufficiently large / it will become much smaller than Ep. In this limit the soliton lattice is so deformed that solitons occupy the nearest minima of the Peierls relief. As the spacing / decreases, the interaction of solitons becomes stronger, and at a certain lp the in commensurate phase becomes energetically favorable rather than the phase with localized solitons. Let us estimate the value of / p. Trapping of the soliton in the minimum of the relief leads to an energy gain of the order of Ep9 which should be compared with the energy loss due to deformation of the soliton lattice, b(dEinJdl)9 where 2s i nt is interaction energy of adjacent solitons and b is a 2 we period of the substrate lattice. Assuming that Eint ~ Xb y/vexp(—ZN/tf), obtain Zp~-. p ν
(6.67)
In this case δρ — Sc ~ Epjbk9 where δρ is the value of δ at which depinning of solitons from the substrate takes place. The pinning energy vanishes, the soliton lattice becomes (almost) equidistant and solitons form a lattice incommensurate with the substrate. In mathematical terms this is a transition from the stochastic region to the region of quasiperiodic continuous curves. It
212
INCOMMENSURATE CRYSTALS AT 7 = 0
is reminiscent of the Mott or Anderson transition between the states with low and high mobility of carriers. Since the solitons are the mass carriers, the kinetic properties should be expected to change sharply at this point (see Section 10.4). The character of the excitation spectra is an important difference between the pinned and unpinned soliton lattices. Obviously, in the pinned phase there are no acoustical modes. The gap in the spectrum of this phase is proportional to Ep. The unpinned phase possesses an acoustical mode. This difference, though exponentially small (see (6.66)) and elusive in any order of expansion in v, has a dramatic effect on the properties of the ground state. All the above reasoning can be repeated for the ground state of the uniaxial two-dimensional crystal, with the same result (quantum and thermal corrections being neglected). Allowance for the thermal fluctuations of the linear soliton similarly results in depinning (for detailed analysis see Chapter 8). This description of the depinning transition has been slightly simplified. As has been mentioned, in the phase plane in the vicinity of any trajectory corresponding to the incommensurate phase, domains of the existence of commensurate phases lie infinitely close to it. In these domains solitons are still pinned to the lattice, though with very small pinning energy. Depinning in the strict sense is possible only if one can neglect this small pinning energy and the corresponding commensurate phase domains. The thermal fluctua tions and, on top of that, quantum fluctuations in the one-dimensional crystal wipe out these commensurability domains; that justifies our approximate consideration a posteriori. At a given chemical potential the ground state of the system in the pinning phase is a periodic soliton lattice. The dependence of the period of this lattice is described by a function of the "devil's staircase" type, described in Chapter 5. In real systems, structures with long periods can be realized only at sufficiently low temperatures. The observation of transitions between them can be hampered by large diffusion times. As a result we shall experimentally observe numerous metastable states with solitons stochastically distributed over the wells of the Peierls potential at large distances one from another.
6.6. Orientational Epitaxy The experimental observations of orientational epitaxy—reorientation of the adatom lattice axes relative to the substrate ones—are described in Chapter 3. Below we present a model of the phenomenon in more detail. In the harmonic approximation, the energy of the two-dimensional lattice
213
ORIENTATIONAL E P I T A X Y
in the potential relief of the substrate can be written as v
E = \l
- ' > , ( r ' ) + Σ (r + u(r)).
Ζ
Γ,Γ'
(6.68)
Γ
Here u(r) is the displacement vector at the point r, D 0-(r — r') is the dynamical matrix, and V(r) is a periodic function. Let bx,b 2 and a 1 ?a 2 be the basis vectors of the substrate and adatom lattices, respectively. Let us start with the perturbation theory relative to V. Assuming that u(r) is small, neglect it in the argument of V(r + u(r)). The displacements can then be easily found in the momentum representation. The Fourier transforms u(q) are nonzero only for q equal to the reciprocal vectors of the substrate lattice. Let q x and q 2 denote its basis vectors. Minimization of (6.68) with respect to u(q) gives the solution uM=
(6.69)
-^'(qtaiW
where *(q) is the matrix inverse to the Fourier transform of Z)y(r), and V(q) is the Fourier transform of V(r). The energy corresponding to this solution is Δ* =
~Σ<ΐΡΓΑ^\ν(<ύ\ . 2
z
(6.70)
q
Recall that the matrix Du(q) is a periodic function with periods equal to the reciprocal lattice vectors p x and p 2 of the adatom lattice. With fixed magnitudes of the vectors p x , p 2 and qu q2 and fixed angles between p x, p 2 and q l 5q 2 , the energy (6.70) will be a function of the relative orientation of the pairs (Pi,p 2) and ( q i , q 2) . The minimum-energy condition determines this relative orientation angle. As the coverage θ changes, the vectors p t and p 2 and the matrix Z) 0(q) change too, which alters the relative orientation of the lattice and substrate as well. The case of simple triangular substrate and adatom lattices was studied by Novako and McTague (1976). Considering central forces and only the nearest-neighbor interaction, they found that the transition from the sym metric to asymmetric orientation of the substrate and adatom lattice axes should take place at certain coverage. This phenomenon was observed experimentally by Shaw et al (1978) in the system K r - G r . The case of rectangular centered substrate and triangular adatom lattices was studied by Uimin and Shchur (1978). Such a situation is realized experimentally, for example, in C s - W (110). Uimin and Shchur considered the nearest-neighbor interaction in the harmonic approximation and the lowest harmonics of the substrate potential preserving its symmetry. Their
214
INCOMMENSURATE CRYSTALS AT Γ = 0
numerical calculations show that the orientation angle changes abruptly at a certain critical value of the coverage. The values of the relative orientation angle before and after the jump agree well with the experimental results obtained by Fedorus and Naumovets (1970). Burkov (1979) has shown that with the same symmetry of the system a second-order transition is admissible too. Near the transition point the difference between the energies of the two phases is small. Thus, if the coverage is close to the critical value, the orientation of the adatom lattice can also change with temperature at fixed coverage.
6.7. Two-Dimensional Models of Incommensurate Crystals Of particular interest is the case of q values close to a vector p, of the reciprocal adatom lattice. When q -+ p„ the dynamical matrix D 0(q) vanishes a s — 2 so e Ifl Pl > *h applicability of the perturbation theory is doubtful. However, if the misfit δ = |q — p|/|p| is small, the resonance approximation formulated in previous sections for the one-dimensional case can be em ployed. In this limit the displacements of adatoms change slowly from site to site, which permits using the continuous approximation. Contrary to the onedimensional problem, different cases occur in two dimensions. We shall discuss some of them.
6.7.1. H e x a g o n a l L a t t i c e The best-known hexagonal lattices are lattices of noble gases with a simple triangular lattice on graphite. In this case q! being close to pl implies that simultaneously q 2 is close to p 2. Then, retaining only the resonance harmonics in the substrate potential, one can replace the potential with a certain function Κ(φ), where φ is the displacement of adatoms from their positions in the commensurate phase. The function Κ(φ) is a periodic one with the periods of the substrate and has a sixfold symmetry axis for the hexagonal substrate lattice. The triangular lattice is isotropic in its elasticity properties. The formula (1.17) for the elastic energy was given in Chapter 1. The displacement vector u is related to the vector φ by φ = u+
+ η 2δ 2,
(6.71)
where δ,· = a f — bi9 and ni and n2 are integers defining lattice sites. It is
TWO-DIMENSIONAL
M O D E L S OF INCOMMENSURATE
CRYSTALS
215
convenient to express these numbers in terms of the radius vector r = n^bi -f w 2b 2: 2
φ = ϋ + ' ^ - δ 1+ ^ - δ 2.
(6.72)
Vectors φ and u will be treated as functions of the continuous variable r. Notice that in (6.72) q x and q 2 have been replaced by p x and p 2. Now, substituting φ into the elastic energy (1.17) and adding j Κ(φ)ώ·, we obtain the total energy. For Κ(φ), the simplest form compatible with the substrate symmetry is Κ(φ) = K 0[3 - cos q x · φ - cos q 2 · φ - c o s ^ + q 2) · φ].
(6.73)
Minimization of the total energy with respect to φ yields the system of equilibrium equations (ij = 1 , 2 )
dridrj
or]
δφί
Some particular solutions of the system (6.74) are known. The com mensurate phase is described by the trivial solution φ = 0. It is easy to obtain solutions describing the soliton line or the striped structure of aligned solitons. However, analytic solutions, in particular for the hexagonal lattice of solitons have not been obtained as yet.
6.7.2. A n i s o t r o p i c S u b s t r a t e s Two different situations can be distinguished here. The first one was described in Chapter 1: a uniaxial incommensurate crystal, where the lattice is commensurate in one direction and incommensurate in the other. Change of variables from u to φ transforms the energy of the uniaxial crystal (1.1), (1.2) into
»-M*(£-'H(S)'*™} ™ In the case of the uniaxial crystal the initially commensurate crystal becomes incommensurable in one of the two directions. Another experiment ally observed transition occurs when such a crystal becomes incommensurate in the second direction too. Since in one of the two directions the crystal is
216
INCOMMENSURATE CRYSTALS AT 7 = 0
already incommensurate, in this direction the substrate effect can be neglec ted, and the substrate potential can be supposed to depend on one direction, i.e., on only one component of the displacement vector.
6.7.3. U n i a x i a l P e r i o d i c S o l u t i o n s In the simplest case of a uniaxial crystal with arbitrary potential Υ(φ)9 the counterpart of the equation (1.3) is 2
λ
2
δφ
Λ λ
δ φ _ άΥ(φ)
ι ΤΖϊ + 2 -ι τ - 2 = — — · ~^? + Ί ? ~ άφ
(6.76)
This equation contains no δ. We shall search for a solution of (6.76) as a function of one variable ζ = χ cos θ + .y sin 0. As a result, (6.76) will take the form familiar from the one-dimensional case: 2
ά φ _ άΥ{φ) ο-ΓΤ 2 = — τ τ ^ dz " άφ
λ
λ
2
2
ο = *2 sin θ + λχ cos θ.
(6.77)
Repeating the procedure of Section 6.2, we can readily obtain the energy per unit area, 2
U = - S + \λχδ 9
(6.78)
where & is related to δ and θ by
.
(6.79)
v W ^ t a n ^ + l Here £(
(6.80)
In view of (6.78), the energy is minimal when δ takes its maximal value. By virtue of (6.80), E(S) is a monotonically increasing function of δ9 and, as follows from (6.79), the energy per unit area is minimal at θ = 0. Thus, solution of (6.76) will depend only on the argument x. Denote the minimum of E(S) by £ c . Now, the critical value of
217
E L A S T I C SUBSTRATE
structure are readily found: (6.81) where K(S) is a counterpart of the complete elliptic integral of the first kind:
For Υ(φ) from (1.2), λί=
λ2 = λ, and / » / 0 we have (see Section 6.2) (6.82)
6.8. Elastic Substrate The formation of solitons is followed by the displacement of adatoms from the minima of the potential relief, which produces elastic deformations of the substrate. As Gordon and Villain (1979) have noticed, these deformations decay with distance by a power law, which must result in a power-law interaction of solitons. An accurate analysis of this interaction mechanism necessitates the self-consistent solution of the problem of an incommensurate crystal on an elastic substrate. For simplicity, the substrate will be supposed to have isotropic elasticity properties, with rigidity much higher than that of the overlayer (the criterion will be given presently). This corresponds to the experimental situation with overlayers on metallic substrates. In this case the substrate deformation is much smaller than that of the overlayer and can be taken into account in the overlayer-substrate interaction by the substitution u' = u — uX9 where ux is a deformation of the substrate along the OX axis. Then the elastic energy of the adatom lattice will be H = ^dxdyip
XI^J
X+2(^J
+K(u-ux + ^ x ) j ,
(6.83)
where a and b are the periods of the overlayer and substrate. Since in the considered system the solitons are aligned along the OY axis, it suffices to consider deformations of the substrate only in the XOZ plane. The unde-
218
INCOMMENSURATE CRYSTALS AT
T=0
formed surface of the substrate is ζ = 0; the bulk of the substrate is the half space ζ > 0. Then the energy of the elastic deformation of the substrate can be written as (Landau and Lifshitz 1986) dz {oxxuxx
+ azzuzz
+ 2axzuxz)
(6.84)
where aik is the stress tensor, and uik is the deformation tensor. We now have to solve the equations of two-dimensional elasticity theory subject to the boundary conditions uxx(x90) = dujdx, uxz(x90) = 0, and then to find ux as a function of x. The equations are (Landau and Lifshitz, 1986) _ i ^ + — ± i = 0, ox dz
—^- + — ^ = 0. ox dz
6.85
The solution of these equations for the Fourier transforms in the variable χ is ajq,z)
= E(l - σ Γ ^ β , Ο χ ΐ - |«|z)exp(-\q\z\
aZ2(q9z)
= E(l - σ Γ ^ β , Ο χ ΐ + \q\z)^-\q\z)9
z) = - ^ ( 1 -
(6.86)
ayHqzuxx(q90)Qxp(-\q\z).
Here Ε is Young's modulus, and σ is the Poisson coefficient. Substituting (6.86) into the energy (6.84) and integrating over z, we can obtain the energy of the elastic deformation of the substrate induced by interaction with the overlayer: He = E(l-a)~
2
\dy\dq'^t.
(6.87)
Making the standard substitution φ = u — ux + (b — a)x/a and minimizing the elastic energy at fixed φ9 we readily obtain the final formula for the elastic energy of the adatom lattice adsorbed on the deformable substrate (compare with (6.75)):
219
E L A S T I C SUBSTRATE
where 2
λ0 =
4Ε^-\1-σ)- 9 2
g(x - χ') = λ\
~ σ) 2 4E(x - x')|2*
The principal feature of this formula (apart from the presence of the nonlocal term) is a renormalization of all the parameters of the overlayer Hamiltonian due to the substrate deformation. As follows from the above formula for renormalization of the elastic modulus of the overlayer, the rigid-substrate criterion is λ0 » λν In the first approximation in λχ/λ0 the nonlocal energy of the substrate-mediated interaction can be estimated by substituting the solution of the equation (6.76) for φ (with the renormalized constants λ\ and δ'). If the spacing / is large, / » / 0, the integration over all χ and χ' in (6.88) can be replaced by an integration over one period and a summation over all solitons. As a result, we obtain the contribution of the interaction of solitons to the energy density:
The power law (6.89) and, particularly, the specific value of the exponent are of importance in the thermodynamics of the soliton lattice. One striking feature of the interaction mediated by the elastic deformations of the substrate is its independence of the amplitude of the substrate potential (see (6.89)). This property was noticed by Lyuksyutov (1982) and Talapov (1982). The closely related problem of the interaction of steps on a crystal surface mediated by its elastic deformations was considered by Lau and Kohn (1977), Gordon and Villain (1979), and Marchenko and Parshin (1980). The energy 2 of the interaction of steps decreases with increasing spacing like Γ .
Incommensurate Crystals at Γ = 0
The subject of this chapter is the ground state of incommensurate crystals in the classical approximation. It is convenient to introduce the problem with one-dimensional models. The presentation will be organized as follows: the first three sections are concerned with the ground state and the excitation spectrum in such models; the effects of discreteness of the incommensurate crystal lattice on its translational properties (pinning of the soliton lattice) are considered in the subsequent two sections; then in Sections 6.6-6.9 we proceed to describe the ground state of the two-dimensional incommensurate crystal—orientational epitaxy and the soliton lattice.
6.1. Resonance Approximation The simplest one-dimensional model of a crystal in a periodic potential was proposed by Frenkel and Kontorova (1938) to describe the structure of dislocations. In this model one considers a one-dimensional chain of springcoupled particles placed in a periodic potential relief with period b (Fig. 6.1). 193
194
INCOMMENSURATE CRYSTALS AT
Γ=0
a
b
FIGURE 6.1.
A chain of particles coupled by springs in a periodic potential.
The potential energy U of such a system is
U = Σ [|(*n+1 " *n - af + K(x„)],
(6.1)
where a is the period of the free chain, xn is the coordinate of the nth particle, and λ is the stiffness of the spring. Note that due to the different dimension ality, the definitions of λ and V are different from those of Chapters 1 and 4. Let the periods a and b be close: a = b + δ,
6«b.
(6.2)
We shall assume that V is small: 2
|K(x)|«Afc .
(6.3)
Let us introduce the variable un = x„-
na.
(6.4)
Then minimization of (6.1) with respect to un results in the equation V\dn + un) un+x
K
- 2un + «„_! = — —-—"A
(6.5)
In view of the smallness of δ, the quantity ν\δη + un) is a slowly varying function of n. As follows from the weak-potential condition (6.3) and
195
RESONANCE APPROXIMATION
Equation (6.5), un is a slowly varying function of η as well. Therefore, the difference equation (6.5) can be approximated by a differential equation. This approximation. approximation will be called the resonance It is convenient to introduce a new variable φ(η) = δη + uw
(6.6)
which is the displacement of the nth particle from its position in the free chain in the absence of the potential. In terms of this variable the potential energy (6.1) takes the form
υ
=
Μ ί '
)
δ
+
η
φ
λ
ά
η
·
( 6
·
7 )
The resonance approximation can be also used at a = (M/N)b + <5, where Μ and Ν are integers and δ « b. Consider the particular case of a = (b/2) + δ. For simplicity, assume that Υ(φ) = — V cos qφ9 where q = 2n/b. Then the energy (6.1) can be written as
- Kcos qx„ + Kcos # „ >,
(6.8)
where χ„ = u2n + 2n<5, ψη = M 2 „ + I 4- (2n + 1)<5. First consider a commensurate phase in which φ and χ are n-independent. Then the energy (6.8) takes the form 2
2
U = ?j- Ιλ(φ - χ) + λδ - Vcos qx + Kcos # ] .
(6.9)
Minimization of (6.9) gives equations V X
" ^
=
~ 2λ
Q S in qX
'
m Sί Ζ
=
Sm
The solutions of these equations are of the form V
π X
=
2^-u °
qC Sm7t
mn +
T'
^
β Λ )ϋ
196
INCOMMENSURATE CRYSTALS AT Γ = 0
where m is an integer. This shows that if the period of the potential relief is double the period of the crystal, then atoms are trapped at places where the absolute value of the potential derivative V\x) is maximal, rather than at bottoms of wells. In the incommensurate phase φ and χ will be slowly varying functions of n. It is convenient to introduce new variables φη = (χ„ + φη)/2 and 2 ω„ = χ„ — φη> Then the formula (6.8) for I/, up to the K -order terms, in the new variables takes the form υ
= J
2δ
)
+
Ω
+ (
-
δ
+
vqVsmq^dn.
+
(6.12)
Minimization of ( 6 . 1 2 ) in ω yields I do ω
=
V
- 2 Λ Γ - 2 ΐ
9
8
ί
η
·
#
)
Substituting ( 6 . 1 3 ) in (6.12), we obtain after some transformations 0
-Μϊ- ) Μ
,+
Τ3Ρ"" ·*]*· 1
· Ι , 46)
In the more general case of a = (M/N)b + δ, similar calculations result in
U=
A
-
hj + ν^^Νφ^άη,
(6.15)
2
l
where A is a certain constant, hoc δ, and VN oc V(Vlkb f~ . Thus, for any degree of incommensurability the potential energy can be transformed to the universal form (6.7). The presented results were obtained in papers by Pokrovsky and Talapov ( 1 9 7 8 ) and Teodorou and Rice ( 1 9 7 8 ) .
6.2. Soliton Lattice A qualitative picture of the incommensurate phase and the simplest es timations of the soliton parameters were given in Section 1.2. Let us consider this problem in more detail. Different terms in the energy (6.7) correspond to opposite tendencies in the arrangement of the particles. On the one hand, particles tend to occupy the minima of the potential relief ν(φ). On the other hand, they tend to form a lattice with period a different from the period b of
( 6 1 3
197
SOLITON L A T T I C E
the potential. As a result of the tradeoff, at a certain critical value <5C the purely commensurate phase becomes unstable, and domains with broken in commensurability, or solitons, will form. The value of <5C can be found from the equilibrium equation for φη, which, in its turn, can be easily derived from (6.7): 2
, ά φ _ dV(4>) « -2 ^ · dn " άφ
(6.16)
This equation has the first integral
ffl-
λ 2
\ J - I
<*+ν(φ).
(6.17)
The equations (6.16) and (6.17) have a simple mechanical analogy. The equation (6.16) describes the deflection φ of a pendulum in the field ν(φ), η being the time. The equation (6.17) is the energy conservation law. The case of S > — Vmin, when the pendulum energy exceeds the potential energy at the top point of the trajectory, corresponds to the incommensurate phase. When δ = Sc = — Vmin, the pendulum stops at the top point. It leaves this point after an infinite time, then passes through the bottom point after a finite time and climbs again to the top after an infinite time. This trajectory corresponds to the formation of one soliton in the particle chain. Namely, at a very large distance from the soliton core all the particles occupy the minima of the potential Υ(φ). Let 0 0 as η — oo. Then at large distances from the soliton core φ is exponentially small. In the core φ = b/2 and φ' = άφ/dn « y/V/λ. On the right of the soliton core, as η - > + oo, the deviation of φ from b is exponentially small too (see Figs. 1.5, and 6.2). The soliton energy Es is the difference between the total energy (6.7) and the energy Uc of the com mensurate phase: t>c
=N |^+K
Ea =
0
j^2[V(4>)
m i n
J,
(6.18)
+ / J - δλ^άη
0)
= £< - Xdb,
(6.19) 0)
Here J V 0 is the number of particles in the chain, Sc = — VmiB, and £ < is the soliton energy at δ = 0. As follows from (6.19), the critical value δΒ, at which soliton creation becomes energetically favorable, equals <5C =
E^/Xb.
(6.20)
198
INCOMMENSURATE C R Y S T A L S AT 7 * = 0
Rewriting (6.19) in terms of the variable φ, using (6.17), we obtain
^
0)
= £ V 2 A [ ^ ) - K m i n] # .
(6.21)
In the case of Υ(φ) = - Kcos(27r/i# we find £ < = (4b/n)y/Xv. Solving Eq. (6.17) with S =
ν(φ):
do
, oc e x p ( - \ n - η0\^/φ)9
where κ =
ά ν(φ)/άφ 2
2
at the minimum of
Υ(φ).
(6.22)
199
SOLITON L A T T I C E
The formation of solitons implies a transition from the commensurate phase to the incommensurate one ( C — I transition). Such a transition should occur in the vicinity of any rational value of a/b = M/N. Hence, with change of <5, the commensurate structures must replace each other with arbitrarily high frequency. Hence, a situation of the "devil's staircase" type (Section 5.1) occurs. As shown in the previous section, the amplitude of the effective N potential is proportional to V , so that intervals of dc correspond to a given NI2 commensurate phase, and <5C oc V . Obviously, thermal fluctuations will smear out the fine structure of transitions. Thus, the formation of solitons becomes energetically favorable at δ > <5C. number of solitons produced at δ > <$c will be controlled by the The interaction between them. This results in the formation of a periodic lattice with the period determined by
da
(6.23)
y2[
The
energy of the soliton lattice per unit length can be easily found from
and (6.17):
U ΤΓ~
^2^+ν(φ)2άφ-δ^
=
λδ
Τι
Ν0
-
(6.24)
2
© + —τ—.
άφ
(6.7)
2
(Ο.
.24)
Minimizing with respect to the parameter S, we obtain δ as a function of $ and consequently, by virtue of φ as a function of δ. For instance,
(6.17),
δ
= Γ y/2[*
+ V(4>)]d
(6.25)
Near the C - I transition δ — <5C has the following dependence on S:
where α is a certain numerical constant. The energy of the incommensurate and phase can be derived from
(6.24)
(6.25):
l/ = No|^--^+^<5 J. 2
(6.27)
200
INCOMMENSURATE CRYSTALS AT Γ = 0
At S = SG = - Vmin, the energy (6.27) coincides with the energy (6.18) of the commensurate phase. In the incommensurate phase 8 > &C9 and con sequently, at the same value of δ the energy of the incommensurate phase is always lower than that of the commensurate phase. For the potential of the form V(
where amn denotes the Jacobi amplitude, and k = y/2V/(V+ S) is the modulus of the elliptic function (Akhiezer, 1970). If Vmin= — V, then k = 1 at S = Sc = Κ the degenerate elliptic function coincides with the hyperbolic function, and the solution (6.28) describes an isolated soliton and coincides with (1.4). The relationship between δ and k is given by equation
(6.29)
where E(k) is the complete elliptic integral of the second kind. A plot of φ(η) for the soliton lattice is shown in Fig. 6.2. In the vicinity of the C - I transition the spacing / of solitons increases as
(6.30)
The corresponding formula for the energy, U
—
=
- 6bX + ^ q ^ e x p t - l / y i d ) j
0)
λδ
2
4 +^-,
(6.31)
has a simple physical meaning. The term 2s£ — 6bX is the energy gain for the formation of one soliton, whereas the exponential term describes the energy increase due to the exponentially weak repulsion of solitons. Minimization of (6.31) with respect to I leads readily to the formula (6.30) for the spacing of the soliton lattice.
201
SPECTRUM OF S M A L L OSCILLATIONS
6 3 . Spectrum of Small Oscillations As mentioned in Chapter 1, a principal feature of the incommensurate phase is a group of continuous transformations—translations of the soliton lattice leaving the energy unchanged. Indeed, the exact solution of the equation (6.17) can be written in the form (6.32) This solution contains an arbitrary parameter n 0, on which the energy of the incommensurate phase does not depend. Invariance under the group of continuous translations implies the existence of an acoustical branch (the Goldstone mode) in the spectrum of small oscillations in the incommensurate phase. In order to find a spectrum of the small oscillations it is necessary to add the kinetic term to the energy (6.7). The total energy is (6.33)
Η =
where m is the particle mass. Let φ(η, ί) be represented as φ0(Η) + Ψ(η, T), where φ0(Η) satisfies Eq. (6.16) and Ψ(Η, T) is small. In the linear approximation Ψ(π, T) satisfies the equation Δ
2
Ψ
m — 2^ — DT
Δ
2
Λ——*-2 DN
ά ν(φ ) Ψ—ΤΡΓάφ% 2
Ψ
+
0
= 0·
(6.34)
Evidently, the dispersion law in the commensurate phase will be of the form (6.35) where Ω is the frequency, Q is the wave vector, and Κ is defined in (6.22). In this case the spectrum has a gap Ω 0 = Y/Κ/ΙΗ. In the incommensurate phase the equation for oscillations of frequency Ω reads as (6.36)
202
INCOMMENSURATE CRYSTALS AT Γ = 0
2
Here we have replaced ά ν/άφο with λφ%/φ'0 using (6.16) (a prime denotes differentiation with respect to n). The equation (6.36) has the form of the Schrodinger equation with a periodic potential. The existence of a zero-frequency oscillation mode follows from the equation (6.36) at once. The corresponding solution is Ψ(Ω = 0, η) = φ'0, describing the translation of the soliton lattice. To simplify the notation we introduce the function 1,2
ι 2
τ = ΦΌ = {2[
(6.37)
The zero-frequency mode has zero quasimomentum, q = 0. For small q we search for a solution of the form Ψ = τ β χ ρ ^ Jjx^,
(6.38)
where χ is a periodic function of η with period / satisfying the condition (6.39) This constraint follows from the Flouqet-Bloch theorem on the properties of the solutions of equations with periodic coefficients: Ψ(η + Ζ) = βχρ(^Ζ)Ψ(η),
(6.40)
and from the periodicity properties of τ. Assume that at small q the dispersion law is of the form Ω = cq. Then the equation for χ resulting from (6.36) is 2
iq(x'z + 2χτ') + q (^j-
2
- χ ) τ = 0.
(6.41)
Now, multiplying (6.41) by τ, integrating between zero and /, and using the periodicity of χ and τ, we obtain
ioW"* )' ^ ' 2
2
0
(642)
This equation permits determining the sound velocity in the small-q limit. As
203
SPECTRUM OF S M A L L OSCILLATIONS
2
follows from (6.41), in the small-g limit χ = η/τ , where η is given by the equation
C'dn
(6.43)
Jo τ
Employing dn = άφ/τ, which follows from the definition τ(φ) = φ', from (6.42) and (6.43) we obtain
J^\
fil^
U2
C
(
6
4
4
)
In the incommensurate phase far away from the C - I transition the value of c is close to y/X/m. On the other hand, near the C - I transition,
c oc Near the C - I transition the spacing of solitons is large and the soliton lattice can be treated as a chain of interacting particles, the solitons themselves being regarded as particles. The frequency of oscillations in such a chain is determined by the standard formula Ω = Asm j .
(6.46)
The constant A can be easily determined from the small-*? limit when Ω = cq. Therefore, ^
2c . ql
T
s l n
T
( 6
·
4 7 )
The equation (6.47) describes the entire acoustic branch of the spectrum. Evidently, it is valid only near the C - I transition. The maximum frequency of the acoustic excitations, Ω ^ = 2c//, vanishes for δ - • <5C as
Ω™*(<5-<υ
1 / 2
m
[ΐη^-] •
These results were obtained by Pokrovsky and Talapov (1978).
(6.48)
204
INCOMMENSURATE CRYSTALS AT 1 = 0
6.4. Lattice Discreteness Effects The following two sections are mostly of a mathematical character. Readers not interested in the mathematical formalism can skip them and proceed to the end of Section 6.5, where a qualitative picture of pinning is presented. The discreteness of the lattice has a strong influence on the properties of the incommensurate lattice. One of them has already been mentioned: the infinite number of commensurate and incommensurate phases (the "devil's staircase"). Another such phenomenon—the pinning of solitons—is the subject of this section. We shall use the Frenkel-Kontorova model (see Section 6.1). Taking 2 λφ/2π) ν(\ — cos φ) for the potential, with fybjln instead of φ in (6.6), we rewrite the second-order finite-difference equation (6.5) as a system of two first-order equations: ω η+ ι =con
+ vsm((un + φη\
Φη+l π + Φη· =
ω
(6.49)
The second of these equations can be regarded as a definition of ωη. The system (6.49) can be treated as a mapping of the two-dimensional vector space (ωη9φ„) onto another two-dimensional vector space ( ω π + ,1 φ „ + 1) . Suppressing the subscripts, we rewrite (6.49) once more: ω' = ω + ν ύη(ω + φ\ φ' = ω + φ.
(6.50)
This formulation of the problem was suggested by Aubry (1979). In our presentation we shall follow the method developed by Pokrovsky (1981). Mappings of the type (6.50) were studied by mathematicians long ago and are known as canonical mappings. The above formulation of the problem makes it similar to those of the nonlinear dynamics. The basic theorems were derived by Poincare, Birkhoff, Kolmogorov, Arnold, and Moser (for a review see the book by Arnold and Avez (1968)). The principal ideas will be described here in a somewhat heuristic form without rigorous proof, to make the mathematical formalism more accessible to nonexperts. The reader can find more details, e.g., in the book by Lichtenberg and Liberman (1983). The mapping (6.50) is continuous and analytic, and its Jacobian equals unity, so that it is an area-preserving mapping having an inverse trans formation with the same properties. We shall call a curve Γ an invariant curve of the mapping Τ if any point of this curve (ω, φ) is mapped onto
205
L A T T I C E DISCRETENESS E F F E C T S
another point (ω', φ') belonging to the same curve Γ. If ω = (ο(φ) is an equation of the invariant curve, then by the definition (6.50) of the mapping Τ we can write ω'(ω(φ%
φ) = ω(φ'(ω(Φ),
Φ)1
(6.51)
Substituting the functional form of ω ' , φ' from (6.50) into (6.51), we obtain the equation for the function ω(φ): ω(φ
+ ω(φ))
= ω(φ)
+ νύη\_φ
+ α#)].
(6.52)
It is worthy to note that the invariant curve is not uniquely defined. Indeed, suppose a point (ω 0, φ0) is mapped into the point ( ω ΐ 9 φ ι ) . If we draw an arbitrary continuous curve joining these two points, it will be mapped under Τ into a continuous curve joining (ωϊ9φ1) and its map (ω 2,φ 2)· By virtue of the continuity of the mapping, the second curve can be continued in both directions. According to the method of its construction this curve is an invariant one. With some effort one can provide for the continuity of the derivatives at the junction points. Let us search for the infinitely differentiable solutions of Eq. (6.52) in the form of power series (v is supposed to be small): ω
= Σ
k
(6.53)
cokv .
k= 0 For the zero-order function ω 0 we obtain the equation ω 0 ( φ + ω 0) - ω 0 ( φ ) = 0,
(6.54)
with the obvious solution ω 0 = constant. The first-order term satisfies the equation ω χ ( φ + ω 0) - ω ^ φ ) = sin(<£ + ω 0) ,
(6.55)
which has the solution
ο>ι(Φ)
=
\
^sin y - ^
cos( + £ω 0).
(6.56)
The t?-linear term in the expansion (6.53) is small compared with the zeroorder term ω 0, if ω 0 is outside the narrow strips with width y f v along the lines
206
INCOMMENSURATE CRYSTALS AT Γ = 0
ω 0 = 2nm9 where m is an integer. We shall call these strips the first-order danger zones. The solution for the second-order term in (6.53) is
°>2(Φ) =
2
TS(1 — 2 cos ω 0) [sin (a> 0/2) sin ω 0 ] "
1
eos(2$ + §ω 0).
(6.57)
The second-order term in (6.53) is small compared with the first-order term if ω0 lies outside the first- and second-order danger zones. The latter are strips of width ν along the lines ω = n(2m + 1). It can be easily checked that in the nth approximation the denominator acquires a new factor sin(n
(6.58)
which has the solution 2
co = 2i?(C-cos<£),
(6.59)
where C is a certain constant. This approximation is equivalent to the continuous (resonance) approximation described above in Section 6.1. The values of C > 1 correspond to the periodic curves. The interval — 1 < C < 1 corresponds to the closed curves which at C = — 1 contract to the points at φ = (2m + 1)π, ω = 0. The points φ = mn, ω = 0, are the fixed points of the mapping T. At small v9 even values of m correspond to hyperbolic points, and odd ones to degenerate foci, the invariant curves being closed around them (see Fig. 6.3). Of particular interest is the case of C = 1, which corresponds to a separatrix joining the adjacent hyperbolic fixed points. This separatrix corresponds to the isolated soliton carrying one extra particle or hole. As long as the separatrix joins two fixed points, the corresponding soliton possesses a continuous degree of freedom. Indeed, starting with an arbitrary
207
L A T T I C E DISCRETENESS E F F E C T S
ω
FIGURE 6.3.
The pattern of the first-order invariant curves.
1
point on the separatrix and applying successively transformations Τ or f " , one can move gradually to the right or left, approaching φ = 2π or φ = 0 as w —• οο or η - • — οο, respectively. Geometrically this degree of freedom describes precisely the soliton center of gravity. We recall that if there is a continuous degree of freedom the soliton energy is independent of position. To proceed to the higher approximations we represent ω as ο; = ω where ω
( 1)
( 1)
+ ω
( 2)
+ .··,
satisfies the equation (6.59). Then ω (2)
(1)
-^-(ω ω ) + ^ ( ω αφ 2
( 1 ) 2
) τ^(ω αφ
( 1 )
( 2)
(6.60) will satisfy the equation (ί)
) = νω οοϊφ.
(6.61)
The solution should be chosen so as to avoid singularities at C < 1. Such a solution proves to be independent of new parameters: (2)
ω
= ^ιηφ
(6.62)
At small ν the second-order corrections do not change the qualitative pattern (η) n/2 of the invariant curves. The nth approximation term ω is of the order of v and is a linear combination of sin kφ and cos kφ with k running from 1 to n/2 for even η and from 1 to (n + l)/2 for odd n. As already mentioned, this structure of the higher-order terms can be most easily verified by a straightforward analysis of the equation (6.5). Thus, the qualitative pattern of
208
INCOMMENSURATE CRYSTALS AT Γ = 0
the invariant curves is unaffected by any higher-order corrections to the continuous approximation (6.59). By considering the second-order danger zone along the line ω = π, we obtain the equation for the function ω2(φ) = οο(φ) — π — (v/2) sin φ similar to 2 (6.58). The difference between (6.58) and co2 is in the substitutions ν ν and sin(0 + ω) -> sin2( + ω). The above method can be also used to obtain the zeroth-approximation invariant curve (ο{φ) = π + (υ/2) sin φ. A secondary structure of closed and open invariant curves separated by separatrices develops around this curve. The period of this superstructure is half the modulation period and half the period of the structure in the firstorder danger zone. The emerging pattern of the invariant curves is shown in Fig. 6.4. It consists of danger zones bounded by separatrices and layers of open curves in between them. 2 The separatrix intersection points are fixed points of the mappings 7^ T , 3 f , etc. From the physical point of view they correspond to the com mensurate phases. The separatrices correspond to solitons near a given commensurate phase. Open curves describe the periodic soliton lattices. The
FIGURE 6.4.
The global pattern of the invariant curves.
P I N N I N G OF THE SOLITON L A T T I C E
209
A
Β FIGURE 6.5.
Intersections of separatrices.
existence of layers of open curves is guaranteed by the Kolmogorov-Arnold Mozer theory. We have already explained how these layers are formed. Such trajectories can be shown to correspond to the acoustical branch in the excitation spectrum.
6.5. Pinning of the Soliton Lattice The power series in υ cannot converge at all v. Indeed, an arbitrarily small change of ν can transform a closed trajectory into an open one and vice versa. We can estimate the accuracy of the power expansions. The nth-order term contains sin[( + ω)η]. Let us expand it in powers of ω. The corrections will be small only if ηω « π/2. For the first-order danger zone ω ~ y/v, that yields the bound η ~ n/(2y/v). Up to terms of this order the expansion represents the invariant curve more and more accurately; then it diverges. This situation is typical of asymptotic expansions. A characteristic feature of such expan sions is that one may add exponentially small terms of the form exp(—const/^/ϋ) without changing expansion coefficients. One can prove to polynomial accuracy that some separatrix starting at one fixed point A passes through another fixed point B. However, in the general case with exponential accuracy, that cannot be guaranteed. This means that separatrices in general can intersect (Fig. 6.5). The splitting of separatrices takes place in an exponentially narrow region with width of the order of exp( —const/y/v) near the separatrices of the continuous approxi mation. The integral curves squeeze through this region from one fixed point
210
INCOMMENSURATE CRYSTALS AT 7 = 0
to another in a manner reminiscent of a random walk. This phenomenon is called Arnold diffusion. The intersection of separatrices is closely connected with the pinning of the solitons. Let O 0 be a certain separatrix-intersection point. Then the points 1 0 ± 1? 0 ± 2, . . . generated from O 0 by successive mappings Τ and f " will be the separatrix-intersection points as well. Thus, the generated sequences of intersection points tend asymptotically to the points A and S, respectively. Hence, the only way to leave A at η — οο for Β at η + οο is to follow precisely this sequence of intersection points. The only degree of freedom for the soliton is a choice of the initial point in this sequence. A typical solution in the stochasticity interval has the form of randomly distributed solitons and antisolitons. Outside the stochasticity interval there exist quasiperiodic invariant curves that never intersect. Starting with any point of this curve, we can obtain a periodic soliton structure with a continuous group of trans lations. The connection between the pinning of solitons and the stochasticity mode was first noted by Aubry (1979, 1980). There are special potentials in which solitons always possess a continuous degree of freedom. One potential of this class is the potential Vx defined by its derivative 1
2
1
νί(φ) = 4 t a n - { ( t a n h ^ ) s i n ^ [ l - ( t a n h ^ c o s ^ ] " } .
(6.63)
The finite-difference version of the equilibrium equation (6.5) has the solution <£w = 4tan
1
{expert
+ n 0) ] } ,
(6.64)
where n0 is an arbitrary constant that is a soliton center of gravity. In the limit of ξ = y/ϋ « 1 the potential νλ(φ) tends to v(l — cos φ). But the small difference between these potentials is of paramount importance, as it is a source of the pinning of solitons. It is easy to check that the soliton energy is n0-independent. To estimate the soliton pinning energy it is convenient to use the Poisson summation method. The total energy of the chain, U9 can be written as (see (6.1), (6.7)) (6.65) where
The m = 0 term in (6.65) corresponds to the continuous approximation. To
211
P I N N I N G OF THE S O L I T O N L A T T I C E
estimate the discreteness effects the m = 1 term suffices, since at small ν the higher harmonic contributions will decrease rapidly. In the special potential νχ(φ) contributions of two components of U cancel each other in (6.65). Then, for estimation purposes, ρ(φη) should be replaced by the difference ν(φ„) — ν^φη). The integration in (6.65) makes it possible to estimate the pinning energy: 1
2
Ep oc lb exp( - π / Jx>\
(6.66)
This asymptotic estimation is valid only at ν - • 0. 6.5.1. Qualitative P a t t e r n o f Pinning The above pattern of discreteness effects on the incommensurate phase can be presented in a more transparent, though cruder, way. For the isolated soliton the discreteness of the lattice gives rise to periodic potential relief, similar to the Peierls relief for dislocations (see, e.g., Hirth and Lothe, 1967), with amplitude Ep (see Pokrovsky, 1981). In the continuous approximation solitons form a lattice with spacing Z. The discreteness effect gives rise to the Peierls relief. Since the soliton-soliton interaction decreases exponentially with increasing / (see Section 6.2), at sufficiently large / it will become much smaller than Ep. In this limit the soliton lattice is so deformed that solitons occupy the nearest minima of the Peierls relief. As the spacing / decreases, the interaction of solitons becomes stronger, and at a certain lp the in commensurate phase becomes energetically favorable rather than the phase with localized solitons. Let us estimate the value of / p. Trapping of the soliton in the minimum of the relief leads to an energy gain of the order of Ep9 which should be compared with the energy loss due to deformation of the soliton lattice, b(dEinJdl)9 where 2s i nt is interaction energy of adjacent solitons and b is a 2 we period of the substrate lattice. Assuming that Eint ~ Xb y/vexp(—ZN/tf), obtain Zp~-. p ν
(6.67)
In this case δρ — Sc ~ Epjbk9 where δρ is the value of δ at which depinning of solitons from the substrate takes place. The pinning energy vanishes, the soliton lattice becomes (almost) equidistant and solitons form a lattice incommensurate with the substrate. In mathematical terms this is a transition from the stochastic region to the region of quasiperiodic continuous curves. It
212
INCOMMENSURATE CRYSTALS AT 7 = 0
is reminiscent of the Mott or Anderson transition between the states with low and high mobility of carriers. Since the solitons are the mass carriers, the kinetic properties should be expected to change sharply at this point (see Section 10.4). The character of the excitation spectra is an important difference between the pinned and unpinned soliton lattices. Obviously, in the pinned phase there are no acoustical modes. The gap in the spectrum of this phase is proportional to Ep. The unpinned phase possesses an acoustical mode. This difference, though exponentially small (see (6.66)) and elusive in any order of expansion in v, has a dramatic effect on the properties of the ground state. All the above reasoning can be repeated for the ground state of the uniaxial two-dimensional crystal, with the same result (quantum and thermal corrections being neglected). Allowance for the thermal fluctuations of the linear soliton similarly results in depinning (for detailed analysis see Chapter 8). This description of the depinning transition has been slightly simplified. As has been mentioned, in the phase plane in the vicinity of any trajectory corresponding to the incommensurate phase, domains of the existence of commensurate phases lie infinitely close to it. In these domains solitons are still pinned to the lattice, though with very small pinning energy. Depinning in the strict sense is possible only if one can neglect this small pinning energy and the corresponding commensurate phase domains. The thermal fluctua tions and, on top of that, quantum fluctuations in the one-dimensional crystal wipe out these commensurability domains; that justifies our approximate consideration a posteriori. At a given chemical potential the ground state of the system in the pinning phase is a periodic soliton lattice. The dependence of the period of this lattice is described by a function of the "devil's staircase" type, described in Chapter 5. In real systems, structures with long periods can be realized only at sufficiently low temperatures. The observation of transitions between them can be hampered by large diffusion times. As a result we shall experimentally observe numerous metastable states with solitons stochastically distributed over the wells of the Peierls potential at large distances one from another.
6.6. Orientational Epitaxy The experimental observations of orientational epitaxy—reorientation of the adatom lattice axes relative to the substrate ones—are described in Chapter 3. Below we present a model of the phenomenon in more detail. In the harmonic approximation, the energy of the two-dimensional lattice
213
ORIENTATIONAL E P I T A X Y
in the potential relief of the substrate can be written as v
E = \l
- ' > , ( r ' ) + Σ (r + u(r)).
Ζ
Γ,Γ'
(6.68)
Γ
Here u(r) is the displacement vector at the point r, D 0-(r — r') is the dynamical matrix, and V(r) is a periodic function. Let bx,b 2 and a 1 ?a 2 be the basis vectors of the substrate and adatom lattices, respectively. Let us start with the perturbation theory relative to V. Assuming that u(r) is small, neglect it in the argument of V(r + u(r)). The displacements can then be easily found in the momentum representation. The Fourier transforms u(q) are nonzero only for q equal to the reciprocal vectors of the substrate lattice. Let q x and q 2 denote its basis vectors. Minimization of (6.68) with respect to u(q) gives the solution uM=
(6.69)
-^'(qtaiW
where *(q) is the matrix inverse to the Fourier transform of Z)y(r), and V(q) is the Fourier transform of V(r). The energy corresponding to this solution is Δ* =
~Σ<ΐΡΓΑ^\ν(<ύ\ . 2
z
(6.70)
q
Recall that the matrix Du(q) is a periodic function with periods equal to the reciprocal lattice vectors p x and p 2 of the adatom lattice. With fixed magnitudes of the vectors p x , p 2 and qu q2 and fixed angles between p x, p 2 and q l 5q 2 , the energy (6.70) will be a function of the relative orientation of the pairs (Pi,p 2) and ( q i , q 2) . The minimum-energy condition determines this relative orientation angle. As the coverage θ changes, the vectors p t and p 2 and the matrix Z) 0(q) change too, which alters the relative orientation of the lattice and substrate as well. The case of simple triangular substrate and adatom lattices was studied by Novako and McTague (1976). Considering central forces and only the nearest-neighbor interaction, they found that the transition from the sym metric to asymmetric orientation of the substrate and adatom lattice axes should take place at certain coverage. This phenomenon was observed experimentally by Shaw et al (1978) in the system K r - G r . The case of rectangular centered substrate and triangular adatom lattices was studied by Uimin and Shchur (1978). Such a situation is realized experimentally, for example, in C s - W (110). Uimin and Shchur considered the nearest-neighbor interaction in the harmonic approximation and the lowest harmonics of the substrate potential preserving its symmetry. Their
214
INCOMMENSURATE CRYSTALS AT Γ = 0
numerical calculations show that the orientation angle changes abruptly at a certain critical value of the coverage. The values of the relative orientation angle before and after the jump agree well with the experimental results obtained by Fedorus and Naumovets (1970). Burkov (1979) has shown that with the same symmetry of the system a second-order transition is admissible too. Near the transition point the difference between the energies of the two phases is small. Thus, if the coverage is close to the critical value, the orientation of the adatom lattice can also change with temperature at fixed coverage.
6.7. Two-Dimensional Models of Incommensurate Crystals Of particular interest is the case of q values close to a vector p, of the reciprocal adatom lattice. When q -+ p„ the dynamical matrix D 0(q) vanishes a s — 2 so e Ifl Pl > *h applicability of the perturbation theory is doubtful. However, if the misfit δ = |q — p|/|p| is small, the resonance approximation formulated in previous sections for the one-dimensional case can be em ployed. In this limit the displacements of adatoms change slowly from site to site, which permits using the continuous approximation. Contrary to the onedimensional problem, different cases occur in two dimensions. We shall discuss some of them.
6.7.1. H e x a g o n a l L a t t i c e The best-known hexagonal lattices are lattices of noble gases with a simple triangular lattice on graphite. In this case q! being close to pl implies that simultaneously q 2 is close to p 2. Then, retaining only the resonance harmonics in the substrate potential, one can replace the potential with a certain function Κ(φ), where φ is the displacement of adatoms from their positions in the commensurate phase. The function Κ(φ) is a periodic one with the periods of the substrate and has a sixfold symmetry axis for the hexagonal substrate lattice. The triangular lattice is isotropic in its elasticity properties. The formula (1.17) for the elastic energy was given in Chapter 1. The displacement vector u is related to the vector φ by φ = u+
+ η 2δ 2,
(6.71)
where δ,· = a f — bi9 and ni and n2 are integers defining lattice sites. It is
TWO-DIMENSIONAL
M O D E L S OF INCOMMENSURATE
CRYSTALS
215
convenient to express these numbers in terms of the radius vector r = n^bi -f w 2b 2: 2
φ = ϋ + ' ^ - δ 1+ ^ - δ 2.
(6.72)
Vectors φ and u will be treated as functions of the continuous variable r. Notice that in (6.72) q x and q 2 have been replaced by p x and p 2. Now, substituting φ into the elastic energy (1.17) and adding j Κ(φ)ώ·, we obtain the total energy. For Κ(φ), the simplest form compatible with the substrate symmetry is Κ(φ) = K 0[3 - cos q x · φ - cos q 2 · φ - c o s ^ + q 2) · φ].
(6.73)
Minimization of the total energy with respect to φ yields the system of equilibrium equations (ij = 1 , 2 )
dridrj
or]
δφί
Some particular solutions of the system (6.74) are known. The com mensurate phase is described by the trivial solution φ = 0. It is easy to obtain solutions describing the soliton line or the striped structure of aligned solitons. However, analytic solutions, in particular for the hexagonal lattice of solitons have not been obtained as yet.
6.7.2. A n i s o t r o p i c S u b s t r a t e s Two different situations can be distinguished here. The first one was described in Chapter 1: a uniaxial incommensurate crystal, where the lattice is commensurate in one direction and incommensurate in the other. Change of variables from u to φ transforms the energy of the uniaxial crystal (1.1), (1.2) into
»-M*(£-'H(S)'*™} ™ In the case of the uniaxial crystal the initially commensurate crystal becomes incommensurable in one of the two directions. Another experiment ally observed transition occurs when such a crystal becomes incommensurate in the second direction too. Since in one of the two directions the crystal is
216
INCOMMENSURATE CRYSTALS AT 7 = 0
already incommensurate, in this direction the substrate effect can be neglec ted, and the substrate potential can be supposed to depend on one direction, i.e., on only one component of the displacement vector.
6.7.3. U n i a x i a l P e r i o d i c S o l u t i o n s In the simplest case of a uniaxial crystal with arbitrary potential Υ(φ)9 the counterpart of the equation (1.3) is 2
λ
2
δφ
Λ λ
δ φ _ άΥ(φ)
ι ΤΖϊ + 2 -ι τ - 2 = — — · ~^? + Ί ? ~ άφ
(6.76)
This equation contains no δ. We shall search for a solution of (6.76) as a function of one variable ζ = χ cos θ + .y sin 0. As a result, (6.76) will take the form familiar from the one-dimensional case: 2
ά φ _ άΥ{φ) ο-ΓΤ 2 = — τ τ ^ dz " άφ
λ
λ
2
2
ο = *2 sin θ + λχ cos θ.
(6.77)
Repeating the procedure of Section 6.2, we can readily obtain the energy per unit area, 2
U = - S + \λχδ 9
(6.78)
where & is related to δ and θ by
.
(6.79)
v W ^ t a n ^ + l Here £(
(6.80)
In view of (6.78), the energy is minimal when δ takes its maximal value. By virtue of (6.80), E(S) is a monotonically increasing function of δ9 and, as follows from (6.79), the energy per unit area is minimal at θ = 0. Thus, solution of (6.76) will depend only on the argument x. Denote the minimum of E(S) by £ c . Now, the critical value of
217
E L A S T I C SUBSTRATE
structure are readily found: (6.81) where K(S) is a counterpart of the complete elliptic integral of the first kind:
For Υ(φ) from (1.2), λί=
λ2 = λ, and / » / 0 we have (see Section 6.2) (6.82)
6.8. Elastic Substrate The formation of solitons is followed by the displacement of adatoms from the minima of the potential relief, which produces elastic deformations of the substrate. As Gordon and Villain (1979) have noticed, these deformations decay with distance by a power law, which must result in a power-law interaction of solitons. An accurate analysis of this interaction mechanism necessitates the self-consistent solution of the problem of an incommensurate crystal on an elastic substrate. For simplicity, the substrate will be supposed to have isotropic elasticity properties, with rigidity much higher than that of the overlayer (the criterion will be given presently). This corresponds to the experimental situation with overlayers on metallic substrates. In this case the substrate deformation is much smaller than that of the overlayer and can be taken into account in the overlayer-substrate interaction by the substitution u' = u — uX9 where ux is a deformation of the substrate along the OX axis. Then the elastic energy of the adatom lattice will be H = ^dxdyip
XI^J
X+2(^J
+K(u-ux + ^ x ) j ,
(6.83)
where a and b are the periods of the overlayer and substrate. Since in the considered system the solitons are aligned along the OY axis, it suffices to consider deformations of the substrate only in the XOZ plane. The unde-
218
INCOMMENSURATE CRYSTALS AT
T=0
formed surface of the substrate is ζ = 0; the bulk of the substrate is the half space ζ > 0. Then the energy of the elastic deformation of the substrate can be written as (Landau and Lifshitz 1986) dz {oxxuxx
+ azzuzz
+ 2axzuxz)
(6.84)
where aik is the stress tensor, and uik is the deformation tensor. We now have to solve the equations of two-dimensional elasticity theory subject to the boundary conditions uxx(x90) = dujdx, uxz(x90) = 0, and then to find ux as a function of x. The equations are (Landau and Lifshitz, 1986) _ i ^ + — ± i = 0, ox dz
—^- + — ^ = 0. ox dz
6.85
The solution of these equations for the Fourier transforms in the variable χ is ajq,z)
= E(l - σ Γ ^ β , Ο χ ΐ - |«|z)exp(-\q\z\
aZ2(q9z)
= E(l - σ Γ ^ β , Ο χ ΐ + \q\z)^-\q\z)9
(6.86)
ayHqzuxx(q90)Qxp(-\q\z).
Here Ε is Young's modulus, and σ is the Poisson coefficient. Substituting (6.86) into the energy (6.84) and integrating over z, we can obtain the energy of the elastic deformation of the substrate induced by interaction with the overlayer: He = E(l-a)~
2
\dy\dq'^t.
(6.87)
Making the standard substitution φ = u — ux + (b — a)x/a and minimizing the elastic energy at fixed φ9 we readily obtain the final formula for the elastic energy of the adatom lattice adsorbed on the deformable substrate (compare with (6.75)):
219
E L A S T I C SUBSTRATE
where 2
λ0 =
4Ε^-\1-σ)- 9 2
g(x - χ') = λ\
~ σ) 2 4E(x - x')|2*
The principal feature of this formula (apart from the presence of the nonlocal term) is a renormalization of all the parameters of the overlayer Hamiltonian due to the substrate deformation. As follows from the above formula for renormalization of the elastic modulus of the overlayer, the rigid-substrate criterion is λ0 » λν In the first approximation in λχ/λ0 the nonlocal energy of the substrate-mediated interaction can be estimated by substituting the solution of the equation (6.76) for φ (with the renormalized constants λ\ and δ'). If the spacing / is large, / » / 0, the integration over all χ and χ' in (6.88) can be replaced by an integration over one period and a summation over all solitons. As a result, we obtain the contribution of the interaction of solitons to the energy density:
The power law (6.89) and, particularly, the specific value of the exponent are of importance in the thermodynamics of the soliton lattice. One striking feature of the interaction mediated by the elastic deformations of the substrate is its independence of the amplitude of the substrate potential (see (6.89)). This property was noticed by Lyuksyutov (1982) and Talapov (1982). The closely related problem of the interaction of steps on a crystal surface mediated by its elastic deformations was considered by Lau and Kohn (1977), Gordon and Villain (1979), and Marchenko and Parshin (1980). The energy 2 of the interaction of steps decreases with increasing spacing like Γ .