Generalizations of the Frenkel-Kontorova model

Mobility & diffusivity goto top

Mobility. Mass or charge transport in one-dimensional anharmonic lattices is described by the flux J = ρBF,  where ρ  is the density, B  is the mobility, and F is the driving force (F → 0).

In the FK model, the mobility has minima at rational (commensurate) GS configurations

Figure: simulation results for the mobility B versus the coverage for the 1D model with V0=100 at different temperatures: T=0.005 (diamonds and dotted line), T=0.05 (asterisks and dashed line), T=0.1 (triangles and solid line). The dash-triple-dotted line is for the perturbative approach at T = 0.1,


Chemical diffusion.  Definition:    image001.gif

where <<...>> stands for average over mesoscopic distances.

Method  I:  Use the Einstein relation  Dc = kBTB / χ,  where  χ  is the (dimensionless) static susceptibility,


μ is the chemical potential, n is the atomic concentration, and Π is the pressure



Then we may use the ideal kink-gas ideology, where  χ  is given by

image002.gif,  andimage003.gifimage004.gif


Method IIKink diffusion ≈ chemical diffusion.  Idea:

Dc = flux of atoms

gradient of atoms
flux of kinks

gradient of kinks
= Dkink +
small corrections due to
kinkkink interaction
Proof:   ul(t) = ul(phonon)(t)  + Ntot

j (kinks) 
ul(kink)(laXj  Dc θkDk + θakDak

Therefore, at low T  (kBT << εPN) we have  Dkink= Dk0 exp(–εPN /kBT),  where

ηl < η < ωPN
η > ωPN
and for high T  (kBT >> εPN) we obtain  Dkink≈ (kBT /mkη)[1 – (1/8)(εPN/kBT )2]


Simulation results for exponential interaction (see below):

              PN barrier vs concentration Dc  versus concentration
theta-fig1.gif theta-fig8.gif


More details may be found in the following papers (pdf files may be found here):

The anharmonic FK model goto top

A. kink ≠ antikink

d2ul/dt2 + sin ul + V′(a+ulul–1) – V′(a+ul+1ul) = 0

continuum approximation:  xl(t) = la + ul(t),  lx=laul(t) → u(x,t)

anharmonic interaction:  V′(a+u) ≈ V′(a) + V′′(a) u + ½V′′′(a) u2

utt + sin uuxx(1 – αux) = 0,  d2 = a2V′′(a),  α = – (a/d)V′′′(a)/V′′(a)

u(x) = uSG(x) + Δu(x),  Δu(x) ≈ (4α/3) tanh–1sinh(x)/cosh(x)

m ≈ (8/ad)[1 (π/6)α],  EPN = EPN0 δEPN



owing to anharmoncity of Vint(x)

mantikink > mkink

εantikink < εkink

εPN antikink > εPN kink



B. PN barriers – Devil's staircase:  for any rational concentration θ0 = p/q, jump in the PN barrier:

EPN(θ0–0) – EPN(θ0+0) = δEPN > 0



See also the papers (pdf files may be found here):

Two-dimensional scalar FK model goto top

Two coupled FK chains goto top

Requirements on Hint{u1(x), u2(x)}:

(a)  for |u1(x) – u2(x)| << as  it must be  Hint{u1,u2} ~ ∫dx [u1(x) – u2(x)]2


(b)  Hint = inv  at  u2u2 + as

old configuration                                                               new configuration




(c)  Hint ≠ 0  if  u1(x) ≡ u2(x) ≠ const



      Hint{u1,u2} ~ ∫dx dxu1′(x) Vint(xx′) u2′(x′),  because  ρ(x) = –as–1u′(x)  is the density of "extra" atoms

      Local approximation: Vint(x) → δ(x)


{u1, u2} = as–1

dx {–α[1 – cos(u1u2)] + γu1u2′}


α ≈ –



      and      γ





repulsion: α, γ > 0;     attraction: α, γ > 0


2D system of coupled FK chains goto top

anzats:  un(x, t) = uSG(x; Xn(t)) = 4 tan–1exp{–[xXn(t)]/d}

Effective Hamiltonian:

Heff = n

ε + 1/2m



+ V(XnXn–1) + 1/2εPN(1 – cos Xn)

 where  ε = md2  is the X-kink rest energy

            m = 4/πd  is the X-kink rest mass

            εPN  is the amplitude of PN barrier for X-kinks


Interaction between the kinks:  V(X) = –ε[αW1(X/d) + γW2(X/d)],  where

            W1(Y) = (1 + Y/sinhY) tanh2Y    and    W2(Y) = (1 – Y/sinhY)

small displacement (|X| << as):  V(X) ≈ ½GX 2,  where G = –m(α + (⅓)

large shift (X → ∞):  V(∞) = –ε(α + γ) = Edissociation


2D structures of kinks goto top


(a) If  G < 0  (repulsion between the adatoms,  α,γ > 0) and  Ediss < 0,  then repulsion between X-kinks:

c(2x2) structure of X-kinks

εA ≈ εPN(Y)< εPN  so that  D > D0


Cs or K on W or Mo(112)


(b) If  –3α < γ < –α < 0  then  G < 0  and  Ediss > 0:

oblique Y-chains of X-kinks

D > D0  if  x0as(n+½)

D < D0  if  x0asn  (n  is integer)


Li or Na on W or Mo(112)  at   0.5 < θ << 1


(c) If  G > 0  (α + ⅓γ < 0)  then X-kinks attract each other:

Y-chains of X-kinks

εA ≈ εPN + (EY + EY)  or  εA ≈ εPN + Ediss


Li on W or Mo(112)  at  θ ~1


See also the following papers (pdf files may be found here):

The secondary FK model goto top


Parameters of Y-kinks:

width:  DY ≈ b(2G/εPN)1/2

Y-kink energyEY ≈ π2G  if  DY << b

      ≈ (8/3)[–2πd(3α+γ)]1/2exp(–πd/4)  if  DY >> b

PN barriersEPN(Y) ≈ εPN – π2G  if  DY << b

            ≈ (16/3)π4G exp[–π2(2G/εPN)1/2]  if  DY >> b

The zig-zag FK model goto top

The model with a transverse degree of freedom goto top


trivial GS for Vint(as) < Vbif fk-zigBa.gif
zigzag GS for Vint(as) > Vbif fk-zigBb.gif

Example: for Coulomb repulsion   as2ωy2 + 4Vint′(as) = 0,  Vbif ≈ 0.25ωy2aA3


Existing of a transversal degree of freedom modifies

and leads to predictions

Aubry transitions goto top

displacements from
nearest minima of  Vsub(x)
average elastic constant
g = N–1 N

l,l′=1 (ll′)
2Vint (rlrl+l)/∂xl2


reverse Aubry transition

f = 10–4 along x

xl0 = displacements at  f = 0

xl = displacements at  f = 10–4

Δxshift = N–1 ∑(xlxl0)

c = Δxshift / f

f09-06a.gif f09-06b.gif f09-06c.gif

Competition:  Aubry VAubry= f(θ)  ↔  Vbif ~ ωy

(a) ωy < ω*gbif < gAubry ,  no Aubry transition

(b) ω* < ωy < ω**,  gmin< gAubry< gbif ,  normal/reverse/normal Aubry transitions

(c) ωy > ω**,  gAubry< gmin= min g(V),  normal Aubry transition

Kinks goto top

  "massive" antikink (vacancy)   "massive" kink (extra atom)




(m.e. cnf)








trivial GS

V < Vbif (k)






zig-zag GS

V > Vbif

"nonmassive" (phase) kink f09-10.gif min-en. cnf
saddle cnf

The case of  Vbif (k)< V < Vbif :  the bifurcation begins at the kink's core


Peierls-Nabarro barriers goto top


     utt + sin u d2uxx(1 – αdux) = 0

trivial GS:  kink width  d0= (2π)(g + 4g1)1/2α0= –(2π/d0)3[V ′′′(r0) + 8V ′′′(r1)] > 0

zigzag GS:  kink width  d = (2π)(4g1– ¼ ω2)1/2,  α = – d –3[(4π)3V ′′′(r1) + (π4/b2)(ω2+4g)] < 0

     d << d0


Simulation results for exponential interaction (θ=1, V0=200=fixed, ωy is varied):





Exchange-mediated diffusion mechanism goto top

conventional surface diffusion:  DD0 exp[–(EsaddleEm.e.)/kT]



complicated exchange diffusion:  DDkink ρkink

      where  DkinkkT /mkinkηkink  and  ρkink= C exp[–(EkinkEm.e.)/kT]








The zigzag FK model was introduced by Braun and Kivshar (PRB 44 (1991) 7694) and then studied in details in the series of papers (pdf files may be found here):

  • Oleg Braun, Maxim Paliy, and Bambi Hu, Phys. Rev. Lett. 83 (1999) 5206 "Fuse safety device on an atomic scale"
  • O.M. Braun, O.A. Chubykalo, Yu.S. Kivshar, and T.P. Valkering, Physica D 113 (1998) 152 "The Frenkel-Kontorova model with a transverse degree of freedom: kinks structures"
  • O.M. Braun, Thierry Dauxois, and M. Peyrard, Phys. Rev. B 54 (1996) 313 "Solitonic-exchange mechanism of surface diffusion"
  • O.M. Braun, O.A. Chubykalo, and T.P. Valkering, Phys. Rev. B 53 (1996) 13877 "Structure of kinks for a complex ground state"
  • O.M. Braun and M. Peyrard, Phys. Rev. B 51 (1995) 17158 "The Frenkel-Kontorova model with a nonconvex transverse degree of freedom: a model for reconstructive surface growth"
  • O.M. Braun and M. Peyrard, Phys. Rev. E 51 (1995) 4999 "Ground state of the Frenkel-Kontorova model with a transverse degree of freedom"
  • O.M. Braun, O.A. Chubykalo, and L. Vzquez, Phys. Lett. A 191 (1994) 257 "Dimerized ground state of the Frenkel-Kontorova model with a transversal degree of freedom"
  • O.M. Braun, O.A. Chubykalo, Yu.S. Kivshar, and L. Vzquez, Phys. Rev. B 48 (1993) 3734 "Frenkel-Kontorova model with a transversal degree of freedom: Static properties of kinks"
  • O.M. Braun and Yu.S. Kivshar, Phys. Rev. B 44 (1991) 7694 "Zigzag kinks in the Frenkel-Kontorova model with a transversal degree of freedom"

2D vector FK model (the springs & balls model) goto top

2D substrate potential 2D lattice of atoms
f09-16a.gif f09-16b.gif


Next: The driven FK model goto top

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Last updated on October 5, 2008 by Oleg Braun.                             Translated from LATEX by TTH