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
theta-fig3.gif

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,

bf.gif

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,

                   Eqn08.gif

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

 

f06-05.gif

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

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

a2ωPN/2π
  if   
ηl < η < ωPN
a2ωPN2/2πη
  if 
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

 

anharmak.gif

owing to anharmoncity of Vint(x)

mantikink > mkink

εantikink < εkink

εPN antikink > εPN kink

anharmk.gif

 

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

 

f05-17.gif

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

scalar1a.gif

scalar1b.gif

 

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

scalar2.gif

 

      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)

Hint

{u1, u2} = as–1

dx {–α[1 – cos(u1u2)] + γu1u2′}

where

α ≈ –

2Vint(x)


x2

      and      γ

dx


as

Vint

(x)

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

.
X
 

2
n

+ 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

furrowed.gif

(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)

scalar2a.gif

(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

scalar2b.gif

(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

scalar2c.gif

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

The secondary FK model goto top

second.gif

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

fk-zig1.gif

fk-zig2.gif
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)

 

 

 

 

 

 

f09-09.gif

trivial GS

V < Vbif (k)

 

 

 

 

 

zig-zag GS

V > Vbif

f09-08.gif
"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

f09-12.gif

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):

f09-15.gif

 

dzig.gif

 

Exchange-mediated diffusion mechanism goto top

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

exchange1.gif

 

complicated exchange diffusion:  DDkink ρkink

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

 

exchange2a.gif

 

exchange2b.gif

 

exchange2c.gif

 

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