Bhattacharjee J.K., Bhattacharyya S. Non-Linear Dynamics Near and Far from Equilibrium
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7.5 KPZ Equation and Mode Coupling Theories 203
7.5 KPZ Equation and Mode Coupling Theories
In this section, we show how to treat the KPZ equation within the self consistent
mode coupling scheme shown in Chapter 4. To do this we note that the Green’s
function G(k, ω) can be written as
δh(
k, ω)
δη(
k
,ω
)
1
δ(
k +
k
)δ(ω +ω
)
or as the correlation
h(
k, ω)η(
k
,ω
)
2Nδ(
k +
k
)δ(ω +ω
)
From the expansion written down following Eq.(7.4.14), we find in the expansion,
G(k, ω) = G
0
(k, ω) +λG
1
(k, ω) +λ
2
G
2
(k, ω) +...
G
−1
0
(k, ω) =−iω +νk
2
G
1
(k, ω) = 0
G
2
(k, ω) =
1
2
1
−iω +νk
2
p.(
k −p)
−iω
1
+νk
2
q.( p −q)h
0
( p −q,ω
1
−ω
2
)
×h
0
(q,ω
2
)h
0
(
k −p, ω −ω
1
)η(k
,ω
)
=−
1
2
1
−iω +νk
2
p.(
k −p)
k.(
k −p)G
0
( p, ω
1
)
×C
0
(
k −p, ω −ω
1
)
1
−iω +νk
2
= G
0
(k, ω)
p,ω
1
p.(
k −p)
k.(
k −p)G
0
( p, ω
1
)
×C
0
(
k −p, ω −ω
1
)
G
0
(k, ω)
=−G
0
(k, ω)(k, ω)G
0
(k, ω)/λ
2
(7.5.1)
where
(k, ω) = λ
2
d
D
p
(2π)
D
dω
1
2π
p.(
k −p)
k.(
k −p)G
0
( p, ω
1
)
×G
0
( p, ω
1
)C
0
(
k −p, ω −ω
1
) (7.5.2)
The full Green’s function has the form
204 7 Surface Growth
G = G
0
−G
0
G
0
+.........
= G
0
(1 −G
0
+.........)
= G
0
(1 +G
0
)
−1
⇒ G
−1
= G
−1
0
+ (7.5.3)
The self consistent approximation involves replacing the G
0
and C
0
in the expres-
sion for by the dressed quantities G and C and we have (as in Chapter 3)
G
−1
(k, ω) =−iω +νk
2
+λ
2
d
D
p
(2π)
D
dω
1
2π
p.(
k −p)
k.(
k −p)
×G( p, ω
1
)C(
k −p, ω −ω
1
) (7.5.4)
We now turn to the correlation function C(k,ω) and write down the corresponding
expansion for it.
C(k,ω) =C
0
(k, ω) +λC
1
(k, ω) +λ
2
C
2
(k, ω) +... (7.5.5)
where
C
0
=
2N
ω
2
+ν
2
k
4
=2NG
0
(k, ω)G
0
(−k, −ω) (7.5.6)
C
1
=h
1
(k, ω)h
0
(−k, −ω)=0 (7.5.7)
C
2
= 2h
2
(k, ω)h
0
(−k, −ω)+h
1
(k, ω)h
1
(−k, −ω) (7.5.8)
We note that the first term on the R.H.S of Eq.(7.5.8) dresses the term G(k, ω) in
Eq.(7.5.6). It is the second term on the R.H.S of Eq.(7.5.8), that dresses the noise
term and the mode coupling approximation rests on the expansion
C(k,ω) =
2N +
λ
2
2
d
D
p
(2π)
D
dω
1
2π
[p.(
k −p)]
2
×C
0
(p, ω
1
)C
0
(
k −p, ω −ω
1
)
|G
0
(k, ω)|
2
(7.5.9)
Once again the self-consistent approximation involves replacing G
0
by G and C
0
by C, so that
C(k,ω) =
2N +
λ
2
2
d
D
p
(2π)
D
dω
1
2π
[p.(
k −p)]
2
×C(p,ω
1
)C(
k −p, ω −ω
1
)
|G(k, ω)|
2
(7.5.10)
Now returning to Eq.(7.5.4), we note that the self energy dresses the diffusion rate
ν and if z<2, then the self energy dominates νk
2
for small wave numbers and we
can write Eq.(7.5.4) as
7.5 KPZ Equation and Mode Coupling Theories 205
G
−1
(k, ω) =−iω +λ
2
d
D
p
(2π)
D
dω
1
2π
p.(
k −p)
×
k.(
k −p)C(p, ω
1
)C(
k −p, ω −ω
1
) (7.5.11)
The equal time coordinate space correlation function scales as r
2α
and hence
h(k, t)h(−k, t) scales as k
−D−2α
which implies that h(k, ω)h(−k, −ω)scales
as k
−D−2α−z
. Looking at the second term on the bracketed part of the R.H.S. of
Eq.(7.5.10), it scales as k
−D+2α−z
and for small k will dominate the first term so
long as D +2α>z. In this situation, we will have
C(k,ω) =|G(k, ω)|
2
λ
2
2
d
D
p
(2π)
D
dω
1
2π
[p.(
k −p)]
2
×C(p,ω
1
)C(
k −p, ω −ω
1
) (7.5.12)
The mode coupling approximation is defined through the coupled integralequations
for C(k,ω) and G(k, ω) in Eqns.(7.5.11) and (7.5.12). These equations can also be
viewed as the spherical limit of a N-component model defined by treating h(x, t)
as a N-component vector in the manner shown in Chapter 8.
To exploit these equations fully, we proceed to make a scaling Ansatz for
the Green’s function and the correlation function. The Green’s function has the
dimension of the relaxation rate and we can write
G
−1
(k, ω) =−iω +(k, ω)=−iω +k
z
¯
f
ω
k
z
= γk
z
f
ω
k
z
(7.5.13)
The correlation function has the form
C(k,ω) =Ck
−(D+2α+z)
g
ω
k
z
(7.5.14)
Using Eq.(7.5.11) at zero frequency, we get
(
k, 0) = λ
2
d
D
p
(2π)
D
dω
1
2π
[p.(
k −p)][
k.(
k −p)]
×G( p, ω
1
)C(
k −p, −ω
1
)
which leads to
¯
f(0) = C
λ
2
2
d
D
p
(2π)
D
dω
1
2π
[p.(
k −p)][
k.(
k −p)]
p
z
|
1 −p|
D+2α+z
×f
ω
1
p
z
g
ω
1
|
1 −p|
z
(7.5.15)
206 7 Surface Growth
while from Eq.(7.5.12) under the same conditions
C(
k, 0) =|G(k)|
2
λ
2
2
d
D
p
(2π)
D
dω
1
2π
[p.(
k −p)]
2
C( p, ω
1
)C(
k −p, −ω
1
)
(7.5.16)
leading to
g(0) = C
λ
2
2
2
d
D
p
(2π)
D
dω
1
2π
[p.(
1 −p)]
2
p
D+2α+z
|
1 −p|
D+2α+z
×g
ω
1
p
z
g
−ω
1
|
1 −p|
z
(7.5.17)
It should be noted that in Eqns.(7.5.15) and (7.5.16), both ω
1
and p are scaled
variables, scaled by k
z
and k respectively. These equations would yield solutions
for f(x)and g(x) if α, z and λ are known. More interesting would be to treat α and
z and λ as unknowns and make a reasonable Ansatz about f(x)and g(x). The ob-
vious candidate for the frequency dependent shape is Lorentzian. This corresponds
to
¯
f(x)=1 and g(x) =(1 +x
2
)
−1
. Under these approximations, Eq.(7.5.15) be-
comes,
1 =C
λ
2
2
2
d
D
p
(2π)
D
[p.(
1 −p)][
1.(
1 −p)]
|
1 −p|
D+2α
(p
z
+|
1 −p|
z
)
(7.5.18)
while Eq.(7.5.16) leads to
1 =C
λ
2
4
2
d
D
p
(2π)
D
[p.(
1 −p)]
2
|
1 −p|
D+2α
p
D+2α
1
p
z
+|
1 −p|
z
(7.5.19)
Eliminating C
λ
2
2
from Eqns.(7.5.17) and (7.5.18), we get
d
D
p
(2π)
D
[p.(
1 −p)]
2
|
1 −p|
D+2α
p
D+2α
×
1
p
z
+|
1 −p|
z
= 2
d
D
p
(2π)
D
[p.(
1 −p)][
1.(
1 −p)]
|
1 −p|
D+2α
(p
z
+|
1 −p|
z
)
(7.5.20)
The above relationship gives a connection between α, z and D and using α +z =2
which is a consequence of Galilean invariance, we can determine α and z separately.
One of the interesting issues is at what D does α become 0. We note that the R.H.S of
Eq.(7.5.19). has a logarithmic divergence at α =0. We explore this fact to evaluate
the integrals as expansions about α =0 and retain the first term only. The first
term involves evaluating the integrals in the high momentum limit i.e. p 1 and
extracting the appropriate ‘pole’ terms.
7.6 Growth with Surface Diffusion 207
The integrand on the L.H.S of Eq.(7.5.19) is approximately
p
4
2p
2D+4α+z
=
1
2p
2D+3α−2
.
The divergence occursif D=2 and α =0 and hence in the leading log approximation,
we can write the R.H.S. as
K
D
2(D+3α−2)
. The integral on the R.H.S is more tricky. It
is not symmetric in p and
1 −p and hence to correctly extract the ’leading log’,
one needs to work with the symmetric form which can be done by adding to the
integral, the one with p and
1 −p interchanged and dividing by two. Alternatively,
one can work with the symmetric variables
1
2
−p and
1
2
+p instead of p and
1 −p. The integrand becomes
α
2D
1
p
D+α
and the integral in the ’pole’ approximation
becomes
K
D
2D
, where K
D
=
S
D
(2π)
D
, S
D
being the surface area of the D-dimensional
hypersphere. We now have from Eq.(7.5.19)
[2(D +3α −2)]
−1
= D
−1
or α =
4 −D
6
(7.5.21)
This shows that there is no rough phase for D>4 or the upper critical dimension
is 4. The upper critical dimension is controversial. A set of authors have obtained
4 as the upper critical dimension. Incidentally, Eq.(7.5.20) is exact in D =1.
7.6 Growth with Surface Diffusion
We now consider a variant of the KPZ problem, which is relevant for the growth of
crystals by a process of deposition. In this case there has to be a conservation law for
the number of particles. The atoms fall on a substrate and the crystal grows over the
region covered by the atomic beam. The random fluctuations in the beam intensity
would make the surface rough, but surface diffusion is capable of smoothening out
holes and bumps. This is important. If we imagine the growth process occurring
on a substrate of dimension R (in atomic units) then for a mean height h, the
number of atoms deposited is proportional to R
2
h. For random fluctuation in the
beam intensity the fluctuation in the mean number is
√
r
2
h and this will reflect in a
fluctuation δh in h in the absence of any surface diffusion. Consequently, δh ∼
√
h
R
.
For R h 100 atomic units, δh ∼0.1 atomic unit, which is significant. Hence,
the rate of surface diffusion is worth studying.
The conservation law for particles will have the structure
˙
h +
∇.
j =0 (7.6.1)
where the current is the gradient of a chemical potential and can be written as
j =
∇µ. The chemical potential will be determined by the propensity of the particle
208 7 Surface Growth
to diffuse which will be controlled by the local curvature. The local curvature will be
determined by ∇
2
h and hence
j =κ
∇∇
2
h, where κ is a constant. Thus Eq.(7.6.1)
becomes
˙
h =−κ∇
4
h (7.6.2)
We now need to add the effect random fluctuations and we have the stochastic
growth law,
˙
h =−κ∇
4
h +f (7.6.3)
where
f(r,t)f(
r
,t
)=2Nδ(r −
r
)δ(t −t
) (7.6.4)
The above growth equation is known as the Mullin’s-Sekerka equation. Scaling
distances by b and t by b
z
, so that
x →x
= bx
t →t
= b
z
t
h →h
= b
α
h
and f →f
= b
µ
f
the primed variables satisfy
∂h
∂t
=−κb
4−z
∇
4
h
+b
−z+α−µ
f
(7.6.5)
where
b
−2µ
f(
r
1
,t
1
)f (
r
2
,t
2
)=2b
D+z
Nδ(
r
1
−
r
2
)δ(t
1
−t
2
) (7.6.6)
The noise is unchanged in character if
2µ +D +z =0 (7.6.7)
and Eq.(7.6.5) remains the same as Eq.(7.6.3) if
z = 4
and α = z +µ =4 −(
4 +D
2
) =2 −
D
2
(7.6.8)
Thus we find that for substrate dimensions less than 4, the interface is rough.
The same results can be obtained by writing Eq.(7.6.3) in momentum space as
˙
h(k, t) =−κk
4
h(k, t) +f(k,t)
The propagator G(k, ω) is (−iω +k
4
)
−1
and the correlation function
7.6 Growth with Surface Diffusion 209
C(k,t
21
) =h(k, t
2
)h(−k, t
1
)
= e
−κk
4
(t
2
+t
1
)
t
2
0
dt
t
1
0
dt
e
κk
4
(t
2
+t
1
)
f(k,t
)h(−k, t
)
= e
−κk
4
(t
2
+t
1
)
t
2
0
dt
t
1
0
dt
2Ne
κk
4
(t
2
+t
1
)
δ(t
−t
)
=
N
κ
1
k
4
e
−k
4
|t
1
−t
2
|
(7.6.9)
In real space,
C(r
12
,t
12
) =
N
κ
d
D
k
(2π)
D
e
i
k. r
12
k
4
e
−k
4
|t
1
−t
2
|
= r
4−D
12
f
t
12
r
z
12
(7.6.10)
which when compared with the scaling form
C(r
12
,t
12
) =r
2α
12
f
t
12
r
z
12
shows that α =2 −
D
2
and z =4, as in Eq.(7.6.8)
We now ask the question about the nonlinear terms in Eq.(7.6.3). Recalling,
that the structure of the deterministic term in Eq.(7.6.3) is ∇
2
µ, we note that in the
dependence of µ on derivatives of h,
∇h did not figure because the sign of the slope
cannot determine the probability of falling into or escaping from that local region.
However, when nonlinear terms are allowed, (
∇h)
2
is the obvious candidate for
the most relevant nonlinear term. Accordingly, the nonlinear equation for growth
with surface diffusion becomes
∂h
∂t
=−κ∇
4
h +λ∇
2
(
∇h)
2
+f (7.6.11)
Carrying out the scalings
x →x
= bx
t →t
= b
z
t
h →h
= b
α
h
and f →f
= b
µ
f
we have
∂h
∂t
=−κb
4−z
∇
4
h
+λb
4−α−z
∇
2
(
∇
h
)
2
+b
−z+α−µ
f
(7.6.12)
As in the KPZ equation, the exponents α and z can no longer be determined by
requiring scale invariance. Instead one has to argue that the relaxation rate κ,
210 7 Surface Growth
the coupling constant λ and the noise correlator N will renormalize under scale
transformation. In keeping with the KPZ case, if the coupling constant does not
renormalize (the extra ∇
2
in the interaction term removes the guarantee that there
will be no renormalization of the coupling constant in this case either) we have
from Eq.(7.6.12)
α +z =4 (7.6.13)
We now provide a simple argument for deriving α. Let us imagine the formation of
a bump or a hole of lateral dimension R and height h. The gradients in Eq.(7.6.11)
are in the substrate and hence they will be proportional to R
−1
.Ifτ is the time
to form the bump, then the action of the deterministic part of Eq.(7.6.11) yields τ
from
h
τ
∼
κh+λh
2
R
4
(7.6.14)
Turning to the stochastic part, the mean number of particles falling over an area
R
D
in time τ is
¯
fτR
D
. The fluctuation in this number gives rise to the bump which
is R
D
h. Thus
[
¯
fτR
D
]
1/2
∼ R
D
h
yielding τ ∼R
D
h
2
(7.6.15)
Combining with Eq.(7.6.14), we find
R
4
κ +λh
∼R
D
h
2
If λ =0, then h ∼R
4−D
2
, giving α =2 −
D
2
as in Eq.(7.6.8) If the nonlinear term
dominates i.e. κ ≈0, then (provided λ has no dependence)
α =
4 −D
3
with z = 4 −α =
8 +D
3
(7.6.16)
We now explore the effect of carrying out a RG treatment on this model. The
equations of motion in momentum space are
˙
h(
k, t) =−κk
4
h(
k, t) +λ
p
k
2
p.(
k −p)h( p, t)h(
k −p, t) +f(
k, t) (7.6.17)
with
f(
k
1
,t
1
)f (
k
2
,t
2
)=2Nδ(
k
1
+
k
2
)δ(t
1
−t
2
) (7.6.18)
7.6 Growth with Surface Diffusion 211
Splitting h(
k, t) into h
<
(
k, t) and h
>
(
k, t) as before and working in frequency
space
−iωh
<
(
k, ω) =−κk
4
h
<
(
k, ω)+λ
[k
2
p.(
k −p)]{h
<
( p, ω
1
)
×h
<
(
k −p, ω −ω
1
) +2h
>
( p, ω
1
)h
<
(
k −p, ω −ω
1
)
+h
>
( p, ω
1
)h
>
(
k −p, ω −ω
1
)}+f
<
(
k, ω) (7.6.19)
and
−iωh
>
(
k, ω) =−κk
4
h
>
(
k, ω)+λ
[k
2
p.(
k −p)]{h
<
( p, ω
1
)
×h
<
(
k −p, ω −ω
1
) +2h
>
( p, ω
1
)h
<
(
k −p, ω −ω
1
)
+h
>
( p, ω
1
)h
>
(
k −p, ω −ω
1
)}+f
>
(
k, ω) (7.6.20)
The task is to solve h
>
(
k, ω) perturbatively from Eq.(7.6.20) and insert in
Eq.(7.6.19) to obtain an effective equation of motion for h
<
(
k, ω). We expand
h
>
(
k, ω)=h
(0)
>
(
k, ω)+h
(1)
>
(
k, ω)+h
(2)
>
(
k, ω)+...... (7.6.21)
and find
h
(0)
>
(
k, ω)=
f
>
(
k, ω)
−iω +κk
4
(7.6.22)
(−iω +κk
4
)h
(1)
>
(
k, ω) =
[k
2
p.(
k −p)]{h
<
( p, ω
1
)h
<
(
k −p, ω −ω
1
)
+2h
<
( p, ω
1
)h
(0)
>
(
k −p, ω −ω
1
)
+h
(0)
>
( p, ω
1
)h
(0)
>
(
k −p, ω −ω
1
)} (7.6.23)
and
(−iω +κk
4
)h
(2)
>
(
k, ω) =
[k
2
p.(
k −p)]{h
<
( p, ω
1
)h
(1)
>
(
k −p, ω −ω
1
)
+2h
(1)
>
( p, ω
1
)h
(0)
>
(
k −p, ω −ω
1
)} (7.6.24)
Up to O(λ
2
) Eq.(7.6.19) becomes
−iωh
<
(
k, ω) =−κk
4
h
<
(
k, ω)+λ
[k
2
p.(
k −p)]
×{h
<
( p, ω
1
)h
<
(
k −p, ω −ω
1
)}
+2λ
[k
2
p.(
k −p)]h
(0)
>
( p, ω
1
)h
<
(
k −p, ω −ω
1
)
+λ
p,ω
1
[k
2
p.(
k −p)]h
(0)
>
( p, ω
1
)h
(0)
>
(
k −p, ω −ω
1
)
+2λ
2
[k
2
p.(
k −p)]h
(1)
>
( p, ω
1
)h
<
(
k −p, ω −ω
1
)
+2λ
2
[k
2
p.(
k −p)]h
(1)
>
( p, ω
1
)h
(0)
>
(
k −p, ω −ω
1
) +f
<
(7.6.25)
212 7 Surface Growth
Substituting from Eqns.(7.6.23) and (7.6.22) for h
(1)
>
and h
(0)
>
, we average over f
>
.
The nonvanishing term of interest is
= 4λ
2
p,ω
1
q,ω
2
[k
2
p.(
k −p)][p
2
q.( p −q)]
−iω +κp
4
h
<
( p, ω
2
)
×h
(0)
>
(
k −p, ω −ω
1
)h
(0)
>
( p −q,ω
1
−ω
2
)
= 4λ
2
p,ω
1
q,ω
2
N[k
2
p.(
k −p)][p
2
q.( p −q)]
−iω +κp
4
×
δ(
k −q)δ(ω −ω
2
)
[(ω −ω
1
)
2
+κ
2
(
k −p)
8
]
=−8λ
2
N
p,ω
1
[k
2
p.(
k −p)][p
2
q.( p −q)]
(−iω +κp
4
)[(ω −ω
1
)
2
+κ
2
(
k −p)
8
]
=−4λ
2
N
p
[k
2
p.(
k −p)][p
2
q.( p −q)]
(−iω +κp
4
)[−iω +κp
4
+κ(
k −p)
4
]
=−
2λNk
2
κ
p
p.(
k −p)
−iω +κp
4
+κ(
k −p)
4
×[
k.(
k −p)p
2
(
k −p)
4
+(
k −p)
2
(
k. p)
p
4
] (7.6.26)
To this order, Eq.(7.6.25) can be cast as
−iωh
<
(k, ω) =−k
4
(κ +δκ)h
<
(k, ω) +f
<
(k, ω)
+λ
p
[p.(
k −p)]h
<
( p, ω
1
)h
<
(
k −p, ω −ω
1
)
where
δκ(k, ω) =
2
k
2
λ
2
N
κ
2
d
D
p
(2π)
D
p.(
k −p
−iω +κ[p
4
+(
k −p)
4
]
×[
k.(
k −p)p
2
(
k −p)
4
+(
k −p)
2
(
k. p)
p
4
] (7.6.27)
The low frequency, low momentum contribution to δκ is
δκ(k, ω)
2
k
2
λ
2
N
κ
2
b
d
D
p
(2π)
D
6(
k. p)
2
−p
2
k
2
p
2
2p
4
=
λ
2
N
κ
2
6
D
−1
b
d
D
p
p
4
(7.6.28)