Free-fermion solution for overall equilibrium crystal shape

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Free-fermion solution for overall equilibrium crystal shape

L. Mikheev, V. Pokrovsky

To cite this version:

L. Mikheev, V. Pokrovsky. Free-fermion solution for overall equilibrium crystal shape. Journal de

Physique I, EDP Sciences, 1991, 1 (3), pp.373-382. �10.1051/jp1:1991139�. �jpa-00246337�

J.

Phys.

I1

(1991)

373-382 MARS 1991, PAGE 373

Classification

Physics

Abstracts

68.40 68.42

Free-fermion solution for overall equilibrium crystal shape

L. V. Mikheev

(I, *)

and V. L.

Pokrovsky

_(~,3)

(')

Institute of

Crystallography

of the

Academy

of Sciences U-S-S-R-, Leninski

Prosp.59,

Moscow 117333, U-S-S-R-

~)

Landau Institute for Theoretical

Physics,

Ul.

Kosygina

2, Moscow 117940, U.S.S.R.

(3) Institut fur

Festk6rperforschung, Forschungszentrum

Jfilich

GmBH,

Postfach1913 D-5170 Jfilich, Germany.

(Received16

July 1990,

accepted

in

final form

8 November 1990)

Abstract. We

generalize

the random walk _or free-fermion method of Yamamoto, Akutsu and Akutsu to obtain _a

simple explicit

solution for the overall

equilibrium crystal shape

of a

simple

cubic

crystal

at temperatures, T, low with respect to the nearest~neighbour

coupling,

E. The thermal

rounding

of the _comers and of the

edges

of the cube _{appears to} be

qualitatively

different _:

at the _comers the width of the rounded, vicinal surface and its radius of curvature are estimated to

be of order

(k~ TIE)

R, where 2 R is the diameter of the

crystal.

As the distance I _{from the}

comer

along

_an

edge

increases, the width of the vicinal surface and transverse radius of curvature decrease _as

exp(- ik~ _{TIER) reaching}

_an

exponentially

small value at the middle of the

edge.

On the other hand, the radius of curvature

parallel

to an

edge

_{grows as exp}

(ik~ TIER),

_so that the product of _two

principal

curvatures remains constant up ^to a numerical factor. A

qualitative explanation

of the results is

presented,

based _on the strong

dependence

of the stiffness of the steps

on their orientation with respect to the _axes of the

crystal.

1. Introducdon.

The

problem

of the statistical mechanical

description

of

equilibrium crystal shapes (ECS)

has

recently

attracted considerable attention

[I]. However,

most authors have restricted themselves _to

(I)

the close

vicinity

^of _a ^{facet and}

(it) high synunetry

directions. A noticeable

exception

^{is the work of Abraham}

[2]

^{in which the ECS of} a BCC

crystal

has been found for _a wide range of directions normal to

(110). Beyond

that _some advances have been achieved in two directions _:

(I)

Kashuba and _one of _us

[3]

have found _an

interpolation expression

for the ECS at an

arbitrary angle along

a

high

symmetry ^direction

(for example,,perpendicular

to the

(100) plane

in _a

simple

cubic

crystal)

_;.

(*)

Now at : Institute for Physical Science and

Technology, University

of Maryland,

College

Park, MD 20742, U.S.A.

JOURNAL DE PHYSIQUE T I, M 3, MARS 'WI '8

374 JOURNAL DE

PHYSIQUE

I N 3

(it) Yamamoto,

Akutsu and Akutsu

(YAA) [4-6] investigated

the

vicinity

of _a facet

edge scanning

_over azimuthal

angles.

In the framework of the free-fermion

approximation (I.e.

noncrossing

random

walks) they

found that the Gaussian curvature,

K,

has _a

jump

_across the

facet

edge

from _zero to a universal value K=

A~p~gr _~~d~,

_{where A is the}

_Lagrange

multiplier

in the Wulff construction

(see

Ref.

[5]

and the next

section), p

m

I/kB T,

and d is

the

height

of _an

elementary step.

Later Saam [7]

pointed

out that the universal value of K is related to the universal

properties

of correlations in the free-ferrnion

system

or in _a dilute ensemble of

steps.

In this paper we show that the random walk

approach

of YAA

yields

_{a very}

simple explicit expression

for the overall ECS if the

fugacity

of _a

kink, exp(- pE),

is small.

Another

approach

to the calculation of the overall

ECS,

similar to that of reference

[2],

has been

developed by Kashuba[8].

2. The model.

We consider the

simplest possible

model of _a

simple

cubic

crystal

which still captures ^the essential features of real

crystals

: lattice symmetry ^and strong fluctuations of

steps.

We ascribe _{an energy} E to each

nearest-neighbour

bond in the

crystal.

In the

bulk,

all the bonds _are in their

low-energy

_or satisfied state, while _some bonds _are broken

by

the surface.

Thus,

the energy of the

crystal

is

equal

to E _x

(number

of broken

bonds).

We consider _a surface tilted _on average ^at an

angle

_<

I

to the z-axis and

specify

its

configurations by

the 2

(in general, multiple-valued) height

function

zo(x, y).

The

length

of _a bond is taken to be

unity,

_so both the local

height

_zo and the lateral coordinates _{x, y are}

integers.

We define steps

as lines

separating

sites with local

heights differing by unity.

Then any

configuration

_can be

represented

as a set of

steps.

We will ascribe _a direction to each

step.

The direction _may be indicated

by

_{an arrow, so} that the z-coordinate is

larger

to the

right

of the _arrow

(Fig. I).

We consider the

low-temperature

limit in winch the

fugacity exp(- pE)

m exp

(-

_E

)

« ~. This is

typically

realized for

crystal-vapor

interfaces at temperatures below the

melting point.

All effects of order

exp(-

2

e)

will be

neglected.

In

particular,

if _one step passes over another one, two additional vertical bonds have to be broken and the

probability

is diminished

by

a

factor

exp(-

2 _e

).

Thus

configurations

with

overlaps

_may be

ignored,

_so that _zo is _a

single-

valued

ftc~ction

of _x and _y. Now it is easy ^to see that the energy of _a

configuration

is _: H

= Const. ₊ EL

(I)

where L is the total

length

of all

steps.

^{Note that} two steps may run

along

the _same link _: in

our

nearest-neighbour approximation

the energy of steps is additive. Now

following

YAA

[4, 6]

we

represent

a

step

as a random walk. The direction of the

step

^{becomes the time direction}

of the walk. At any lattice vertex the step ^{has four}

possibilities

_{to move}

(Fig. I).

^We ascribe _a

weight

Y~±~ ^m

exp(-

_E ± p

j)

to _a

jump along

the

positive (negative)

direction of the

j-axis

=x,y).

We will _see below that at the rounded

parts

of the surface the

largest

Y~~

is of order

unity (otherwise

_a flat facet is _more

favourable).

The

probability

of two consecutive

jumps

^back and

forth, namely,

_{Y~~ j}

Y~ j ⁼ exp

(-

2 _E «

I,

will be

neglected,

_so

that there is _no double

counting

in _our

approach.

We consider two

regimes

_:

A.

Y~ ~

«

Y~ ~ <

Y~~~

«

Y~~ ~ ⁼

l. In this _case the

probability

of _two consecutive

jumps

forwards and back

along

the y axis is O

(Y~))

=

O

(exp(-

2 _e

))

and thus

negligible.

The step goes

along

the x-axis with _rare

jumps

in the

perpendicular

direction

(Fig. 2a).

B. _~

~ w ~

~ « ~

~~ w ~

~ ~

O I

).

In this _case

p~

= e, so that both

probabilities

of

jumps

in the

negative

direction _{~_~,}

~_~

^are

O(exp(-2e))

ml and thus

negligible

(Fig. 2b).

N 3 SOLUTION FOR OVERALL

EQUILIBRIUM

CRYSTAL SHAPE 375

I

x

Fig.

I.

Representation

of _a

single

step as a random walk.

t

Y

y ~

i~y~~~)

a) ~=t (tLw~~j b)

~

Fig.

2.

Representations

of steps as fernfionic world fines _:

a) Regime

A _;

b) Regime

B.

The _crossover between

regime

A and B _occurs when Y~(~ = Y~_~ which

corresponds

to

p~

=

e/3

_; both Y~(

~

and Y~ ~ are then of order exp

~

e which is

negligible

at our level of 3

precision.

The crucial

point

is that at low

temperatures (E

_» I

)

the two

regimes

A and B have

a domain where

they overlap.

All other _{cases are} obtained from these two

by

the _use of symmetry. Thus the

simple explicit expressions

which _we obtain below for these two

regimes

will allow for _a

description

of the overall ECS. It should be noted that in both _{cases our} model satisfies the usual

terrace-step-kink conjecture.

Thus _we arrive at the

grand partition

sum £

=

jj

exp

jj (pj

_{E x}

(total

number of

j~type jumps)],

^where the first _{sum runs}

J +x,y

over all the

configurations

of the steps.

Only

sets of

nonintersecting

_{steps are} considered. The chemical

potentials

_{p~ are}

adjusted

_{so as} to

provide

the desired _mean orientation of the surface. It is

easily

shown that the derivatives of the free energy

f

=

log (£)/N~N~

with

respect

to the p~

yield

the _mean

slopes

of the surface _:

@~

/

=

p~

=

(dzo/dy)

d

~

/

₌ p~ =

(dzo/@x)

Here

N~

and

N~

are the lateral sizes of the system. ^{The chemical}

potentials

_p~ introduced above _are,

thus, simply

related to the

commonly

defined chemical

potentials [I], conjugate

to the

slopes.

Thus Andreev's

duality

transformation

(see

in

[I]) f-AZ, p~-Ay,

p~-

^{Ax allows us to convert the}

partition

sum

£(p~p~)

into the ECS function

Z(x, y).

Here A is the _common scale factor

proportional

to the size of the

crystal.

376 JOURNAL DE

PHYSIQUE

I M 3

3.

Equilibrium crystal shape

: the

explicit

solution.

The

partition

_sum

corresponding

to the Han~iltonian

(I)

is

explicitly

evaluated .within _our

assumptions by

the transfer-matrix method of YAN. The

representations

needed for the

regimes

A and B _{are a} bit different.

REGIME A. Here the x-axis is taken as the time direction. The lines _x

=

Const. _are crossed

by

each

step

once and

only

_once

(Fig. 2a).

l&then the time _goes from

x to x +

I,

_{a step} either rests at the _same

site,

_y, with the

probability

_Y~~~

m Y~~ ^or it

jumps

to a

neighbouring site,

y ⁺

I,

with

probability

_{Y~±~ Y~~.} In order _{to account} for the

nonoverlapping

of

steps

we

introduce _an additional _row of sites between each

pair

of steps so that for _a state of _m steps

N~

_-

N~

+ m and forbid steps ^to meet on the _same site. Thus _we arrive

precisely

at the system

considered in reference

[4].

The transfer-matrix used

by

YAA is

I

=

fl T(k

l~~^~~ _; T

(k )

₌

Y~~

+ Y~~~ Y~~ exp

(

ik

)

+

Y~ ~ Y~ ~ exp

(

ik

(2)

k

where

at,

_{a~ are} the fermion creation and annihilation

operators.

Note that after the extraction of _{a common} scale

factor,

_Y~~ the values

T(k)

I _are the

eigen~Jalues

of the

nonhermitian Hamiltonian

z _17l+y

^a~

(Y

+ I

a(Y)

_{+ 7l}

-y a~ ~Y)

a(Y

₊ i

)1 (3)

which is _a natural

generalization

of the Hamiltonian used in reference

[3].

The

partition

_sum His determined

by

the

eigenvalue

of T of

largest

modulus. For _a fixed number of steps m it

corresponds

to

filling

all ferrnion states with k

<

k~

=

«ml (m

+

N~).

Then

f(Y~~

_Y~~) is obtained

by

the nfinimization of the function

f(m,

_p

~,

p~)

=

log (£(m))/fIV~N~

_over m. Thus _we find

f(m,

_p

~, ~c

~)

= ruin

(~ )

i~~ log T(k)

dk

=

(4)

k~ ^"

kF Jo

REGIME B. Now the time axis _goes

along

the

diagonal

_x

= y

(Fig. 2b).

At each instant of time _a step

jumps

either to the

right

_or to the left with

probability

_Y~~~

m Y~~, Y~~~ _m Y~~

respectively.

We introduce

again

additional «time» _rows between each

pairs

of steps to arrive at the system ^{of reference}

[6]

with

T(k)

defined

by

T(k )

= 7~_~ exp

( tk/2 )

₊

7~_~ exp

ik12 (6)

Now

(4) yields f

= 0

=

Const. when

T(0)

< 1, which

corresponds

to the flat

(001)

facet.

Hence the

boundary

of the facet is

given by T(0)

= 1. On

combining

the two

expressions, (2)

and

(6),

and

using

the result in

(4),

_we obtain _an

interpolative equation

for the facet

boundary, namely,

Y~~~+ Y~_~+

Y~~~+

Y~_~ =

2exp(- e) (coshAx+coshAy) =1, (7)

which satisfies all the

symmetries

of the system. Note that _we have

proved

_a

posteriori

the

conjecture

_max

Y~j ⁼

O(I)

used when

defining regimes

A and B in the

previous

section.

N 3 SOLUTION FOR OVERALL

EQUILIBRIUM

CRYSTAL SHAPE 377

Thus _we have

developed

_a method of

constructing

the rounded surface

adjacent

to an

(100)

facet.

However,

this

approach clearly

breaks down _near another

(say (010)) facet,

where many steps run

along

the _same

edge

and

overhangs

cannot be

neglected.

The natural way of

avoiding

this

difficulty

is to match the constructions

starting

from the

(100)

and

(010)

facets.

The mirror

plane

of the

crystal parallel

to

(l10)

should be crossed

by

the surface at a

right angle,

I.e. @~Z = @~~

f

=

I at the intersection. The last

equality

defines _{a curve on} the surface

f(~c~ _~c~)

defined

by (4), (5), (6).

It must be _a

plane

_curve.

Straightforward

calculations

(see

the

Appendix)

_prove that this is indeed _{so :} up to corrections of order e~ ~, the _curve lies in the

plane

'Og(71x)+f=AY+Az-E=0 (8)

which is

parallel

to

(011) (Z

+ y =

I,

if _we take the scale factor A

=

e).

Thus _{we can}

smoothly

match the surface

Z(x, y)

with

X(y,

_z and others obtained

by permutations

and

changes

of

sign

of the three coordinates. In this way the overall

crystal shape

is obtained. Note that

(8)

defines the absolute size of the

crystal

_: the center lies at _a distance

e

IA

below the

(001)

facet.

Thus A

=

e/R,

where R is the distance from the center of the

crystal

to _a facet.

We have _up to _now set the

origin

of z-axis at the

(001) plane.

It is _more natural to shift the

origin

to the center which

requires

the additive redefinition _: Z

=

~f

₊

e) IA.

For small

Y~~

(regime A), (8) corresponds

to

k~= gr/2 (see

the

Appendix). Therefore,

Y~~

changes

_across the curved

region

from I-

(Y~~~+

_{Y~_~)} ^{at the facet}

edge

to I (Y~~_i ⁺

Y~_~)/gr

^{at the middle of the}

region.

Thus it is close to

unity everywhere

which

provides

_a smooth _crossover between

regimes

A and B. Now it is

easily

checked that

by introducing

the _new variables

l~

= Y~~j ⁺ Y~-j,

0

= x,

y),

_{we may use} the

interpolative expression

f

kF

=

nfin

log (Hj

₊

H)

₊ 2

H~ H~

cos k

) dk/2 (9)

kF ^"

~F

o

for all values of

H~ H~,

^thus

defining

_a universal function

f(H~ H~).

^After

^that,

^for any

given

e =

E/kB T,

the ECS for _x

~ y ~

0,

Z

~ x is obtained

through

the

mapping

_:

Ax

= cosh~

(e~ H~/2),

_{A y}

=

cosh

(e~ H~/2)

,

Az

=

f

_{+ e}

(10)

and then the _rest

by

^rotations and reflections. The whole

procedure

is accurate up ^to corrections of order

exp(-

4

e/3).

The results for the

projection

of the ECS onto the

(l10) plane

for _E

=

3,

5 and 10 _are shown in

figure

3.

4. The correlation functions and Gaussian _curvahtre.

We start

by calculating

the

density-density

correlation function for _our random walkers. The standard transfer matrix

technique [9] yields

(p (xi

_p

(x~))

=

jai

: a+

(xi a(xi P

a+

(x~) a(x~)

_:

(o) /(o Pi _{o) (I I)}

where

(0(

stands for the

ground

state

corresponding

to the

largest eigenvalue

of

I,

_{while fi}

=

(q, tj),

where _r labels the sites in the _«

space-direction

_{» rows}

(x

=

Const. in the

A-regime

and _{x y}

= Const, in the

B-regime),

and _t

(time)

counts these _rows the

symbol

_:

denotes

time-ordering

of the operators and t _w t~ ti

(.

^For

I given by (2)

the correlation of

378 JOURNAL DE

PHYSIQUE

I N 3

,

' 1

0

j

' /

', _/ '

~

~ 0

~fm/fizz~'

_~

'mfi---=fi$

1' ~', /~ '

~~ ^'

-o ⁱ

'

-i , -1

,

~'

~~ 5 -1o -05 oo 05 -i 5 o oo 5

x x

a)

b)

5

o

0.5

~ oo

-o 5

-1 0

-l 5 -1 00 05 10 15

x c)

Fig.

3.

Projections

of the ECS onto the (l10)

plane.

Solid fines represent the

profile

of the

shape

_;

dashed lines denote facet

edges. a)

_e

= pE

= 3_;

b)

_e

=

5 c) _e

=

10.

the

density

fluctuations

3p

_{m p}

lpi

= p

-k~/gr

is determined

by

the creation-annihi- lation of

particle-hole pairs

which

yields

(3p (xi)

BP

(X2))

=

z eXPli(k2 ki)

_(X2

Xi)I (I Pk~) PkilT(k2)/T(ki)l~ ^('2)

After

setting 3p(x)= jjp~exp(iqx)

it is

easily

_seen that the

asymptotic

form of

q

G(q)

w

3p~(~)

^{for small} q is determined

by

the close

vicinity

of k

= ±

k~. Introducing

w =

d

log T(k)/dk(~ (so

that w*

= d

log T(k)/dk(_~ )

we get

G(q) =k)Rew/gr(k)(w(~+k)+2k~k~Imw). (13)

N 3 SOLUTION FOR OVERALL

EQUILIBRIUM

CRYSTAL SHAPE 379

Since each

step

crosses each _space-row

only

_{once one} finds _p

=

(dz/dr)~.

Hence _one obtains

zq

~)

"

[G (q )/k/]~

= ~r

(k)(

_w ^~ ₊

k/

+ 2

k~

k~ ^Im w

)] /Re

w

"

~<,J

k<

kj ⁽¹⁴⁾

which

implies

det r

;,~ ^{= gr}

~

(l 5)

So _even

though

the transfer-matrix

I

_is nonhermitian and _w is not

real,

the value of det r is real and universal. Now,

following

the arguments ^{of YAA and Saam}

[7]

we find that the correlation function

given by (14)

_can be obtained from the

capillary

Hamiltonian

H

=

id~x ^{r;~ d;z}

^d~z,

(16)

2

where _z denotes the fixed z-axis rather than the normal to the

given

_average orientation of surface. Hence

r;j

=

d; d~g(p'),

where g is the

Legendre

transform of the

grand potential log E/N~(N~

+

m)

and

depends

on the _mean

tangent slope p]

=

(dz/dJ~)

of the

auxiliary N~

_x

(N~

+ m

) lattice, p(

=

k~/gr. ^Finally f(l~

" fllinp>

(i pl)~ lpi

_l~

r

+

PI

_{l~r g}

(p')I

>

so that

dot

o, oj f

= dot

rj ~/(i pl)~

~

(i

₊

pr)~

"

~

>

(17)

where

pj

=

p(/(I

₊

p()

are the real

slope

variables

conjugate

to ~c~.

Combining (17)

^and

(I)

and

keeping

ⁱⁿ nfind that for both

regimes (I

₊

pr)~/(l

_{+ p )~}

=

z

_n~ ~,

j

= x, y, z, we get

K

= A ^~

p

~( _nx +

ny

⁺ nz )~/~T~

=

R~ ~(

nxl

+

~y

+ ~z1

)~(P El"

)~

(18)

where R is the distance from the center to the

(100) plane.

Note the

explicit

cubic

symmetry

of the final

expression

in contrast to the

asymmetric

_way in which it _was derived.

5. Discussion.

Equations (9)

and

(10)

contain _a

complete

solution of the ECS

problem

for the _nearest

neighbour simple

cubic

crystal

in the

low-temperature

limit. Some

examples

of

shapes

_are

presented

in

figure

3. The model

clearly

^involves two

major simplifications

_:

(I)

the total

neglect

of the

long-range

interatonfic forces other than NN bonds. This

precludes

the formation of facets other than of

(100)

type.

(ii)

The

low-temperature appoximation,

I.e. the

expansion

in powers of

fugacity exp(- e).

Because of this _we have

completely neglected

all

overhangs, particularly

the appearance of closed finite terraces winch renormalize the

step

free _energy and the effective

step-step

interaction.

These

assumptions

_are

physically justified

for _{a range} of

temperatures

_{T~~ <} T<

T~~,

where T~~

~

are the

roughening temperatures

^of

(100)

and

(l10)

faces

respectively.

For the

crystal-valor

_system

T~~ is

usually higher

than the

melfing

temperature. The effect of _{a sea} of closed

step loops

_on the free energy of _a

single (infinitely long) _step

has been

recently

considered

by

Holzer and Wortis

[10].

In this _way the

shape

of a

single

^facet has been determined at

temperatures

closer to T~~. ^The effect of the

loops

_on the

step-step interaction,

380 JOURNAL DE

PHYSIQUE

I N 3

which determines the

shape

of the vicinal surface

connecting adjacent facets,

was not

investigated

to _our

knowledge.

The _appearance of _{a «} devil's staircase _» of

higher

order faces

provided by long-range _step-step (equivalently, fermion-fermion)

interaction _was demonstrat- ed

by

Kashuba and _one of _us

[3].

To

summarize,

the main results of the

present

^work are :

(I)

If 2R is the distance between the

opposite faces,

the width of the rounded vicinal surface between the facets

obeys

the

exponential

law

(R/pE) exp(- pEilR),

where

I

_{is the} distance

along

the

edge

to the nearest _comer. Tl1us while the

edges

of the cube _are smeared at Tm 0 with _a small relative width _a

(R/pE) exp(- pE) [3],

^the comers are smeared with _a much

larger

relative width

aR/pE.

(ii)

The model under consideration _cannot be reduced to _an

equivalent

fermionic system because the transfer-matrix and the Hamiltonian _are not Hermitian. Nevertheless the determination of the

generalized

stiffness matrix r is

universal, determining

the critical

exponent

Y~ ⁼

I/4

_gr

[7].

(iii)

The Gaussian curvature K is

given by

the rather

simple expression (18).

It is

clearly

seen that

K~'~

=

(Ki K~)~/~

₌

O(pE/R)

varies

by

_a factor 9 _on the

rounded,

vicinal

surface,

while the

principal

curvatures

Ki,

~ vary more

strongly

from

Ki

#

K2

_" R~

pE

at the _comers

to

Ki

= exp

( pE) pE/R

and

K~

_{= exp}

(- pE) (pE/R)

at the

edges.

The numerical

prefactor

in

(18)

is

obviously model-dependent.

Its

experimental

determination could _serve to check the

applicability

of the model.

(iv)

The

shape

of the vicinal surface _near the

(001)

facet

edge

is well described

by

the

simple expression

z =

(2~'~/3 «) (nx fly 1)~/~/(nx ny)~'~ ⁽¹⁹⁾

which follows from

(9)

and

(10).

Thus the universal law Az ₌

Cp

^3/~

[ll],

_p

being

the normal distance from the facet

edge,

is fulfilled

everywhere.

However the coefficient C

depends

_on the location

along

the

edge

of the

facet, varying

from of order

(pE)~'~

at the _x

= ± y comers

to much

larger values,

of order

(pE)~/~

_exp

(pE),

at the nfiddle of _an

edge.

Thus the vicinal surface rises much _more

steeply

_near the center of the facet

edges

than _near the _comers.

The

qualitative picture arising

from _our

explicit

calculation _can be obtained

by simple physical

_arguments.

First,

the form of the surface

along

the

edge

of the cubical

crystal

is

qualitatively

sinfilar _to the form of the

boundary

of the facet _see

(7).

The left hand side of this relation

simply reproduces

the free energy of _a

single

_step.

At low

temperatures

^{it is convenient} to

represent

any

configurafion

of the

step

as a

perturbation

of the lowest energy

configuration

in which the

step

runs

parallel

to the

edge (I.e.

the

x-axis).

The

elementary

excitations _{are «} kinks _», that

is,

links of

perpendicular,

_y-

orientation. Each kink carries _a

quantum

^of

slope, dy/tlx

= ± I.

According

to the

duality

transformation which _{we are}

exploiting,

the coordinate

along

the

edge,

_x, is

proportional

to the cheInical

potential

of _a kink. The chemical

potential

is

conjugate

to the

slope

it is of

opposite sign

for kinks of

opposite slopes.

The

exponential dependence describing

most

parts

^of the _curve

(7) corresponds

to the Boltzman statistics

describing

_{a gas} of kinks of _one

sign, dominating

at a

given

_part of the

edge.

Both _near the centers of the

edges

and _near the _comers these

exponential

curves

join smoothly owing

to the

mixing

of two different sorts of kinks.

Second,

the form of the vicinal surface _transverse to _an

edge

is

given by

the free _energy of the

system

^of many

steps compressed by

an extemal _«pressure»

proportional

to the transverse coordinate _p. The

compressibility

is determined

by

the

entropic repulsion

of the

steps,

^{which for}

large

step-step

separations, L~,

varies like

DkB T/L(.

Here D is the

N 3 SOLUTION FOR OVERALL

EQUILIBRIUM

CRYSTAL SHAPE 381

diffusivity ~proportional

to inverse

stiffness)

of _a

single step, determining

its _mean square

wandering (w() _=DLjj,

_{as a} function of the distance

Lj along

the step. Tile law AZ

=

Cp ~/(

^cited

above, immediately

follows with C _oz D _~/~. The width

Ap

of the rounded

region

_may be estimated

by simple scaling

_{arguments as}

Ap

oz C ^~~ _oz D _: if the

repulsion

is

weak/strong

the

compressibility

is

large/small

and the characteristic «pressure»,

Ap,

necessary ^to compress the system ^to the

density

of order

unity

is

small/large.

Now the

steps

^diffuse

by propagation

of kinks and D is

easily

shown to be

merely proportional

to the average number

density

of kinks

(not

to the _«

slope density

»

m

slope

_: kinks of both

signs

contribute to the

diffusion).

The crucial

point

here is that if the _average

slope

of the

steps

^with

respect

to the

edge,

_q, is smaller than the

fugacity

of _a

kink,

exp(- pE),

then the

diffusivity

is determined

by

the

thermally

activated kinks. So _one has

Ap

_oz D _oz exp

(- pE)

which is very small _at low temperatures. ^{On the other}

hand,

for q ^» exp

(- pE),

the finite

density

of

kinks,

which is of order q, is determined

just by

the

geometrical

constraint. In

particular,

_near the _{comers q}

=

O(I)

and

Ap

_{oz q} is

relatively large. Finally, combining

the

Ap

oz q

dependence

with the

exponential approximation

q oz exp

(- pEilR)

from

(7),

_we obtain the

exponential dependence

of

Ap

on the distance

along

the

edge

from the _comer that _was noted above.

Finally,

notice that the

only quantity determining

the results of the

preceding

discussion is the

fugacity

of _a kink _{on a} step. ^In our case this is

just

_exp

(- pE),

but it _can be defined _more

generally

for any

lattice, interaction,

_or facet

orientation,

and is

expected

to be _a small

parameter

at low

temperatures.

^{In this} way our results _can be

applied

much _more

widely,

to _a vicinal surface between _any two facets well below their

roughening temperatures ~provided

the direct

long-range repulsion

between

steps [12]

^is

negligible).

The

major problem

here is that in most of the _cases, the facets _are not

equivalent,

_so that two different

fugacities

of kinks

determine the behavior from the

opposite

sides of the rounded _zone. The

description

of such

an

asymmetric

situation remains _a task for future work.

Acknowledgments.

One of _us

(LM)

is indebted to Noriko Akutsu for

drawing

his attention to references

[4-6].

The kind

help

of Curtis A.

Doty

and

especially

of Michael E. Fisher in the final stage ^{of work}

on this paper is

gratefully acknowledged.

The work _was

completed

with the aid of financial

support

from the National Science Foundation

(under

Grant NO. DMR

90-07811).

Appendix.

Here _we prove that the

points satisfying

the

equation

_d~~

f

=

I

lie,

within the

prescribed

accuracy, on the

plane (8).

In

regime

B _we find

d

f/d~l~

= d

f/d log

_{1~~ =}

(fir kF)~ ~~

_~

X ~" ~ dk

o + X^~ + 2 _{X cos} k

~~r

kF) l~+

^tan

i

tan

l~ ^(Ai)

where _X

= 71y/71x. ^On the other hand the condition for _{an extremum to} be

present

ⁱⁿ

(5)

is

equivalent

to

log

[Y~j+ Y~

j

₊ 2

Y~_~ Y~~ ^cos

kF]

₊

f

=

log

Y~ ~ +

f

₊

log [I

_{+ X}^~ ₊ 2 _{X cos} k

=

0

(A2)

382 JOURNAL DE PHYSIQUE I N 3

Setting

the r.h.s. of

(Al)

to

unity

_we find _{x +} 2 _cos

k~

= 0 and after substitution into

(A2)

obtain

(9).

The

corresponding

calculations for the

regimeA

_are much

simpler. Setting df/d log

Y~~ ⁼

kF/(gr k~)

=

I,

_we obtain

k~

=

gr/2,

while minimization of

(6)

leads _to

f

₊

log

_{Y~~ +}

log

_{[1 +}

_2(Y~~

_{+ Y~~) cos}

k~]

=

0,

which for

k~

=

gr/2 again

reduces to

(9).

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