Marder M.P. Condensed Matter Physics

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Dislocations

383

Figure 14.5. The Burgers vector, in this case for an edge dislocation, is constructed by fol-

lowing a counterclockwise path that would close in the perfect crystal. When it surrounds

a dislocation, however, the path is open. The Burgers vector is the vector needed to close

the path.

of defect costs vastly less energy at any given time than would be needed to pick

up a chunk of crystal all at once and place it down at a new location.

There are two basic types of dislocations. The one pictured in Figure 14.3 and

in Figure 14.4(A) is an edge dislocation. A second type is the screw dislocation,

shown in Figure 14.4(B).

A basic characterization of a dislocation is its Burgers vector. This vector is

obtained by going to the vicinity of a dislocation, and then traversing a counter-

clockwise path around it which would be closed in a perfect crystal—for example,

6 atoms up, 7 to the left, 6 down, and 7 to the right, as in Figure 14.5. When such a

path surrounds a dislocation, it does not close, and the Burgers vector is the vector

needed to close the path. For an edge dislocation, the Burgers vector is perpendic-

ular to the dislocation line, while for a screw dislocation it is parallel, as shown in

Figure 14.4.

From Figure 14.3 it should be clear that there is a unique plane in which it is

particularly easy for a given dislocation line to move. This plane is known as the

slip plane or glide plane. Motion perpendicular to that direction, which can only

be achieved by difficult rearrangements of atoms, is called climb.

14.2.1 Experimental Observations of Dislocations

Theoretical consideration of the sort considered above were far from able to con-

vince practical metallurgists that dislocations were

real.

Instead, dislocations seemed

like elaborate theoretical fictions intended to rescue the improbable theory that

384 Chapter 14. Dislocations and Cracks

Figure 14.6. In order to add a new layer of atoms to a crystal, it is typically necessary to

add a large clump of

new

atoms simultaneously. (A)

single atom attaching to the surface

will be unstable and detach. (B)

large enough clump is stable.

matter was made of atoms from its embarrassing failure to account for the obvi-

ous practical properties of solids. Even some physicists felt that "the reality of

the 'dislocations', which form the basis of [Taylor's] theory, seems very doubtful."

[Frenkel and Kontorova (1938), p. 2]. Starting in the 1950s new techniques imaged

dislocations directly. The first was in response to theories of crystal growth. If one

has a flat crystalline surface, it is hard to deposit a new layer of atoms on it. A

single atom sitting on a surface is not by and large stable, because only by being

surrounded by neighbors on as many sides as possible does the bonding that holds

the solid together keep the surface atoms glued on. In Figure 14.6 the atom in (A) is

likely

jump off the surface, and a clump as large as the one shown in (B), or even

larger, might well be required for stability. However, forming large clumps simul-

taneously is statistically very unlikely. The calculations of these probabilities were

due to Volmer: "Keith Burton showed how you could put numbers into Volmer's

algebraic expressions: and it turned out that while Volmer's curve for dependence

of growth-rate on supersaturation was qualitatively similar to what was observed,

there was a quantitative disagreement by a factor of

1000

which we said was the

largest factor of discrepancy we had ever called agreement." [Frank (1985), p. 9]

Figure 14.7. Mechanism due to Frank, showing how the presence of a screw dislocation

should speed the process of crystal growth, by providing open regions to which atoms can

easily attach.

Dislocations 385

Figure 14.8. Experimental observation of growth spirals on the (0001) plane of silicon

carbide. [Source: Amelinckx (1964), p. 5.]

The idea that Frank proposed to rescue the situation was that atoms could easily

attach to a screw dislocation at the top of a crystal, as shown in Figure 14.7. Such

spiral patterns, greatly magnified in scale by chemical etching, are now routinely

observed on crystal surfaces, as shown in Figure 14.8. It is possible, by having

impurity atoms diffuse to dislocations, to image them directly with electron, X-

ray, and neutron diffraction. A picture showing various dislocations surrounding

silica particles in brass appears in Figure 14.9(A). Finally, high-resolution electron

microscopy makes it possible to view dislocations at the atomic scale, as shown in

Figure 14.9(B).

Figure 14.9. (A) Experimental observation of individual dislocation loops surrounding

N13SÌ particles in nickel-silicon alloy. (Courtesy of

Humphreys, Manchester University.)

(B) High-resolution electron micrograph of

CdTe,

showing dislocation loops imaged at the

atomic scale. [Source: Cullis et al. (1985), p. 205.]

386 Chapter 14. Dislocations and Cracks

14.2.2 Force to Move a Dislocation

A virtual work argument can be used to determine the effectiveness of external

stresses in moving a dislocation. Viewing the dislocation as a line being pushed

through the crystal, define /, the force per length a needed to make the dislocation

move. If

the

dislocation in Figure 14.5 moves from right to left past N atomic sites,

then a layer N atoms long moves by the Burgers vector b. If an external shearing

force

ext

has been applied to the top and bottom of the sample, then the total work

done in the process is

ext

The work required to move the dislocation over one

lattice spacing is then

ext

/N,

and the force f

per length a required to move the

dislocation must be

<7xyb

(14.6)

where

(I4J)

is the external stress appi' ~

to the crystal, with a the lattice spacing. Notice that

putting the crystal under tension, either with σ

χχ

(pulling along the x axis), or with

Oyy

(pulling along the y axis) has no power to move the dislocation. If L is a unit

vector pointing along the dislocation line (L = z in the case of Figure 14.5), then

Eq. (14.6) can be rewritten in coordinate independent form as

f=(a-b)xL (14.8)

The first term on the right-hand side of

Eq.

(14.8) indicates that one forms the 3x3

tensor σ and takes its dot product with the Burgers vector b. The direction of L is

chosen according to the right-hand rule; the Burgers vector is defined by taking a

counterclockwise path around the dislocation, and the direction L of

the

dislocation

is the direction the thumb points when the right-hand fingers curl around it. The

advantage of describing the force on the dislocation in this form, due to Peach

and Köhler (1950), is that it is independent of one's choice of coordinate system,

is equally true for edge or screw dislocations, and remains true for a dislocation

which curls about or forms a loop.

Equation (14.8) is useful when one supposes that there is a critical force f

needed per unit length to make the dislocation move. Equation (14.8) then de-

scribes the external stress that will be required for a crystal to begin to flow, if it is

populated with dislocations. It can also be used, for example, to decide when one

dislocation will begin to move because of stresses created by another dislocation.

14.2.3 One-Dimensional Dislocations: Frenkel-Kontorova Model

In order to understand how dislocations allow crystals to flow, it is necessary to

determine the critical force f

. This task is somewhat complicated in two- and

three-dimensional settings, so it is helpful to examine a one-dimensional model for

dislocations, the model of Frenkel and Kontorova (1938), where the job is easier.

Dislocations

387

F = F

crit

/2 ►

-5-4-3-2-10 12 3 4 5

Figure 14.10. The Frenkel-Kontorova model contains a one-dimensional array of

parabolic potentials placed side by side, in which are located a collection of masses. The

masses are acted upon by a force, which tends to push them over the potential, and are

connecting by springs of constant k. (A) Equilibrium dislocation. (B) Dislocation on the

verge of beginning to move because the mass at 0 is balanced on the top of the potential.

In this one-dimensional model, pictured in Figure 14.10, one considers a col-

lection of masses connected by springs, and sitting in a periodic potential, de-

scribed by

U(x) = hX[x

—

a int(x/a +

l\i2

-fx,

(14.9)

where / describes the external forces applied to the particles. The integer part of

xja + \ gives the integer nearest to xja. If mass n is located at x

, then the total

force on mass n is

/„ =k[x

\ -x

-a\+k[x

-\ -x

+a]

(14.10)

When a dislocation is present, one of the wells does not have a particle in it.

Choose that well to be the one centered at 0, so that particle 0 is now sitting in the

well centered at —a, and particle 1 is sitting in the well centered at a. This means

that Eq. (14.10) can be rewritten

k[x

\ —x

—a\ +k[x„-i —x

+a] +f

—

X[x

—

(n— l)a] for« < 0

k[x

—x

— a]

+k[x

—

JC[x

—

na] forn > 0.

(14.11)

In equilibrium, all the forces /„ have to vanish. Guessing the form for n < 0

= f/X + a(n-\)+A

, (14.12)

388

Chapter 14. Dislocations and Cracks

one immediately finds that for n < 0,

k{e

-2 + e-

)-JC = 0 (14.13)

while for n > 0 one again has Eq. (14.13) but the solution is of the form

= f/X + an+A

. (14.14)

The question is, do the two solutions join properly at n = 0? They only match if

computed from Eq. (14.14) is the same as

found in Eq. (14.12), andxi computed

from Eq. (14.12) is the same as x\ from Eq. (14.14). So one must require

To obtain equivalent results more formally, .

a-{-Ai=A

carefully write down the two equations/o = 0 (14.1ja)

and /, = 0, using Eqs. (14.12) and (14.14).

Aie

= a+A

e-\ (14.15b)

which gives

A/ = —°— (14.16a)

ei + \

. (14.16b)

These are the equilibrium solutions. What should correspond to the motion of a

dislocation? The motion should begin as soon as the force on the system is great

enough so that mass 0 has reached the peak at —a/2 and is about to slide down into

well at 0, pulling the mass behind it in its wake. This condition is

Xo =

a+Al

(1417)

=*/

= ^-tanh|. (14.18)

The maximum force that the potential U is able to supply is aX/2—all masses

would be able to slide when / reached this value with or without a dislocation—so

the hyperbolic tangent in Eq. (14.18) describes the advantage to be gained from

having a dislocation. One obtains plasticity with low forces when q is small, which

means equivalently that the dislocation is quite extended. This limit is achieved

when X/k is small, in which case one finds

(14.19)

y te

and

(14.20)

Two-Dimensional Dislocations and Hexatic Phases 389

14.3 Two-Dimensional Dislocations and Hexatic Phases

Dislocations play an interesting role in the statistical mechanics of two-dimensional

crystals. Two-dimensional crystals do not, in fact, technically exist. Long-range or-

der is almost always destroyed by thermal fluctuations in two dimensions. Another

type of order, orientational order, can however exist, and dislocations are central

actors in the phase transition that brings orientational order to an end. The mathe-

matics of this phase transition applies almost unchanged to a broad range of other

physical systems, ranging from two-dimensional electron gases to thin films of he-

lium, and provides a setting in which to gain an introduction to renormalization

group techniques.

14.3.1 Impossibility of Crystalline Order in Two Dimensions

Three dimensions are special in many respects. One surprising property, first re-

alized by Peierls (1934) and Landau (1937), is that three-dimensional space is the

lowest dimensionality in which long-range crystalline order is possible. Certainly,

one can draw pictures of crystals in two dimensions, but if one built them, they

would be destroyed at any nonzero temperature by fluctuations. This fact has al-

ready been demonstrated, because Problem 7 in Chapter 13 shows that sharp Bragg

peaks cannot survive in one and two dimensions as a result of quantum fluctuations.

The same conclusion follows from examining the effect of thermal fluctuations in

a purely classical context.

Consider, for simplicity, a two-dimensional elastic medium, where in terms

of displacement field u introduced in Section 12.3, the potential energy takes the

simple form

■) v-^ du

d r -C y . Here C is a constant with dimensions of en- (14.21)

*—£ drn drg ergy per area.

In order to study the statistical mechanics of

system whose energy is given by

Eq. (14.21), it is best to work with the Fourier transform variable u

(k) such that

(r) = Y^e

rrl

(k). (14.22)

The idea that u describes only fluctuations on a scale larger than some distance

that is bigger than an interatomic spacing may be implemented by requiring that

u(k) = 0 for k> I/T>. (14.23)

Then Eq. (14.21) can be rewritten as

U = J d

r \C £ k

(k)u*

{k') (14.24)

ßakk'

The first time Eq. (14.22) is used, it is substituted directly intoEq. (14.21). To substitute the

second time for «

, however, one uses the fact that u

is real and substitutes the complex

conjugate of Eq. (14.22) instead.

390 Chapter 14. Dislocations and Cracks

y^ k

(k)\

UseV^, =fd

where Vis the

two-dimensional volume of the system.

(14.25)

With this expression for the energy of a slightly deformed system, one can

perform statistical averages. First, examine the expectation value of the average

square displacement of particles from their original location, which is

)

y UßCr)ußCr)) Here V is the two-dimensional volume of the (14.26)

^-—' system, and () is a thermal average.

= Σ<Μ*)Ι

ßk

This calculation is identical to the one that (14.27)

leads to Eq. (14.25), except that the deriva-

tives with respect to rß are missing.

In order to perform the thermal average, one has to integrate over all allowed

values of u^, weighted by the energy of each configuration. Because u(r) is real,

one has to restrict this discussion to cases where

u(k) =

u*(—k)

(14.28)

and also restrict it to long-wavelength excitations in accord with Eq. (14.23). Oth-

erwise, all possible functional forms of u(k) are permitted. Picking one value from

the sum over k indicated in Eq. (14.27), one has to compute

(\u

(k)\

)

I EU du

(k')\u

(k)\

■Ρψ ΣαΡ *'

ΙΜ*')Ι

!Yl

,du

Çk>)e-^T.

v^C

>)?

J du

(k)\uß(k)\

-ßVCk

(k)\

J du

(k)e-P

vck2

[(u

)

(„y]

-roW+(*0

]

/ durdvte-PKWY+wn

VCk

Returning to Eq. (14.27), one has that

VCk

. d

k k

(2TT)

V» dk k

2^k~C~

Regarding all the u(k') for

different k as independent

variables, although subject to (14.29)

constraint (14.28), and

integrating over them.

Most terms cancel between

numerator and denominator.

The factor of

beneath C (14.30)

disappears because, using

Eq. (14.28), two terms in the

exponential depend on

integration over

up(k).

One has to integrate over both

the real and imaginary parts Ur(\A

"Î1

and

up(k).

Use

ΓΊ

à**'"*

v^/2(i4.32)

(14.33)

Using the two-dimensional density of states (14.34)

from Section 6.3.

OO.

Recall Eq. (14.23).

(14.35)

Two-Dimensional Dislocations and Hexatic Phases

391

The integral in Eq. (14.35) diverges logarithmically for small k, just where

linear elasticity should be working the best. Assuming that deviations of the crys-

tal from equilibrium are small has led inevitably to the conclusion that they are

infinitely large and that long-range translational order is impossible in two dimen-

sions.

This result is known as the Mermin-Wagner theorem, after Mermin and

Wagner (1966). It does not mean that real two-dimensional systems are liquid at

any nonzero temperature, nor does it mean that lightning will smite him dead who

tries to adds the last atom to a two-dimensional array. Rather, as shown in Figure

14.12,

it means that Bragg scattering peaks are substantially broadened. In fact, the

conclusion is somewhat pedantic. The divergence in Eq. (14.35) is only logarith-

mic,

so in any finite-sized sample the deviations from long-range order may not be

overwhelmingly large. Two-dimensional crystals do not melt at all temperatures;

they are just somewhat floppy at long distances.

14.3.2 Orientational Order

Although the average fluctuations in positions of

atoms

are large in a two-dimensional

crystal, Mermin (1968) showed that fluctuations in the angles of orientation of

atoms with respect to their neighbors are much smaller. This fact may be deter-

mined by arguments that closely mirror the previous ones.

Figure

14.11.

Illustration of how a displacement

causes the angle φ between points r

and r' to shift.

Suppose again that one has a displacement field «(?). How do these displace-

ments cause angles between nearby points to shift? Referring to Figure

14.11,

let

(dx,dy)

= r'-r. (14.36)

The angle φ between the two points is

φ = tan'

(dy/dx). (14.37)

When they are displaced by a field w(r), the new locations of

and r' are

r + u(r) and r' +

u(r').

(14.38)

392 Chapter 14. Dislocations and Cracks

Therefore, the new angle between them is

=tan

-i

y/°y

(1439)

^ dx + du

/dx dx + du

/dy dy

dx dy

>+■

+ dy

' du

\ dx du

dy du

- dy dx ' dy dx dx dy

(14.40)

,/ , · , /duy du

\ 2

x /Λ

ΛΛ

^φ'

-φ = οο$φ sin<£

(—

-—-

+cos

(^^-sin

φ—

. (14.41)

dy dx ' dx dy

Averaging over φ, one finds that the average change in local orientation δφ at each

point r is given by

Adopting the Fourier representation Eq. (14.22), one has

Φ<?)

= \ Yj&xVh>$) - ikyu

(k)y

. (14.43)

Therefore, the spatial average of orientation fluctuations is

^-(δφ{7)δφ{7)) (14.44)

Ç k

(\u

(k)\

) +k

(\u

(k)\

) -k

((u

(k)u;(k) +u

(k)u*

(k)))(l4A5)

ϊΣ^νΠ^

+ ^) UsingEq.(14.32),andnoticingthat<^(*)M;(A)> (14.46)

must vanish by symmetry.

kßT [

2π

PI"

dkk k

AC Jo Jo

(2TT)

16TTD

(14.47)

In contrast to the amplitude fluctuations, orientation fluctuations do not diverge

in this analysis. Therefore, it is possible that they will remain finite, although long-

range crystalline order has been broken. The crystal has bond-orientatioml order.

14.3.3 Kosterlitz-Thouless-Berezinskii Transition

As the temperature of a two-dimensional crystal increases, there comes a point

where temperature fluctuations destroy the long-range orientational order present

at low temperatures, just as the long-range positional order of ordinary solids dis-

appears at the melting temperature. The manner in which the transition occurs was

first described in detail by Kosterlitz and Thouless (1973) and Berezinksii (1972),

with an additional transition at higher temperature found by Nelson and Halperin

(1979) and Young (1979).