Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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212 6 Semiconductors

1 µm

(a)

1 µm

[–110]

[01–1]

(b)

Fig. 6.4. Dislocation structures in Si-5 at% Ge taken during an in situ straining

experiment in the partly unloaded state. (a) 610

◦

C, (b) 745

◦

C. From the work in

[368]

6.3 Dislocation Dynamics 213

In summarizing, it may be stated that dislocations arrange along the

Peierls valleys at rest and during the motion at low temperatures, that is,

at high stresses. 60

◦

dislocations are more mobile than screws. Dislocations

moving at higher temperatures assume a curved shape. They have many con-

strictions, which are mostly connected with jogs [361, 369]. The latter move

with the dislocations and impede their motion. Between these obstacles, the

dislocations bow out. This behavior is in contrast to the general assumption

that dislocations moving under the control of the double-kink mechanism are

very straight and oriented along the Peierls valleys. In elemental semicon-

ductors, dislocations always move in the dissociated state [370] in a smooth

viscous way.

6.3 Dislocation Dynamics

The dependence of the dislocation velocity v

on the stress and temperature in

semiconductors is commonly described by the power law (4.103), which is an

approximation of the Arrhenius equation with correct behavior at low stresses.

Data of the stress dependence of high-purity silicon by Imai and Sumino [262]

were presented above in Fig. 4.42. These data were obtained by observing the

dislocation motion on a mesoscopic scale by X-ray topography (Sect. 2.5.1).

As observed in the above videos, 60

◦

dislocations are generally slightly more

mobile than screw dislocations. The stress exponent m is close to unity for

both. Other authors ﬁnd higher stress exponents between 1 and 2 by using the

stress pulse-double etching technique (Sect. 2.2) (e.g. [371, 372]). It is argued

in [358] that the higher exponents are due to the pinning of the dislocations

before their motion, thus being an artefact of the etching technique. However,

stress exponents near 1.5 were also observed in germanium and in compound

semiconductors (see Table 3 of [359]).

The activation energy ΔG

is obtained from Arrhenius plots of the

logarithm of the dislocation velocity at constant stress vs. the reciprocal abso-

lute temperature. For high-purity silicon, ΔG

amounts to 2.2 eV for 60

◦

dislocations and to 2.35eV for screws [358].

Special stress pulse-etching experiments were performed by Farber et al.

[373] on 60

◦

dislocations in Si to separate the processes of double-kink for-

mation and kink migration. They applied sequences of load pulses of the

duration t

separated by load pauses of duration t

. The total time of loading



was always equal to the time of static loading necessary to attain a trav-

eling distance of about 30 μm. The duration t

of the load pulses was chosen

in the range of the traveling times of the dislocations from one Peierls valley

to the next, t

≈ h/v

,whereh is the distance between the Peierls valleys

and v

is the dislocation velocity. For a series of experiments at 600

◦

Cand

astressof7MPawitht

= t

, the maximum of the distribution of the trav-

eling distances shifts with t

decreasing from the 30 μm of static loading to

lower values. With t

further decreasing, undisplaced dislocations appear with

214 6 Semiconductors

increasing frequency until at t

< 10 ms there are no more dislocations mov-

ing. In another series of experiments, the duration of the pauses was changed

at constant t

= 94 ms. With increasing t

, the average traveling distances

decrease toward zero for t

being about 3–5.

These observations are interpreted in the following way. The total time t

necessary for the dislocation to pass over to a neighbored Peierls valley is the

sum of the time t

to form a kink pair of the critical size, and the time t

for

the two kinks to travel until they are annihilated by kinks of opposite sign of

other kink pairs, t

= t

+ t

. Diﬀerent cases may appear.

1. If the pulse duration is smaller than the formation time of kink pairs

), double-kinks having already formed do not have time to expand.

They will shrink and annihilate again during t

+ t

, not resulting in

dislocation motion.

2. If the loading time is between the time of a kink pair to generate and the

time necessary to complete the transition to a new Peierls valley (t

), stable kink pairs may form and start to expand but, during the pause,

they become unstable and start to shrink. If the pause is long enough

), they recombine also not causing dislocation movement.

3. If t

, stable double-kinks form and the pulse separation is not

long enough for all double-kinks to recombine so that the remaining ones

can further travel during the next loading pulse. This is the range of

strongly increasing traveling distances.

4. If ﬁnally t

, most kink pairs have enough time to travel and to

annihilate with kinks of adjacent kink pairs. This is the region where the

traveling distance approaches the value of static loading.

The behavior described is possible only if the migration energy of the kinks

is not small. If ΔF

<kT, the kink velocity is very high and controlled by

viscous damping (see Sect. 4.2.2). Then, dislocation motion in case 3 above

would not be possible as all kink pairs would immediately annihilate in the

pause phases. Thus, in silicon, kink migration is also a thermally activated

process and the dislocation velocity of long dislocation segments is controlled

by (4.46), with the activation energy being the sum of the kink formation and

migration energies,

ΔG

≈ ΔF

+ΔF

A quantitative estimate yields ΔF

≈ 1.6 eV, which implies ΔF

≈ 0.6eV

with the value of ΔG

=2.2 eV quoted above [358]. The very high value of

ΔF

is problematic as the secondary Peierls potential cannot be higher than

the primary one. The authors of [373] therefore assume that extrinsic defects

control the mobility of kinks.

Adding foreign atoms to pure elemental semiconductors changes the dis-

location mobility slightly, as reviewed in [358]. Electrically inactive light

elements in Si like C, O, or N reduce the dislocation mobility at low stresses

leading to a higher stress exponent m. Below a critical stress, the motion is

6.3 Dislocation Dynamics 215

550 °C

0.00 0.05

x (Si)

0.10

(10

–6

m s

–1

)

(a)

800°C

0.90

0.95

1.00

x (Si)

(10

–6

m s

–1

)

(b)

Fig. 6.5. Dependence of the dislocation velocities in Ge–Si alloys on the Si content.

(a) Ge-rich alloys at 550

◦

C. (b) Si-rich alloys at 800

◦

C. Stress τ =20MPa.Data

from [374]

totally blocked. Electrically active donor impurities like P, As, and Sb increase

the dislocation velocity in the high-stress region without changing m, but

decrease it at low stresses yielding an increase in m. The eﬀect depends only

on the concentration, but not on the particular element. Acceptor impurities

like B inﬂuence the dislocation dynamics very little. Many models not treated

here have been developed to explain these observations.

In the Ge–Si alloys of elements of equal valency, the dislocation mobility

is reduced with respect to the pure constituents [374], as demonstrated by the

plots in Fig. 6.5, showing the dislocation velocities at ﬁxed stress in depen-

dence on the concentration x of Si atoms. The interpretation will be given

below (Sect. 6.5) in connection with the macroscopic deformation data.

The in situ straining experiments on polycrystalline silicon [163] allowed

the quantitative determination of the activation parameters of the dislocation

mobility by observing individual moving dislocations and by measuring the

dependence of the average dislocation velocity as a function of the applied load

at a constant temperature, and on the temperature. The low stress exponent

of semiconductor materials is of advantage for carrying out such quantitative

in situ straining experiments in a TEM, as changes in the dislocation velocity

are achieved by comparatively strong changes in the specimen load. At each

temperature, the load exerted on the specimen was adjusted to achieve dislo-

cation velocities in a range suitable for video recording. The average eﬀective

stress was measured by matching the shape of temporarily pinned disloca-

tion segments to loops calculated by the line tension model, as described in

Sect. 3.2.7 and in [101] for MgO single crystals. The loops calculated for the

most prominent slip system (111) (a/2)[10

1] were plotted in the projection of

the actual video recordings onto a (

112) image plane. Dislocation segments

were selected, which ﬁtted the ellipticity of the calculated loops. The relation

between the load and the eﬀective stress is shown in Fig. 6.6.

216 6 Semiconductors

200

100

F (N)

∗

(MPa)

530°C

585°C

625°C

690°C

Fig. 6.6. Dependence of the eﬀective stress τ

∗

in polycrystalline Si on the load

applied to the in situ tensile specimen. Data from [163]

The stress exponent was determined to be m =1.6 by applying ﬁve diﬀer-

ent loads at a constant temperature of 530

◦

C and by plotting the logarithm of

the dislocation velocities v

vs. the logarithm of the eﬀective stress τ

∗

. Similar

to the X-ray topography experiments in [262], the velocities were measured on

continuously moving dislocations so that the diﬀerence between the present

value and the exponent of unity found in [262] is not due to pinning as in

the etching studies. It may result from a lower purity of the present solar

grade crystals, the pinning of the dislocations by jogs, or the eﬀects of radi-

ation in the HVEM, as will be described later. To obtain an Arrhenius plot

of the logarithm of the dislocation velocity vs. the reciprocal temperature,

dislocation velocities measured at diﬀerent temperatures and under diﬀerent

loads were normalized to a constant eﬀective stress of τ

∗

=30MPausing

the above stress exponent and (4.103). The resulting Arrhenius plot is pre-

sented in Fig. 6.7. From the slope of this plot follows an activation energy of

ΔG

=(1.6 ± 0.3) eV. This energy is essentially lower than that determined

by other techniques. The dislocation velocities plotted by large open squares

in Fig. 4.42 are approximately equal to those measured by Imai and Sumino

[262] at high temperatures. At low temperatures, however, the dislocation

velocities in the in situ experiments are more than one order of magnitude

higher than in the X-ray topography experiments.

An earlier in situ study of the dislocation mobility in silicon by Louchet

[375] proved the dependence of the dislocation velocity on the length of the

gliding dislocations for short dislocation segments (<0.4 μm),asitispredicted

by the theory of the Peierls mechanism (4.47). More detailed data on this

eﬀect shown in Fig. 6.8 were given in [376] for germanium and in [377] for the

compound semiconductor ZnS. The early in situ experiments on semiconduc-

tors are reviewed in [378]. These papers and theoretical ab-initio calculations

[379] conﬁrm the high kink migration energy in elemental semiconductors

mentioned earlier.

6.4 Recombination-enhanced Dislocation Mobility 217

–4

–5

–6

–7

–8

Fig. 6.7. Arrhenius plot of the dislocation velocities normalized to a stress of

30 MPa. Data from [163]

Fig. 6.8. Dependence of the velocity of screw and 60

◦

dislocations in Ge single

crystals on the length of straight segments taken during in situ straining experiments

in a TEM at 430

◦

C and a stress of 40 MPa. Data from [376]

6.4 Recombination-enhanced Dislocation Mobility

The diﬀerences between the activation parameters determined in the in situ

experiments and the conventional ones may have several reasons. The Peierls

mechanism alone acts only at relatively low temperatures. At high temper-

atures, the dislocations are bowed out between high jogs. This transition

between two mechanisms is similar to that between the Peierls mechanism

and the localized obstacles described in Sect. 4.6. In the transition region,

the strain rate sensitivity decreases more strongly than the stress, which,

according to (4.11), is connected with an increase in the stress exponent m.

The nonconservative motion of the jogs results in the production of intrin-

sic point defects. Vacancy defects have been proved by positron annihilation

studies on Si and Ge crystals deformed under low-temperature high-stress con-

ditions (e.g. [380]). Subsequent annealing causes the vacancies to agglomerate

to clusters of about ten vacancies.

218 6 Semiconductors

The main reason of the high dislocation velocities at low temperatures dur-

ing the in situ experiments in the HVEM and the corresponding low value of

the activation energy (cathodo-plastic eﬀect) is attributed to the recombina-

tion enhancement due to excess carriers [381]. The high-energy electron beam

in the HVEM excites a large number of electron–hole pairs, which recom-

bine in a nonradiative process in dislocation-related places. In principle, the

released energy can enhance both kink formation or migration. Respective

theories have been developed in [382, 383] and reviewed in [384]. Kink for-

mation is more probable since a single process of enhancement is necessary

for the creation of a double-kink, where the kinks afterwards spread over a

longer distance, while many enhancement events are necessary to support each

migration step over only one lattice distance. Recombination-enhanced dislo-

cation motion can also be stimulated by other kinds of radiation, for example,

laser irradiation.

Special experiments to study the inﬂuence of the electron beam on the

dislocation mobility were performed during the in situ straining experiments

in an HVEM on Ge and Si-5 at% Ge single crystals discussed above (Sect. 6.2)

[368]. At 566

◦

C, the average dislocation velocity in Ge increased by a factor

of about three, while the imaging electron beam current was increased by a

factor of ﬁve. Measurements of the dislocation velocities at diﬀerent tempera-

tures but at a constant electron beam current yielded again drastically lower

activation energies than those experiments in which the stress pulse-double

etching method was applied without radiation [385].

6.5 Macroscopic Deformation Properties

The most prominent feature of the stress–strain curves of semiconductor single

crystals oriented for single slip is the strong yield drop between the upper

(UYP) and lower yield points (LYP), as outlined in Fig. 6.9. After the lower

yield point, there follows a region of low work-hardening. The upper and

lower yield stresses show a strong dependence on the temperature, but a

weak one on the strain rate. The yield drop from the upper to the lower

yield point is a typical eﬀect of both rapid dislocation multiplication and

the low stress exponent. At the beginning of deformation, the dislocation

density is low, leading to high dislocation velocities corresponding to a high

ﬂow stress. With increasing dislocation density, the dislocation velocity drops

down and accordingly the ﬂow stress, too. These features and the temperature

dependence of the ﬂow stress of semiconductor crystals can well be modeled

by a variant of the set of constitutive equations described in Sects. 5.1.1 and

5.2 [278, 279]. Figure 6.10 presents the temperature dependence of the lower

yield stress for silicon and germanium in a plot similar to an Arrhenius plot.

The temperature and strain rate dependence of the upper and lower yield

stresses can be described by a phenomenological relation [279,359]

6.5 Macroscopic Deformation Properties 219

Fig. 6.9. Shear stress vs. shear strain curves for high-purity silicon single crystals

grown by the ﬂoating zone method and deformed along a 123 tensile axis. Strain

rate 1.2 × 10

−4

−1

. 800

◦

C (1), 850

◦

C (2), 900

◦

C (3), 950

◦

C (4). Data from [386]

100

0.5

1.5

(MPa)

–1

)

Fig. 6.10. Temperature dependence of the lower yield stress of high-purity silicon

and germanium. Strain rate 2 × 10

−4

−1

. Data from [359]

or τ

=A˙ε

1/n

exp





(6.1)

where A is a constant and Q is an energy.

The latter equation is not an Arrhenius relation as it does not describe

the dependence of a rate on stress and temperature. It can be derived from

the power law (4.103), which is an approximation to the Arrhenius relation

and the Orowan equation (3.5) between the strain rate and the dislocation

velocity. However, the mobile dislocation density enters the prelogarithmic

factor of the Arrhenius equation of the strain rate. As the mobile dislocation

density at the yield points strongly depends on the deformation temperature,

the prelogarithmic factor is not constant. According to [359], the correla-

tion between the microscopic activation parameters m and ΔG

and the

macroscopic phenomenological quantities n and Q is given by

n = m +2,Q=ΔG

/(m +2).

220 6 Semiconductors

The reduction of the dislocation mobility by alloying the elemental semi-

conductors with additions of equal valency, as it was discussed for the Ge–Si

alloys, also yields an increase in the macroscopic ﬂow stress [374]. The yield

stress maximum probably occurs at the concentration x =0.5. With respect to

pure Ge, the hardening is relatively weak at low temperatures but very strong

at higher ones, resulting in an almost athermal behavior at x =0.4. For lower

concentrations, the hardening can be explained by the formation of Cottrell

atmospheres (Sect. 4.11). In the concentrated alloys, ordering phenomena may

also play some role.

6.6 Summary

In the elemental semiconductors, and in most compound semiconductors, too,

the dislocation dynamics is controlled by the double-kink mechanism over the

whole range of stress and temperature that is experimentally accessible. Then,

the character of the dislocation motion is viscous and the stress and temper-

ature dependence of the dislocation velocity can be described by the power

law (4.103), with a stress exponent m between 1 and 2. The in situ experi-

ments reveal that the dislocations are straight as expected from the Peierls

mechanism only at rest and at high stresses or low temperatures. Mostly, the

dislocations contain jogs, from cross slip of some segments as well as from cut-

ting forest dislocations. The latter may be strong obstacles to glide. In mixed

dislocations, the jogs glide along the dislocations in the direction of the Burg-

ers vector. Nevertheless, this process impedes the dislocation motion owing to

the high lattice resistance. Therefore, the dislocations bow out between the

jogs, thus contributing to the ﬂow stress. In addition, they may form dislo-

cation debris. Further obstacles to the dislocation motion are impurities and

their agglomerates. All these processes lead to values of m higher than unity,

which is expected from the pure Peierls mechanism.

During plastic deformation, most dislocations are created in localized

sources, resulting in the concentration of slip in narrow slip bands. The tem-

perature and strain rate dependencies of the macroscopic yield stresses (upper

and lower yield stress) can also be described by a power law. The stress–

strain curves can be modeled by the set of constitutive equations introduced

in Sect. 5.2. The strong yield drop eﬀect is due to the low stress exponents

as the strong initial increase of the dislocation density results in a similarly

strong decrease of the glide resistance. At low temperatures where the glide

resistance is high, semiconductor crystals are brittle.

Ceramic Single Crystals

Ceramic single crystals span a very wide range of materials from ionic crystals

like the alkali halides with ionic bonding over oxides with mixed bonding to

the carbides or nitrides with strong covalent bonding. Correspondingly, the

Peierls stress and the double-kink formation energies are very diﬀerent. On

the one hand, these quantities are low in the alkali halides so that the Peierls

mechanism controls the dislocation mobility and the ﬂow stress only at low

temperatures. On the other hand, the ceramic crystals with strong covalent

bonding become plastic only at high temperatures. Typical examples of ionic

crystals and oxides are presented in this chapter. Heuer and Mitchell recently

presented a review of the dislocation and deformation properties [387]. Struc-

tural applications of ceramic materials are mostly in the form of polycrystals.

At low temperatures, their strength is limited by the brittleness, which is con-

trolled by ﬂaws in the polycrystalline structure acting as crack nuclei. At high

temperatures, the materials creep. The creep rates are governed partly by the

properties of the grain boundaries and partly by the plastic properties of the

crystal grains, the latter of which are discussed in this chapter. A review of

creep parameters and models of polycrystalline ceramics is given in [340].

7.1 Alkali Halides

The dislocation processes during plastic deformation of alkali halide single

crystals were extensively studied in the 1960s and 1970s. An early review of

the plastic deformation is given by Sprackling [388]. These materials are con-

sidered model substances for defect properties of other solids because of several

reasons. Atomistic calculations of the defect properties are relatively easy and

reliable, as, for example, the theoretical estimation of the Peierls stress in [132].

The interatomic potentials to be considered here are the Coulomb attrac-

tion, Born–Mayer repulsion, and Van-der-Waals potentials. High purity alkali

halide crystals can be grown so that the intrinsic deformation parameters can