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On the yield point of floating-zonesilicon single crystalsM'barek Omri a , Claude Tete a , Jean-Pierre Michel a &Amand George aa Laboratoire de Physique du Solide , Unité Associée auCNRS No. 155, Institut National Polytechnique de Lorraine ,ENSMN, Parc de Saurupt, 54042, Nancy Cedex, FrancePublished online: 13 Sep 2006.

To cite this article: M'barek Omri , Claude Tete , Jean-Pierre Michel & Amand George (1987)On the yield point of floating-zone silicon single crystals, Philosophical Magazine A, 55:5,601-616, DOI: 10.1080/01418618708214371

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PHILOSOPHICAL MAGAZINE A, 1987, VOL. 55, No. 5,601-616

On the yield point of floating-zone silicon single crystals I. Yield stresses and activation parameters

By M’BAREK 0m1, CLAUDE TETE, JEAN-PIERRE MICHEL and AMAND GE~RGE

Institut National Polytechnique de Lorraine, ENSMN, Parc de Saurupt, 54042 Nancy Cedex, France

[Received 26 September 1986 and accepted 22 November 19861

Laboratoire de Physique du Solide, Unite Associee au CNRS No. 155,

ABSTRACT In initially dislocation-free silicon single crystals deformed in compression

(single slip) at low strain rates, the lower yield stress zIY exhibits a three-stage temperature dependence with a plateau (1170K5T51320K at js2 x s - l ) between the low- and high-temperature regimes, where zly decreases with increasing II: Pre-strained silicon crystals were deformed in the temperature range of dislocation velocity measurements (820 K 5 T 5 1070 K). With the assumption that pre-straining which minimizes yield-point phenomena, ensures dislocation struc- tures (i.e. internal stresses) that are at worst only weakly dependent on the deformation conditions, the activation Gibbs free energies derived from activation volumes measured at the lower yield point are found to be in very good agreement with activation energies for dislocation velocity at the same stresses.

8 1. INTRODUCTION The mechanical behaviour of silicon single crystals-and particularly of the yield

region-has long been studied only in the rather restricted temperature range 1070- 1270 K (about 0-65-0-75 T,, where T , = 1693 K is the absolute temperature of melting of silicon) (Alexander and Haasen 1968 (a review), Yonenaga and Sumino 1978, Suezawa, Sumino and Yonenaga 1979). In this range, the flow stress depends strongly on the temperature. One purpose of the present work was, therefore, to investigate higher temperatures to see whether such a dependence still holds up to the melting point. Two groups of authors have given new data in the high-temperature range (Mousset 1979, Mousset and Desoyer 1980, Siethoff and Schroter 1978, Schroter, Brion and Siethoff 1979, Brion, Siethoff and Schroter 1981, Siethoff 1983). Their studies were mainly concerned with later stages of the hardening curve. In particular, it was shown that silicon exhibits a five-stage stress-strain curve with two successive hardening and recovery stages. Siethoff and Schroter (1984) proposed that, while a t moderately high temperatures the transition from state I1 to stage I11 is best explained by cross-slip, as is commonly accepted in f.c.c. metals (Escaig 1968), a t very high temperatures cross-slip would instead be responsible for the second recovery stage V while the transition from stage I1 to stage I11 could be ascribed to a diffusion-controlled dislocation mechanism. However, little attention seems to have been paid to the yield region, which is the main subject of the present work.

On the other hand, glide velocities of single dislocations have been measured in Si by several groups (Chaudhuri, Pate1 and Rubin 1962, Erofeev and Nikitenko 1971, George and Champier 1979, Fischer 1975, Patu and He Yi Zhen 1980, Louchet 1981,

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Imai and Sumino 1983, Alexander, Kiesielovski-Kemmerich and Weber 1983). A range of temperature and stress is of special interest since it has been investigated by several authors, with the results being in close agreement. This central range can be delimited by 770 K < T < 1070 K and 3 MPa < T < 50 MPa, where t is the resolved shear stress. More recently, velocity measurements performed at higher temperatures by Farber and Nikitenko (1982) showed an abrupt change in the temperature dependence of the 60" dislocation velocity at T > 1470 K. Yet these data, which have not been confirmed by other groups so far, are given at stresses that are usually higher than yield stresses measured to such high temperatures. Dislocation velocities cannot be safely extra- polated at stresses smaller than about 5 MPa, since Imai and Sumino (1983) have shown that non-electrically active impurities like oxygen, carbon or nitrogen, the concentrations of which are rarely known accurately, do have an appreciable effect on the dislocation mobility. The data obtained in the central (7; t) range defined above, together with a law of dislocation multiplication and a theory of the effective stress, were successfully used by Alexander and Haasen (1968) to explain the yield behaviour of silicon. The fair agreement obtained (see also Schroter, Brion and Siethoff (1983)) can, however, be questioned since an extrapolation of velocities towards higher temperatures and smaller stresses was necessary to fit to the conditions of macroscopic deformation tests. A second purpose of the present work was to deform silicon in the temperature range used for velocity measurements. As will be shown, the range of moderate temperatures can be reached without experimental difficulties with a suitable pre-straining of samples. (Lower temperatures have been investigated recently using hydrostatic pressure to prevent specimen failure; sce Castaing, Veyssiere, Kubin and Rabier ( 1 981).) If an appropriate strain rate is chosen the stress ranges are also put into coincidence.

Velocity measurements show that dislocation glide is thermally activated with an apparent activation energy of - 2.3 eV, weakly stress dependent in the stress range investigated. In many materials in which plastic deformation is thermally activated (b.c.c. metals at low temperatures, for example) activation parameters have been measured in macroscopic experiments using techniques like the stress relaxation test (see Kocks, Argon and Ashby (1975) for a review). Surprisingly, these techniques have not been applied to element semiconductors, which would give a good opportunity to compare the results with direct velocity measurements. Our last objective is to derive activation parameters using the stress relaxation technique in the yield region of silicon, in order to check the validity of this approach.

5 2. EXPERIMENTAL DETAILS Mechanical tests were performed under a continuous flow of 10% H,, 90% N, gas,

with a high-temperature compression stage, similar to that described by Cadoz, Castaing, Dolin, Gervais and Pelissier (1975), mounted on an Instron machine.

Single crystals of n-type ( p > 5 R cm), float-zoned (FZ), dislocation-free silicon were obtained from Wacker, and compression samples with dimensions 425 x 4.25 x 14 mm3 with a (123) axis and { 11 l}, (541) side faces were cut with a wire saw and diamond polished (1/4pm).

Curves of shear stress t against shear strain y were derived from the recorded force/ elongation curves while assuming that the plastic strain was due to primary slip only, and was, homogeneous along the gauge length. The assumption of primary slip was checked by slip trace and TEM investigations. A precise determination of the gauge length is not easy. Clearly it is smaller than the total length of the compression sample.

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Yield point of FZ silicon single crystals 603

The deformed samples exhibited the usual S shape as for any crystal compressed in single slip (fig. 1 (a)). There was experimental evidence that end faces were not deformed. We assumed that dislocations lying in slip planes intersecting end faces could not glide in the area delimited by end faces and the slip direction. Figure 1 (b) shows the volume that accordingly can be plastically deformed. The correction on the gauge length is important (from 14 to 9mm). It has very little influence on the shear stress level for small strains, but it significantly increases the estimated shear strain rate. This must be kept in mind when comparing our results with those of other authors. Shear strain rates, v, of 2 x

For shear strains up to about lo%, no significant differences were observed in the t / y curves according to whether the compression rods were tightly blocked or allowed to move sideways as the crystals were sheared. The results reported here were obtained with blocked rods.

Chemical polishing was observed to increase slightly (at low strain rates) both the upper and lower yield stresses and to give a less homogeneous strain. In view of the comparison with single dislocation velocities, diamond polishing was preferred.

Apparent activation volumes V were derived from stress relaxation tests according to the Guiu and Pratt (1964) analysis. It is useful to stress some points on the application of this widely used technique to the specific case encountered here. The recorded decay of the stress AT after the crosshead has been stopped at t = to and t = 0

s - l to 2 x 10-4s-' were used here.

Fig. 1

(a) Shape of the compression sample after deformation and trace of primary slip lines (schematic). Near end faces, two kinked bands (hatched) roughly normal to primary slip lines appear on (1 11) faces. (b) Sample geometry. The volume assumed to deform homogeneously is situated between planes P and P . (For example, the shadowed part of the (111) plane which is above the plane P is not sheared.) The sample gauge length is then hh'.

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can be linearized in a semi-logarithmic plot of AT against In(t+c), provided that a constant c is added to the time scale. Then

V= kT/A, (1) and A is the slope of the straight line. The constant c is related to the testing conditions by

c = Ml/i,o, (2) where j o is the plastic strain rate at t = 0 and M is the stiffness of the machine-sample assembly (1/M 1: p/20, where p is the shear modulus of Si). In many instances, A can be determined without an accurate determination of c from the slope of the AT against In t curve at t >>c. It is clear, however, that the basic assumptions of Guiu and Pratt are not valid for long times, because in practice the dislocation structure changes during the relaxation test and activation volumes are stress dependent. In the present work, Az/ln t curves exhibited reverse curvatures after a duration of typically 2-3 minutes at T <923 K, i, = 2 x 10- s- ' and 15-20 s at T = 1023 K, i, = 2 x s- ' or T= 1123 K, j=2 x 10-4s-'. Relaxationcurvescouldnot beanalysedat T > 1123K becauseit was obvious that the dislocation structure changed very rapidly. (If the relaxation is no longer logarithmic, a transient peak stress appears on restraining, while the transient is monotonic after short logarithmic relaxations. The peak stress is believed to be due to the de-pinning of formerly mobile dislocations that were trapped in multipoles during the relaxation.)

Equations (1) and (2) show that, in our working conditions, the constant c was larger than the total duration of the relaxation test, because activation volumes wcre small, temperatures high and strain rates low. In such conditions the value of Vdepends strongly on that of c, and the graphical determination is very inaccurate and could lead to errors in Vof more than a factor of 2. Hence c must be determined consistently with the stiffness of the machine-sample assembly; provided this is observed, the direct technique of determining V using strain rate measurements during the relaxation used by Audouard (1979) or Cadoz (1978) was found to give values similar to those obtained by the Guiu and Pratt analysis.

The value of c determined by a computer using best-fit criteria to the logarithmic rule is often misleading owing to intrinsic errors in the data (pen inertia, etc.). if these uncertainties are properly taken into account, the computer determination of c is inaccurate.

Owing to these difficulties, apparent activation volumes could not be obtained with an accuracy better than within 1&20%.

0 3. EXPERIMENTAL RESULTS 3.1. Stress-strain curves and yield stresses in dislocation-free FZ silicon

Curves of shear stress against shear strain were obtained for j=2 x lop5 s - ' and 2 x 10-4s-' from 973 K (0.57 T,,) up to 1623 K (0.95 T,) up to a shear strain of 0.5. (Higher strains cannot be achieved in uniaxial compression because of the increasing shape deformation of single crystals.) The curves (fig. 2) exhibit the usual shape, with an initial peak stress followed by the three stages characteristic of f.c.c. crystals. At the highest temperature the five stages first observed by Siethoff and Schroter (1978) can be detected in our range of strain.

The amplitude of the peak stress as well as the length of the different stages decreases with increasing temperature. Figure 3 shows the temperature dependence of

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Yield point of FZ silicon single crystals

Fig. 2

t ( MPa)

10

0 I I I I

40 Ji"lo) 10 20 30 (a)

605

20 I ("lo) 10 (b)

(a) and (b) Resolved shear-stress/shear-strain curves for initially dislocation-free single crystals of FZ silicon at the shear strain rate j = 2 x 10-5s- ' . Note that (b) has expanded scales.

upper and lower yield stresses, z,,. and zly, for the two strain rates. The peak stress is seen to vanish out at T > 1323 K. zUy decreases with increasing T up to the disapperance of the peak stress. At the lower strain rate the temperature dependence of zly consists of three stages. The low-temperature range, in which zly is a rapidly decreasing function of T, is followed from about 1170K to about 1320K by a plateau where zly remains constant or even slightly increases with ?: At higher temperatures the flow stress extrapolated from stage I of easy glide, z,,, decreases again. In the plateau tly = 2.4 MPa; at T = 1623 K, zy N 0.25 MPa.

At the higher strain rate the plateau no longer appears on the curve of zly against ?: However, the difference of regimes between low and high temperatures is attested to by the double inflexion point at about 1320 K.

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Fig. 3

Ti, t", J 2x10-5s-1

rn 2x10-4s-'

1000 1200 1400 1600 T ( K )

Temperature dependence of upper and lower yield stresses at two strain rates in initially dislocation-free single crystals of FZ silicon.

3.2. Stress-strain curves and yield stresses in pre-strained silicon Even at the lowest strain rate used, deforming dislocation-free silicon was not

possible (with no confining pressure) at T c 973 K, fracture occurring then prior to the upper yield point. It can be seen on fig. 3 that there is only a small overlap between yield stresses and the average range of velocity measurements (3 MPa 6 T 5 50 MPa, 770 K 5 T g 1070 K). In order to extend the temperature range towards lower temperatures, a pre-strain was given to the samples. Pre-straining conditions were looked for to meet the following requirements: (i) to use standard conditions whatever the deformation conditions of the second test; and (ii) to create as many mobile dislocations as possible to reduce the peak stress due to their multiplication but keep hardening as low as possible. The best compromise was found at i, = 2 x 10- ' s- ' and T= 1323 K (in the plateau), with y = 7 x lo-' (before the end of stage I). In practice the sample was cooled to the testing temperature under the stress level attained at the end of pre-straining. After some stabilization, the sample was unloaded and immediately tested.

Figure 4 shows the effect of pre-straining on the stress-strain curve. At j = 2 x lO-'s-' and T=973 K the peak stress, although still apparent, is considerably reduced, and the upper and lower yield stresses are both decreased. At 1123 K, on the contrary, the reduction of the peak stress is due to a slight decrease in zUy and a marked increase in 'sly. The temperature T' above which zlY is increased by pre- straining depends on the strain rate: the higher the strain rate, the higher T'. At i, = 2 x s- l , T' was approximately 1070 K. Therefore the study of pre-strained samples was restricted to lower temperatures.

(The hardening induced by pre-straining at temperatures above T' is readily explained. As stated later, the flow stress in the athermal regime should be essentially equal to the internal stress. Since the dislocation structure, which creates this internal stress, is not likely to be recovered at lower temperatures and stresses, the pre-straining

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Yield point of FZ silicon single crystals

Fig. 4

z

4 0 973 K

( MPa) 7 z . ( MPa)

1123 K

10 -

I I 1 I b I I I 1 I b

607

Effect of pre-straining on the stress-strain curves: (a) T=973 K, ?=2 x 10-'s- '. Pre-straining induced softening in the yield region. (h) T = 1123 K, $= 2 x s- '. Unlike the upper yield stress, the lower yield stress is increased by pre-straining. (p) refers to pre-straining of 7 = 7 x 10 ' at T = 1323 K, $ = 2 x 10 s '; (ap) to deformation after pre-straining; and (wp) to deformation without pre-straining.

should lead to hardening and not softening in the thermal range when the lower yield stress without pre-straining is of the same order of magnitude as the stress attained at the end of the pre-straining treatment.)

Figure 5 shows typical curves obtained with pre-strained samples. At T = 8 18 K, silicon could be deformed without cracks, but this temperature was a limit set by the mechanical strength of our apparatus. The peak stress amplitude increases with respect to both coordinates when the temperature decreases. At 818 K, zu7/zIy= 1.5 and yly -0.1. The temperature dependence of zly measured after pre-straining is given in fig. 6. Values for zly range from 5MPa (T=1073K, j = 2 x 10-5s-1) to 50MPa ( T = 8 1 8 K , j = 2 x 10-5s-1; T=873K, j = 2 ~ 1 0 - ~ s - ' ) , g i v i n g a verygood overlap with the 5, T range of dislocation velocity measurements.

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Fig. 5

T . * (MPa)

,848 K

923 K

1023 K - 1123 K I

8 ( %) 7 0 10 0

Resolved shear-stress/shear-strain curves for FZ silicon single crystals at 3 = 2 x 10- s - after a pre-straining o f y = 7 x lO-’at T=1323K, 3 = 2 x 10-5s-1 .

IY

(MPa)

4 0

2 0

Fig. 6

i = 2x10-5s-1

. f = 8x10-5s-1

0- 800 900 1000 1100

T ( K )

Temperature dependence of lower yield stress in FZ silicon single crystals at different strain rates after a pre-strainingofy=7x at T = l 3 2 3 K , 3 = 2 ~ 1 0 - ~ s - l .

3.3. Activation volumes Figure 7 shows the variation of the apparent activation volume r/; measured at the

lower yield point, as a function of the applied stress. Because of the estimated error bars, the data obtained at different strain rates are not significantly different and may be considered as belonging to a single curve. (It can be noted, however, that at a given stress level the trend is that Vis slightly lower when measured at a higher strain rate.) Apparent activation volumes are small at high stresses ( V S lob3 at T , ~ 240 MPa) and rapidly increase as the applied stress becomes lower than IOMPa. The stress

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Yield point of FZ silicon single crystals

100 I \ Fig. 1

. i = 2x10-5s-1

. zi = Z ~ ~ O - ~ S - ~ . a; = 2 xlo-L 5-1

609

0 20 40 Z(MPa) Stress dependence of the apparent activation volume at the lower yield point (pre-strained

samples).

dependence of Vcould not be satisfactorily described by a law I/- l / t over the entire range investigated.

(Few activation volumes were measured at the lower yield point of samples that were not pre-strained. They were systematically higher, by about SO%, than those obtained at the same applied stress zly on pre-strained samples.)

0 4. ANALYSIS AND DISCUSSION

4.1. Comparison with other results in the literature The stress-strain curves, and particularly the shape of the yield region, agree

qualitatively with published work on FZ silicon (Yonenaga and Sumino 1978, Suezawa et al. 1979, Mahajan, Brasen and Haasen 1979). No Liiders band was found in dislocation-free FZ Si, contrary to what was often observed in Czochralski grown silicon. Homogeneous deformation was certainly favoured by the low strain rates used, but the fact that the state of the surface (chemically lapped or mechanically polished) had little effect on the yield stresses, and particularly on tuy, points to the efficiency of micro-defects as internal dislocation sources. Our crystals, indeed, contained swirl defects (De Kock, Roksnoer and Boonen 1975). As far as it is possible to interpolate between the differences of temperature, strain rate and initial dislocation density, the yield stresses obtained agree quantitatively with those measured by the foregoing authors.

It seems, however, that the three-stage temperature dependence of the lower yield stress has not been reported earlier. These three stages are commonly observed in the variation of the yield stress in many metals, although the plateau is usually much more extended than was found in silicon where it is apparent on zly only at very low strain rates; in previous work the use of higher strain rates prevented its detection. The physical meaning of these three stages is probably the same in Si as in most metals:

(1) In the low-temperature range, the flow stress is mainly governed by thermally activated dislocation glide, limited by the so-called lattice friction (Peierls mechanism) whatever its precise origin (atomic bonding, core anisotropy, etc.).

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610 M’b. Omri et al.

(2) In the plateau the flow stress is controlled by dislocation-dislocation interactions, which are at worst only very slightly dependent on temperature.

(3) In the ‘high-temperature’ range, which in silicon begins very close to the melting point, higher-temperature mechanisms like climb or extensive cross- slipping allow a still lower flow stress.

It is noticeable that at higher strain rates a direct transition occurs between low- and high-temperature regimes without any plateau, and that the use of data from this mixed region to check microscopic models would be misleading.

4.2. Derivation of the activation free-enthalpy In the following discussion only pre-strained samples deformed at the lower yield

point are considered. First the analysis of activation parameters is carried out without any reference to dislocation velocity measurements, theoretical description of disloc- ation glide by double kink formation, or models relating dislocation dynamics to the macroscopic flow stress. The following simplifying assumptions, to be discussed later, are made.

(i) The dislocation velocity is given by

u=v,exp[-AG(~,T)/kT]. (3) In the absence of applied stress the energy barrier AGO = AG(0, T ) is assumed to have the same temperature dependence as the shear modulus, p. The activation volume is defined by

v=-(F). T (4)

r! AG and the corresponding enthalpy variation are related by Schoeck’s formula

A H + ( ; ) ( = ) ” TV

AG =

The shear strain rate is given by Orowan’s formula:

where b is the Burgers vector modulus and pm the mobile dislocation density. (ii) In order to determine AC from measured quantities, an important simplific-

ation can be introduced since p is a weak function of temperature in silicon. In the range o f t and V values measured here, the second term in the numerator on the right side of eqn. (5) can be neglected and AG-0.9AH in the investigated range.

(iii) Owing to the homogenization of the dislocation structure by pre-straining, pm at the lower yield point is a weak function of the deformation conditions, the variations of which can be neglected compared with that of the Arrhknius factor.

(iv) Consistent with the assumption of similar dislocation configurations, the internal stress, t,,, which arises from long-range dislocation4islocation interactions, can be assumed to be constant, and the effective stress, teff, acting on mobile dislocations is simply

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Yield point of FZ silicon single crystals 61 1

T~ should be equal to the stress attained at the end of pre-straining, a value close to the yield stress in the athermal regime.

The experimentally determined activation volume can thus be identified with the actual V defined by eqn. (4), and AG can be determined by several techniques.

First, neglecting the entropy variation, eqn. (4) gives

AG(z)= Vdz, s:" where zo is the Peierls stress (i.e. the effective flow stress at 0 K). zo cannot be determined experimentally but, in a series of tests at a fixed strain rate, eqns. (3) and (6), with the above simplifications, give

(9) AG = kTln ( i 0 /3 ) = akT.

a can be determined by integration in a temperature interval Tl to T2:

AG(Ti)-AG(T,)= Vd~=ak(T2- TI). s:: If the a values derived from data from various temperature intervals within the experimental range are found to be equal, then a value of AG can be obtained by eqn. (9).

In the present work, a can be taken as constant for a given value of i, in the rather narrow investigated T range (see the table). Figure 8 gives the values of AG determined for different values of the effective stress using the integration technique. It may be checked that AG(z) determined from data at two different strain rates are not strictly equal. The difference, however, is not really surprising owing to the uncertainties in activation volume measurements. A more accurate derivation, taking into account the temperature dependence of p, has been proposed by Cagnon (1974). The correction is not significant.

AH can also be determined using the formula

with &,,/aT derived from the data of fig. 6. AH values are shown in fig. 9.

Values of the proportionality coefficient between AG and kT determined by the technique of integration over a limited temperature range (eqn. (10)).

j=2 x 10-5s-1

T(K) 823 848 873 898 923 948 973 998 1023 U 31.5 24.1 25.3 29.2 18.7 24.3 26- 1 20 1 Mean u = 25

j=2x 10-4s-1

873 923 973 1023 1073 24,5 23.8 20.7 19.9

T(K) U

Mean u=22

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612 M'b. Omri et a/.

Fig. 8

I . 5 j

I I I I + 0 10 2 0 3 0 4 0 teff(MPa)

Activation Gibbs free energy, AG, determined by eqn. (9) using average values of a. Full line, f = 2 x 1 0 ~ 5 s ~ ' , ~ = 2 S ; d o t t e d l i n e , f = 2 x 10-4sC1,a=22. Forcomparison,AGvalues for dislocation velocity have been determined from the apparent activation enthalpies, Q, via eqn. (12), which yields AG 1.0.9 Q. A screw dislocations; A 60" dislocations.

V

I

Fig. 9

0 : OP

0

1 I I I I b

0 10 20 30

Activation enthalpy, AH, at the lower yield point versus effective stress: 0 determined by eqn. ( 1 1 ) at j = 2 x 10-5s- ' ; determined by eqn. (11) at 3 = 2 x 10-4sC1; + determined from a plot of In f (interpolated) versus 1/T at given T ' ~ ; V, V activation enthalpies, Q, for screw and 60" dislocation velocities respectively (George and Champier 1979).

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Yield point of FZ silicon single crystals 613

Lastly from the zly( IT; 3) curves, In 3 can be plotted as a function of l/T, a t given zlY values, yielding A H directly. These values are also shown in fig. 9.

It is satisfactory that the three determinations give roughly the same activation enthalpy: slightly higher than 2 eV in the investigated T, T range.

It is usual to estimate the energy barrier at OK from eqn. (9) and the athermal temperature T,, beyond which the flow stress no longer depends on the temperature. At j = 2 x lO-’s-l, T,=1223K and cc=25, so that:

AGO = 2.6 eV.

4.3. Comparison with dislocation velocity measurements Dislocation velocities in Si are conventionally described using an Arrhenius law

with an apparent activation energy Q for the temperature dependence, while the stress dependence approximately obeys a power law with a stress exponent m =(a In v / a In T ) ~ of order unity. From Q and m, the Gibbs free energy can be derived from the analogue to Schoeck’s formula (Kocks, Argon and Ashby 1975):

Q + mkT(d In ,u/d In T ) 1 -(dln,u/dIn 7‘) ’ AG=

which means AG ~ 0 . 9 Q in the range of measurements; so that Q can be compared directly with AG and AH derived from macroscopic parameters. Q values obtained by George and Champier (1979) are shown for comparison in figs. 8 and 9. The agreement is good between AG and the activation energies for 60” dislocation velocities, although the stress dependence of AG is somewhat stronger than that of Q.

4.4. Discussion The Arrhknius type dependence of the strain rate with the same activation energy as

dislocation velocity was predicted by Alexander and Haasen (1968). However, if the temperature and stress dependences of the dislocation velocity are described by two separate laws, as in their model, there should be no reason to get a correct value of the activation free energy from the integration of Vdz. Either the good agreement observed here is accidental, or this is a strong indication that the stress dependence of the dislocation velocity in Si arises mainly from the stress dependence of the exponential term (i.e. the activation volume).

This, however, should not be taken as evidence that dislocation glide obeys such a simple law as in eqn. (3). Detailed investigations of the glide process revealed a rather complex mechanism with several elementary steps thermally activated: kink formation and migration, whose stress dependences are very poorly known (see, for example, Moiler 1978). Yet eqn. (3) can offer a good description of velocity measurements in the range considered here. It is emphasized here that the agreement between micro- and macroscopic activation parameters is obtained in the same range of stress and temperature. (It may be noted, too, that the successful use of the integration procedure does not necessarily imply that the process that appears to be rate controlling in the experimental range (approximately 820-1 100 K) is actually rate controlling over the whole temperature range from 0 K.)

Although the temperature dependence is the same, our assumption of comparable dislocation structures and mobile dislocation densities at the lower yield point, whatever the stress and temperature, strictly contradicts Haasen’s analysis. According to Haasen’s theory, the effective stress at the lower yield point should amount to

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614 M'b. Omri et a/.

roughly one-third of the applied stress, while at low temperatures our simplifying assumption is that nearly the whole applied stress is effective (z, ~ ( 7 , ~ ) .

Obviously the assumption of a constant P,,,,~(T) is not valid since, as displayed by fig. 5, the lower yield point is reached after much larger strains and through much more developed yield peaks at lower temperatures, even in pre-strained samples. There is, however, an experiment which can be conducted to prove that reff is much closer to T~,, than predicted by Haasen's theory.

After the standard pre-straining, a sample is first deformed up to the lower yield point at a low temperature (823 K). Then the deformation is stopped and resumed at a higher temperature (1023 K). If Teff = T , ~ (i.e. internal stresses or hardening are negligible), then the lower yield stress should not depend on the previous test. On the other hand, if some internal stress, T,,, is built up during the first deformation, the flow stress during the second deformation should be 72 >7,,, since T , ~ should not depend much on T and is not likely to recover appreciably at these very moderate temperatures. Figure 10 presents our experimental results. The low-temperature deformation did indeed increase the flow stress during the second deformation (and suppressed totally the upper yield phenomenon), yet this increase points to T,,, 5 10 MPa, to be compared with 71y= 42 MPa. So there must be in practice some increase of the mobile dislocation density with decreasing temperature, but much weaker than predicted by Haasen. It must be recognized that our operating conditions are different from those considered in most previous studies since the dislocation multiplication should be less efficient at the relatively low temperatures considered than in the range 1073-1273K investigated by others. Furthermore, the high dislocation density introduced by pre-straining should decrease the need for a rapid multiplication.

Fig. 10

6ol 40

t 2 0 I-

0 7 0

Stress-strain curves for a two-stage deformation experiment: (a) Deformation up to the lower yield point at T = 823 K after the standard pre-straining (p) at 1323 K. (b) (full line) Deformation at 1023 K after being deformed as in (a). (dotted line) Deformation at 1023 K after only the standard pre-straining (p) at 1323 K. All deformation experiments at the same shear strain rate, j = 2 x 10-5s-1.

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Yield point of FZ silicon single crystals 61 5

It must be stressed, too, that our analysis is consistent with a weak dependence of pm upon deformation conditions. How strong this dependence is cannot be determined from present measurements alone, owing to the above-mentioned inherent inaccuracy of activation volume measurements. Because of the error bars on AG, any prediction about the variation of the prefactor in Orowan’s equation would not make sense.

9 5. SUMMARY AND CONCLUSIONS Provisional conclusions from mechanical tests alone can be stated as follows.

( 1 ) With the use of a low strain rate, three stages can be distinguished in the temperature dependence of the (lower) yield stress in FZ Si, with a plateau between the well-known low-temperature regime and the high-temperature regime. This temperature dependence is similar to that of many metals, except that the transitions occur at much higher homologous temperatures.

(2) A suitable pre-straining allowed us to extend the investigated range of temperature at the lower end in order to put into coincidence the range of macroscopic mechanical tests and of dislocation velocity measurements for both stress and temperature.

(3) Measurements of activation volumes and their analysis, following the classical formalism of thermally activated plasticity, yield activation energies which compare very closely with those derived from dislocation velocities.

(4) There are indications that, in our rather low-temperature experiments on pre- strained samples, the effective stress, z,~~, is closer to the applied stress at the lower yield point that predicted by the model of Alexander a?d Hassen (1968), and that the mobile dislocation density, pm, is a weak function of deformation conditions.

( 5 ) However, the determination of activation parameters from stress relaxation techniques is not accurate enough to allow a precise determination of microscopic quantities such as teff and pm or their dependences on deformation conditions.

Further progress will depend on an experimental investigation of the dislocation structure. Such an investigation is made possible in silicon thanks to the possibility of freezing in the dislocation arrangements in the load-applied state by cooling down samples with full stress applied. This is the subject of the second part of this paper.

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by B. Chalmers, J. W. Christian and T. B. Massalski (Pergamon Press: Oxford).

Phys., 54, 1816.

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