EVIDENCE OF NONLINEAR DISLOCATION MOTION IN INTERNAL FRICTION EXPERIMENTS

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EVIDENCE OF NONLINEAR DISLOCATION

MOTION IN INTERNAL FRICTION EXPERIMENTS

H.-J. Kaufmann, P. Pal-Val, V. Chernov, D. Kamajev

To cite this version:

H.-J. Kaufmann, P. Pal-Val, V. Chernov, D. Kamajev. EVIDENCE OF NONLINEAR DISLOCA-

TION MOTION IN INTERNAL FRICTION EXPERIMENTS. Journal de Physique Colloques, 1987,

48 (C8), pp.C8-113-C8-117. �10.1051/jphyscol:1987813�. �jpa-00227117�

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JOURNAL DE PHYSIQUE

Colloque C8, supplbment au 11-12, Tome 48, dbcembre 1987

EVIDENCE OF NONLINEAR DISLOCATION MOTION IN INTERNAL FRICTION EXPERIMENTS

H

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.

^KAUFMANN'

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^,^P

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.

PAL-VAL, V. M. CHERNOV* and D

.

^A.^KAMA^JEV*

Institute for Low Temperature Physics and Engineering,Academy of Sciences of the Ukrainian SSR, Lenin Prospect 47, Kharkov 310164, USSR

'~nstitute of Physics and Power Engineering, Obninsk 249020, USSR

Resum&: La valeur de la dgformation ultrasonique en fonction de la frgquence dans la rggion de la rgsonance fondamentale dans les monocristaux de Si <110> a &t& gtudige d l'aide de la m&- thode de vibrateur d deux composants d une frgquence de 88 kHz et d 5.9 K. Aux amplitudes glevges on observe un saut de l'amplitude de dgformation quand la frgquence s'approche de la rgsonance d'en bas od d'en haut. On explique les phgnomdnes observgs en supposant un caractsre non-lingdire des vibrations des dislocations dans le champ de tensions intgrieures.

Abstract: Frequency dependences of the magnitude of ultrasound deformation in the vicinity of the fundamental resonance have been studied by the two-component vibrator method in Si <110>

single-crystals at a frequency of about 88 kHz at 5.9 K . At high amplitudes a jump-like change of the deformation amplitude is observed when the resonance is approached from either low or high frequencies. A qualitative explanation of the effects observed is given under the assumption of the nonlinear character of dislocation vibrations in the internal stress fields.

1. Introduction In a number of recent theoretical papers /I-3/it is shown within the framework of the Granato-Lficke model that account of the nonlinear character of internal stresses acting on a dislocation may change our understanding of its behaviour under the action of ultrasound vibrations significantly. However, there are at present only scarce experimental data confirming the nonlinear behaviour of dislocations under the action of ultrasound /4-6/.

In this work some experimental results which may confirm the nonlinear character of dislocation motion in silicon single crystals at helium temperatures and a qualitative theoretical description of the effects observed are given.

2. Experimental Silicon single-crystals of semiconductor purity im- planted with a high dose of arsenic have been studied at 5.9 K by a twocomponent oscillator technique at a frequency of about 88 kHz

/ j / .

The samples were prepared in the form of plates of 0.4 x 7 x 35 mm

.

The longitudinal axis of the samples coincided with the crystallogra- phic direction of <110> with an accuracy

+

l o . The frequency dependencies of the current i via the quartz transducer were measured near the fundamental vibrator resonance as well as the amplitude dependencies of the decrement 6 at the resonance frequency in the forced operation regime. The measurement accuracy of i and 6 was

+

²^% ^and

(')present address : Academy of Sciences of the GDR. Otto-Nuschke-Str. 22/23, DDR-1086. Berlin. GDR

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987813

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JOURNAL DE PHYSIQUE

tained with increasing (direct Pig. 2 Frequency dependences of run; full circles) or decreasing current via quartz transducer at (a) (reverse run; open circles) fre- high and (b) low amplitudes. quency of forced vibrations

(Fig. 2b)

.

that of the vibration frequency was

+

1 Hz. The vibrator has been pla- ced inside the evacuated space installed in a helium cryostat. The pressure of gaseous helium in the space was 300 Pa. The structure of the cooled samples was stabilized by high-amplitude ultrasound irra- diation at a temperature of 6 K for 45 minutes /7/. After struc-

-0 ture stabilization all the dependencies obtained were completely

3

reproducible in numerous measu-

--5

$

^rements-

3. Results Fig.1 shows the amplitude dependence of the decrement (lower curve) and the relative change of the Young modulus (upper curve) (E-Eo) /Eo=2 (f s-f so)

. /fso; where Eo is the modulus value at the lowest ultrasound deformation amplitude & = Eomin;

0

--20 E is the modulus value with the

20 m ,o

15

X)

5

- -

-

- -

- . -

0 ' ^{- ,}

i68 10-7 lob 10-5 lo-L&o given Eo; fso, fs are the reso-

M ¹ 10 i r . @ ~ nance frequencies of the sample

at &omin and the given

cot

^re-

Fig. 1 Amplitude dependence of de- spectively. fso and f were fixed s

crement and relative change of at the maximum value of the Young's modulus. current i via the quartz

transducer: At the resonance fre-

?

P

.i

300

200

100

0

/o

018 96 014 0 2 -

0

on, whether they have been ob-

-

4

o

-

I I

-

I

z:J ,\,

-

quency the ultrasound deformation amplitude is proportional to the current i via the quartz transducerr/7/.

The plots show that in the samples studied a strong dependence of the relative change of

Young's modulus on amplitude is observed. At the same time, the dependence of the decrement on amplitude is weakly pronounced.

To interpret the nature of such an unusual behaviour of the quantity (E-E ) /E

,

the frequency dependenci?s

09

the current via the quartz transducer near the frequency of the fundamental resonance were studied in de- tail (Fig. 2a,b) at two diffe-

-

rent points of the amplitude

-

interval (marked by arrows 1

and 2 in Fig. 1 )

.

-

The shape of the functions i(f)

--

^- measured at comparatively low

-

ultrasound deformation amplitu-

86608 WOO t~~ des practically does not depend

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During the measurements in the high-amplitude region, the direct and reverse branches of the functions i (f) did not coincide (Fig. 2a).

Additionally, at the value i close to the maximum one, a jump-like change of i was observed in the curves i(f) both for the direct and reverse runs of the measurements. The maximum value of i in the direct measurement run was essentially lower than that for the reverse measurement run. However, in a few experiments the amplitude dependence of (E-E )/E has not been found, and symmetric frequency dependence i(f) hag be?n observed up to the highest amplitudes used.

4. Discussion A qualitative explanation of the change of the frequency dependence of i on increasing the forced vibration amplitude may be given under the assumption that the resonance properties of the samples are connected with the motion of dislocations. In the case of low forced vibration amplitudes (Fig. 2b), the dislocation structure of the crystal vibrates with a low amplitude near equilibrium position but the vibrations are of nonlinear character.

The displacement of the dislocation structure of a sample from equilibrium position is described by the functional variable u which belongs to the normalized space with the norm I(*II. The space norm is determined by the exact type of dislocation structure of the crystal.

For further treatment, the space norm is not interesting.

During motion the dislocation structure elements are subjected to viscous drag which is proportional to the velocity of the dislocations. As a general case the equation of motion to determine the displacement of the dislocation structure at the time t has the follow- ing dimension less form:

2

d u 2 du dV(u)

M - = + B - - - + aR [sin (fit)]

dt2 dt du

where M is the mass distribution operator; 'A is the operator of the elastic component of the restoring force; V(u) is the internal potential relief; B is the operator of energy dissipation; R is the operator of the stress distribution along slip planes; a is the forced vibration amplitude reduced to dimensionless f o r m ; n is the dimensionless frequency of the driving force. The potential relief V(u) is pro- duced by the internal stress fields of different nature (Peierls-Na- barro relief, fields of crystal structure defects etc.). As mentioned above, small oscillations of the dislocation structure near equilibrium are considered. For this reason, one may consider only the first nonlinear term in the expansion of V(u) and assume that

a

-

⁽ ^au)= A u

-

^{a W(u)}L

du ⁰

where W is the third order polylinear operator with the nosm I I w ~

.

Because of small-amplitude oscillations, the parameter y = a IIWJI is small (y<<l). The perturbation theory by the parameter y permits one to analyse the effect of the nonlinearity on the character of the frequency dependence in the neighbourhood of the linear resonance.

Consider the total energy of the forced periodic oscillation u(t&,y) at time zero does not limit the generality, since the steady-state oscillation regime is studied,

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C8-116 JOURNAL DE PHYSIQUE

In this expressionR and y are independent parameters but it is convenient to take the forced oscillation energy E as the independent parameter instead of the frequencyR, i.e.nand u may be considered as functions of E and y / a / . Therefore,

Taking y = 0 in ( 1 ) and (3)

,

a system of equations to determine zero terms uo a n d o in the expansion (4) is obtained. The differentiation of eqs. ( I ) an8 (3) with respect to y results in a system of equations providing expressions for u and

R 1

as well as the connection between

n

^and

^flo

with higher accuraJy.

The dependencies E ( Q ) a n d R ( . Q o ) are schematically presented in Figs. 3 and 4 (.A is tRe magnitude of the first linear resonance of the system).

The behaviour of the functionR(fio) is different depending on the magnitude a

.

At sufficiently small values of a, the function

R ( a o ) increases monotonically (curve 1, Fig. 4)

.

On increasing a the dependenceR(no) loses its monotonous behaviour and there is a dip in the vicinity of the pointRo = A(curve 2, Fig. 4).

Fig. 3 Frequency dependence of Fig. 4 Dependence of dimensionless forced oscillator energy (linear frequency on zero term of frequency approximation;

-

magnitude of expansion at (1) low and (2) high the first linear resonance of the amplitudes.

system)

.

Eliminating the variable

fi

from E ( Q ) and 2,

(n

⁾ ^{(curve 2}

Fig. 4) , one may obtain the dependence ofOthe forced vibrgtion energy of the dislocation structure on the external stress frequency (Fig. 5) which appears to be ambiguous. We assume the dislocation system to undergo forced vibrations corresponding to the point a in Fig. 5.

On decreasing the frequency

,

the point showing the state of the system moves along the curve E ( Q ) up to the point b. At the point b, the forced regime loses its stability, and the system falls to the point c , i.e. is transformed into another steady-state regime. When changing the frequency in the opposite direction, the point showing the state of the system moves along the curve E ( R ) from the point d into e, in which the jump to the point h takes place.

The total deformation of the crystal consists of the elastic

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deformation and the deformation due to the displacement of the dislocation system. The jump-like change of

E

f

the oscillation regime in the

points b and e

,

where stability is lost, (Fig. 5) is accompanied by the jump-like change of the dislocation component as maybe evident in the experiment (Fig. 2a).

However, it should be noted that application of this theory to the experimental results obtained

A

n

requires characteristic dimensions of the oscillating elements to be commensurable with the sample size.

Flg. 5 Dependence of the forced Nevertheless it seems unlikely to vibration energy on external find so long elements in the real stress frequency. dislocation structure of the sample

studied. Therefore alternative reasons for the non-linear behaviour are not excluded.

References

/I/ D. A. Kamajev, V. M. Chernov, Preprint, Institute of Physics and Power Engineering, Obninsk, N 1763 (1986)

/2/ D. A. Kamajev, V. M. Chernov, Fiz. tverd. tela,

z,

2077 (1985).

/3/ C. Herring, B. A. Huberman, Appl. Phys. Lett.,

36,

975 (1980).

/4/ P. P. Pal-Val, H.-J. Kaufmann, Fiz. nizk. temp.,

13,

635 (1987).

/5/ G. F. Baker, J. Appl. Phys.,

33,

3366 (1962).

/6/ R. De Batist, Internal friction of structural defects in crystal- line solids, Amsterdam-London-NewYork, N-HPC-Elsevier, 1972,p 44.

/7/ P. P. Pal-Val, H.-J. Kaufmann, Sov. J. Low Temp. Phys.,

9 ,

¹⁶³

(1 983)

.

/8/ Bifurcation theory and non-linear eigenvalues problems, Izd. Mir, Moscow 1974.