INTERACTION OF PHONONS WITH POINT DEFECTSINTERACTION OF PHONONS WITH STRESS-SPLIT ACCEPTOR STATES IN Ge AND Si

HAL Id: jpa-00215097

https://hal.archives-ouvertes.fr/jpa-00215097

Submitted on 1 Jan 1972

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTERACTION OF PHONONS WITH POINT DEFECTSINTERACTION OF PHONONS WITH STRESS-SPLIT ACCEPTOR STATES IN Ge AND Si

C. Elbaum, T. Fjeldly, T. Ishiguro

To cite this version:

C. Elbaum, T. Fjeldly, T. Ishiguro. INTERACTION OF PHONONS WITH POINT DEFECTSIN-

TERACTION OF PHONONS WITH STRESS-SPLIT ACCEPTOR STATES IN Ge AND Si. Journal

de Physique Colloques, 1972, 33 (C4), pp.C4-95-C4-99. �10.1051/jphyscol:1972420�. �jpa-00215097�

IN TERA C TION OF PHONONS WI TH POINT DEFECTS

INTERACTION OF PHONONS WITH STRE S S-SPLIT ACCEPTOR STATES IN Ge AND Si (")

C. ELBAUM, T. FJELDLY (**) and T. ISHIGURO (***) Physics Department, Brown University Providence, R. I. 02912 (U. S. A.)

Rbsume. - On a etudie la propagation des

pulses de chaleur

dans le Si-p et le Ge-p, en fonc- tion de contraintes uniaxiales (jusqu'a 1 x ¹⁰⁹ dynes cm

et de la temperature des pukes. On a obtenu des expressions pour les taux de diffusion des phonons par les processus importants, associts a la separation des etats des accepteurs par les contraintes. Ces taux ont kt6 obtenus pour I'absorption par rtsonance (pour les phonons ayant des energies proches de I'tnergie de separation) ainsi que pour quelques processus du second ordre. En plus, on a evalu6 la diffusion des phonons par les irnpuretes isotopiques, en utilisant le formalisme de Rayleigh. Le taux de diffusion totale calcul6, est en bon accord avec les amplitudes des phonons, mesurees en fonction des contraintes.

Une mise en accord des rtsultats experimentaux et calcules fournit deux groupes de valeurs des constantes du potentiel de deformation ; un groupe pour le cas

statique

(a* q < 1) et le deuxi6me groupe pour le cas

dynamique

(a* q 1) (a* est le rayon effectif de Bohr pour l'impuretk ; q est le nombre d'onde de la composante d'onde considerke). Cette observation rksout le conflit apparent entre les valeurs de ces constantes, obtenues par des etudes diffkrentes dans le regime statique (e. g., piezoreflexion) et dans le regime dynamique (conductibilit6 thermique). I1 faut souligner le fait que la methode des

pulses de chaleur >> convient particuli6rement bien B l'etude de ces differences, car les constantes pour les deux regimes s'obtiennent a partir de la mEme exp&- rience.

Abstract. - Heat pulse propagation has been studied in p-type silicon and p-type germanium as a function of uniaxial stress (up to 1 0 9 dynes cm

and of pulse temperature. The important phonon scattering rates due to stress-split acceptor states were derived. The rates were obtained for resonance absorption (for phonons with energies close to the splitting energy), and for several second order processes. In addition, Rayleigh scattering of phonons by isotopic impurities has been evaluated. The total, calculated scattering rate agrees well with the observed stress dependence of the heat pulse amplitudes. A fit of the experimental results to the calculated ones yields two distinct sets of values for the cr static

(a* q < 1) and dynamic D (a* q 5 1) deformation poten- tial constants (a* is the effective Bohr radius of the impurity ; q is the wave number of the relevant wave component). This observation resolves the apparent conflict in previously reported values of these constants, obtained from separate studies in the static (e. g., piezoreflectance) and dynamic (thermal conductivity) regimes. It should be emphasized that the heat pulse study is particularly well suited for investigating these differences, because the constants for the two regimes are derived within the framework of the same experiment.

Introduction. - The study of heat pulse propaga- tion provides an excellent method for investigations of phonon scattering. The advantages of this tech- nique over studies of phonon transport by means of thermal conductivity measurements have been dis- cussed a t some length previously [I]. In particular, the scattering of phonons by shallow impurity states in Ge and Si, which has been demonstrated in measu- rements of thermal conductivity [2]-[8], can be much better understood and compared more directly with

(*) Research supported in part by the National Science Foundation and the Advanced Research Projects Agency.

(**) Present address : Max-Planck-Institut fur Festkorper- forschung, Heilbronner Strasse 69, 7000 Stuttgart ^1, Germany.

(* * *) NASA International University Fellow. Present address : Electrotechnical Laboratory, 5-4-1 Mukodai, Tanashi, Tokyo, Japan.

theoretical predictions through studies of heat pulse propagation [9]-[I 01.

The coupling of phonons t o these impurities can be related t o the crystal symmetry a t the impurity site. In the case of p-type Si and p-type G e the ground state of shallow acceptors has the T, symmetry of the valence band edge and is thus 4-fold degenerate.

Any strain of lower symmetry acting on the impurity site will split the quartet into two Kramers doublets;

making it possible t o have transfer of holes between the two ievels by elastic or inelastic scattering of phonons [9]-[12]. The necessary level splitting can be caused by strains of any origin, including internal strains, external stressing, o r the lattice waves them- selves.

The experimental study discussed here consists in measuring the amplitude of heat pulses transmitted

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972420

C4-96 C. ELBAUM. T. FJELDLY AND T. ISHIGURO

through single crystals of the appropriate material, In order to verify whether the characteristics of the in a direction perpendicular to the applied uniaxial generator and detector were sensitive to the applied stress. These amplitudes are determined as a function stress, test experiments were conducted on a Si sample of stress, of acceptor content and of pulse tempe- containing - 1013 acceptors ; no effect of stress on

rature

the heat pulse amplitude and width was found.

The theoretical study consists in the following : The matrix elements of an appropriate hole-lattice Hamiltonian are calculated, from which the impor- tant phonon scattering rates due to stress-split acceptor states are derived. Specifically, the rates were obtained for resonance absorption (for phonons with energies close to the splitting energy), and for several second order processes. In addition, Rayleigh scattering of phonons by isotopic impurities has been evaluated.

A comparison of the experimental and theoretical results has led to the concept of static and dynamic deformation potential constants, applicable respecti- vely for static, or low frequency strains, a* q 1, and for high frequency strains, a* q 5 1. Here a*

is the effective Bohr radius, and q is the wave number of the relevant wave components. For low applied stresses (except in the case of the highest acceptor concentrations in Si used), nearly quantitative agree- ment is obtained between theory and experiment by assuming the presence of internal strains in the crystals and a black-body distribution of phonons at the source.

At high stresses the internal strains become irrelevant and the agreement is generally good. For the highest acceptor content in Si (8 x lo1' ~ m - ~ ) , the theore- tical and experimental results have similar qualita- tive features, but quantitative agreement is lacking.

This discrepancy can be accounted for, however, in terms of multiple phonon scattering near the phonon source.

Experimental procedure. - Gallium-doped Ge and boron-doped Si, with acceptor contents ranging from 5 x 1014 ~ m - ~ to 1.7 x 1016 ~ m and 1 - ~ x loi3 ~ m - ~ to 8 x 1015 cm-3 respectively, were studied. The acceptor contents were evaluated by means of room temperature resistivity measurements. In the case of Ge, uniaxial, compressive stress was applied in the [l 1 11 direction, and the heat pulses were propagated along the [liO] direction. In the case of Si, two types of samples were used. In one set the compressive stress and heat pulse propagation directions were the same as for the Ge ; in the other set the stress was applied in the [OOl] direction and the heat pulses were pro- pagated in the [I001 direction ; stresses up to 1 x lo9 dynes cmd2 were used. Evaporated constan- tan and 94 % In-6 % Sn films were used respectively as sources and detectors of heat pulses. A detailed description of the techniques used was given previously by Taylor, Maris and Elbaum [13]. In the present case the experiments were performed with sample temperatures ranging from -- 2

to -- 4

and heat pulse temperatures ranging from -- 5

to -- 17 OK ; the latter being evaluated on the basis of the black body radiation model for phonons.

Phonon relaxation sates. - The coupling of pho- nons to the acceptor holes and the splitting of the ground state quartet by external stressing are dealt with in terms of a strain Hamiltonian constructed from symmetry consicl~rations [14]

H,,,,~, = 2 D,[(L; - + ~ ' ) e , , + ^{C.P.] -t}

+ D:[(Lx L, - L, L,) ex, + ^{C. P.]} . (1) Here L, is the a-component of the angular momentum operator L (L = I), a = x, y or z referring to the Cfold axes ; D, and DL are the valence band deforma- tion potential constants ; C. P. denotes cyclic permu- tation of the indices x, y and z ; ego are strain compo- nents.

When the magnitude of the valence band edge splitting is small compared to the ionization energy of the acceptor and to spin-orbit splitting, the effect of the strain may be considered as a static perturbation potential acting within the ground state quartet. For the purposes of the present study, the energy split for uniaxial compression along the symmetry direc- tions [OOl] and [ I l l ] is considered.

By designating the stress by X, the conventional elastic compliances for cubic crystals by S,,, S,, and S,,, and using the appropriate strain components for the two stress directions [15], the splitting Hamil- tonian (1) becomes :

Hstrain = 2 Du(S1 ¹ - S12) X ( L ~ - 3 L2) ,

for X along [001] (2)

for X along [111] . ₍₃₎

The matrix elements describing the coupling bet- ween thermal dhonons and the acceptor holes are obtained, in principle, in a straight forward manner [l 11 by substituting an expansion in normal modes for the strain components into the strain Hamiltonian, and by using appropriate ground state wave functions. In practice, the calcuIations are quite involved ; a dis- cussion of these problems and explicit results (form factors) have been presented elsewhere [16].

The above results have been used to derive the rates for the important phonon scattering processes in strained p-type Si and p-type Ge. These processes are : a) resonance scattering, zL1, due to direct hole transitions [4] ; b) second order elastic scatter- ing [I I], [12], [17], 7,' ; c) inelastic scattering

-1

z1,2 = 71 + ^{z;' , where 2;'} refers to inelastic scat-

tering. by holes in the upper levels (Mj = f $),

and 2 2 refers to scattering due to

thermally assist-

INTERACTION O F PHONONS WITH STRESS-SPLIT ACCEPTOR STATES C4-97

ed

phonon absorption for o < A/h and to inelastic out using the second Born approximation, following scattering for o > A/& by holes in the lower levels the approche outlined by Kwok [18] for the case of ( M , = + $1. The second order processes all involve n-type Ge, and the extension to p-type Si and Ge transitions of holes via intermediate states by elastic discussed by Suzuki and Mikoshiba [12].

or inelastic scattering, or by thermally assisted phonon In the case of uniaxial compression applied along the absorption, and therefore do not, in general, require [ill] direction and fast transverse phonons propaga- resonance conditions. The calculations were carried ting along the [110] direction, the scattering rates are :

1 - Y (0) [I - exp (- $)I ^{o(hw + A )}

~ ( o f i),

7 1 ~ 2 ' 4 " ^{[ 1 + exp [ -t_ - A)][ 1 - exp ( -} ho&A)] ----

where p is the density, N is the content of acceptors, V2 is the velocity of the fast transverse wave, V, and V , are the velocities of the longitudinal and the transverse waves in the isotropic approxima- tion [Ill, [19], q is the wave number of phonons, D b n d D: are the deformation potential constants for static stress, while DEd and D$ are the deforma- tion potential constants for thermal phonons [12], [20], D = D:~/D",: and a* is the effective Bohr radius.

Analogous expressions for stress along the [OOl]

and phonon propagation along the [loo] direction were also calculated 1161.

In addition to the above scattering mechanisms, Rayleigh scattering of phonons by isotopic impurities was calculated separately, using the expression

7;' = Am4 (7)

where o is the phonon angular frequency and A = 2.4 x s3 in thecase of Ge, and 1.3 x s3 for Si [21] : Phonon-phonon scattering was also consi- dered. In both Si and Ge, at the temperatures used, the contribution due to this mechanism turns out to be negligible compared to the other processes. Finally, then, the total scattering rate for phonons, z-I, is given by

z-I = z,l + ^2;,: + z,l + ^7;'. (8)

In order to compare theory and experiment, the

STRESS [I

I]

7

a I

W

p-Ge I4

3 ^{0 0}

I

0 2 4 6 8

STRESS (10' DYNE SIC^')

FIG. 1 . - Amplitude of heat pulses (circles) vs. stress, for fast transverse phonons propagating in the [If01 direction, in Ge containing 5 x 1014 cm-3 acceptors. The solid curve is a theore- tical fit to the experimental data ; 1 x 10s dynes/cmz corres- ponds to a splitting energy, A, of 2.6 OK. The ((pulse tempera-

ture

is 6.6

and the sample temperature - ^3.5

scattering rates for the various processes considered were computed. Examples of the computed results are given in figures 2, 3, 4, which show the variation of these rates with frequency, at several values of split- ting energy, A (expressed in degrees K), for the accep- tor contents and pulse temperatures indicated.

The theoretical results were computer fitted to the

experimental data ; the solid curve in figure 1 is an

example of such a fit. In general, at low applied stresses,

the calculated amplitudes were somewhat higher

(typically 30 %) than the experimental ones. This

discrepancy can be accounted for by assuming tha.t

C4-98 C. ELBAUM, T. FJELDLY AND T. ISHIGURO

internal strains (due to crystal defects, etc.) cause some splitting of the acceptor states even in the absence of external stresses. A correction was therefore applied

FREQUENCY (Hz)

FIG. 4. - Scattering rates as a function of frequency in Ge containing 1.7 x

cm-3 acceptors, for longitudinal phonons propagating in the [110] direction. The rates are shown for four values of acceptor ground state splitting (expressed in K) due to [ I l l ] stress, and for a crystal temperature of 4.2 OK.

FREQUENCY (Hz)

FIG. 2. - Scattering rates as a function of frequency, in Si containing 1 x 101s cm-3 acceptors, for longitudinal phonons propagating in the [loo] direction. The rates are shown for four values of acceptor ground state splitting, A (expressed in OK), due to [OOl] stress, and for a crystal temperature of 3.36 OK.

FIG. 3. - Same as figure 2, but for transverse phonons.

to the results, by assuming the presence of such internal strains. The magnitude of the correction was selected to be equal, for each case, to the stress corres- ponding to the theoretical (uncorrected) pulse ampli- tude, when the latter is equal to the amplitude obser- ved at zero applied (external) stress. Typical values of this correction are 1-2 x 10' dyne cm-'. At higher applied stress the effect of the correction diminishes rapidly and the uncorrected, theoretical results fit well the experimental data. From the fitting procedure numerical values of D:, D$, D:' , D$ were obtained ; these are : a) for Si D", 3.7 to 3.9 (F 0.4) eV, Dz? = 2.5to3 (+ l.o)eV,~:' = 6.2 to6.8(+ 0.7)eV, Dz, = 4.7 to 5.2 (+ 0.5) eV ; for Ge Df,' = 3.7 to 4.0 (+ 0.4), Dtd ⁼ 5.3 to 6.9 (A 0.5) eV.

In the case of Si containing 8 x 10'' cmP3 accep-

tors, the experimental and theoretical results had the

same qualitative features, but the calculated pulse

amplitudes were much larger than the experimental

ones. This discrepancy can be accounted for in terms

of additional phonon scattering. This additional scat-

tering can be interpreted as a broadening of the reso-

nance interaction and is attributed to the increase

in phonon density near the generator. A numerical

estimate of this effect shows that close agreement

between theory and experiment can be obtained on

the basis of this mechanism. It is noteworthy that the

effect of additional scattering was not seen in even

more highly doped Ge. This observation is consistent

with the much weaker coupling between phonons and

acceptor impurities in the case of Ge.

INTERACTION OF PHONONS WITH STRESS-SPLIT ACCEPTOR STATES

References [I] VON GUTFELD (R. J.), Physical Acoustics, edited by

W. P. Mason (Academic Press, New York, 1968), 5, 233.

121 KEYES (R. W.), Phys. Rev., 1961, ^122, 1176.

[3] GRIFFIN (A.) and CARRUTHERS (P.), Phys. Rev., 1963, 131, 1976.

[4] GOFF (J. F.) and PEARLMAN (N.), Phys. Rev., 1965, 140, A 2151.

151 MATHUR (M. P.) and PEARLMAN (N.), Phys. Rev., 1969, ^180, 833.

[6] CARRUTHERS (J. A.), GEBALLE (T. H.), ROSENBERG (H. M.) and ZIMAN (J. M.), Proc. R. SOC., London, 1957, A 238, 502.

[7] CARRUTHERS (J. A.), COCHRAN (J. F.) and MENDEL-

SOHN (K.), Cryogenics, 1962, 2, 160.

[8] HOLLAND (M. G.) and NEURINGER (L. S.), Proc.

Int. Conf. Semiconductor Physics, Exeter, 1962, 474.

[lo] ISHIGURO (T.), FJELDLY (T.), ELBAUM (C.), Phys.

Rev. Lett., 1971, 27, 667.

[I17 SUZUKI (K.) and MIKOSHIBA (N.), Phys. Rev., 1971, B 3, 2550.

[I21 SUZUKI (K.) and MIKOSHIBA (N.), Phys. Soc. Japan, 1971, 31, No. 1.

[13] TAYLOR (B.), MARIS (H. J.) and ELBAUM (C.), Phys.

Rev., 1971, B 3 , 1462.

[14] HASEGAWA (H.), Phys. Rev., 1963, 102, 1029.

[15] HENSEL (J. C.) and FEHER (G.), Phys. Rev., 1963, 129, 1041.

[16] FJELDLY (T.), ISHIGURO (T.), ELBAUM (C.), Phys. Rev.

(to be published).

1171 KLEMENS (P. G.), Proc. Phys. Soc., London, 1955, A68, 1113.

[18] KWOK (P.), Phys. Rev., 1966, 149, 666.

[19] FEDOREV (F. I.), Theory of Elastic Waves in Crystals, Plenum Press, New York, 1968.

[20] D t , etc., represent acceptor ground state deformation potential constants. The theoretical treatment [9] POMERANTZ (M.) and ^VON GUTFELD (R. J.), Proc. can be carried out either in terms of D, or of

Int. Conf. Semiconductor Physics, Moscow, 1968, Dg.

2, 690. [21] HOLLAND (M. G.), Phys. Rev. 1963, 132, 2461.

DISCUSSION H. F. BUDD. -All your results show that low

frequency phonons are missing with respect to the Bose-Einstein distribution. In addition, as you states, the

temperature

required to fit the data is higher than expected for this distribution. This seems to agree with our calculations which show just this displacement of the phonon distribution towards higher frequencies.

C. ELBAUM. - Your interpretation of our experi- mental data seems to be incorrect in two respects :

1. At low stresses the observed pulse amplitudes are below the value calculated on the

black body

model (without internal strain correction), thus indi- cating additional absorption, rather than the absence of low frequency phonons. Furthermore, approxima- tely the same fractional decrease of the measured amplitude is observed for all pulse powers used (up to

black body

temperatures of 17 OK). This is certainly not consistent with your remark. The internal strain correction happens to be consistent with our observation.

2. The higher cr temperature

required to fit the data occurs only for

black body

pulse temperatures below about 9 OK.

These remarks are not meant to imply that we demonstrated the validity of the ct black body

model, but merely to indicate that your conclusion, as stated, is not born out by our experimental data, except possibly in a restricted temperature range.

A. ZYLBERSZTEJN. - Isn't your distinction between the rr static

and ^cr dynamic

deformation potential constant merely a reflection of the q dependence of the form factor

C. ELBAUM.

A rigorous theoretical treatment should yield a frequency independent deformation potential constant, with the frequency dependence contained in the form factors. In our theoretical treatment we used the effective mass approximation, in which we included d-like contributions, in addition to the s-like terms commonly used. In spite of this extension, a fit of the theoretical to experimental results still yields a frequency dependence in the

((

effective

deformation potential constant. We attri-

bute this outcome to the inadequacy of the effective

mass approximation for treating the problem at hand.