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SELF-ENERGY OF PHONONS INTERACTING
WITH FREE CARRIERS IN SILICON
L. Pintschovius, J. Vergés, M. Cardona
To cite this version:
JOURNAL DE PHYSIQUE
CoZZoque C 6 , supptdment au n o 12, Tome 42, ddcernbre 1982 page C6-634
SELF-ENERGY O F PHONONS INTERACTING W I T H FREE CARRIERS IN SILICON
*
L. Pintschovius, J . A . VergEs and M. cardona*
Abstract.- The dispersion in the self energy of acoustic phonons produced by free electrons and holes in silicon has been measu- red with neutron scattering. The corresponding electron-phonon interaction mechanisms are discussed.
Introduction.
-
Doping, either n- or p-type, is known to change the fre- 1quencies of acoustic phonons in silicon
.
This change is the real part of the self energy of the phonons produced by the free carriers via electron phonon interaction. Similar effects have been observed for the2
Raman phonons
.
Recent measurements in p-type Ge have shown that those self energies can depend strongly on phonon momentum q.3 Since conven- tional ultrasonic and light scattering techniques are not suitable for a detailed study of this q-dependence we have untertaken such study by means of inelastic neutron scattering. The acoustic branches of n-(n=5x10' 'cmh3) and p-type (p=1.7~10*~crn-~ ) Si were measured with respect to intrinsic Si throughout the (100) and (111) directions of the Bril- louin zone. The results yield information about hydrostatic and uniaxi- al deformation potentials for electrons and holes and the g-type inter- valley coupling of electrons.
Results and discussion.
-
The normalized self energy (Aw/w) of the TAphonons of n-Si along (111) at 300K is shown in Fig. 1. The data for q=o were obtained by ultrasonic methods and agree with those in (1). The self-energies are found to be, using second order perturbation theory:
whereE2= 8.7eV(4) is the shear deformation potential of the electron valleys, Cij are the stiffness constants and xe(q) the Lindhard suscep- tibility of a single electron valley (~=&~+24nx,) at T-300K (obtained numerically). The solid curve, calculated with Eq. ( I ) , decays faster than the experiments with increasing q. We believe it is additionally broadened by the finite mean free path R =62
a
(Aq=2n/%=O.O9(2n/a)).The results for LA phonons along (100) in n-Si are in Fig. 2.
They show definite structure for q=O.3(2~/a). The solid curve in Fig.2 was obtained with:
where ( A w / u ) ~ is the effect of g-coupling between (100) and (TOO) val- leys, resonant for q = 0.28 (2 /a), with coupling constant 0.8 e~/g(55, X J J and
XI
are Lindhard susceptibilities parallel and perpendicular to the valley axis, and€, is the "absolute" hydrostatic deformation po- tential of the electrons at A l (we takeEl=
-
7eV(6)). The structure at qe0.3(2.rr/a) is mostly produced by c l with a small contribution of(Aw/w)~. The
GI
effect is completely screened by the free carriers for q=o but becomes unscreened for finite q thus giving the bump observed at q = 0.3 (2a/a).
Fig. 3 discusses TA-phonons along (100) in n-Si. The effect at q=o is due to coupling of the A l with the A2, states along (010) and
(001 ) (deformation potential
&;
(7) ).
The calculated curve explains the q=o effect but fails to account for the observations for qfo.Figure 4 describes TA(111) phonons in p-Si. The effect is main- ly produced by heavy holes along { 110) directions. The solid curve was calculated in this manner with the deformation potentials b=-2.2eV and d=-4.9eV (4). For q=o it agrees with Ref. 8 about it falls short of the measured values for q#o. The additional mean free path q-broade- ning Aq=2a/R
-
0.09 (2r/a) suffices to explain the difference.Figure 5 shows Aw/w for LA(100) phonons in p-Si. In this case the screened effect of the hydrostatic deformation potential av=E1-a=-8.5eV
(a= hydrostatic deformation potential of indirect gap) produces the maximum at q=0.2(2a/a). The calculated curve describes qualitatively this observed effect.
References
1. R. Keyes, in Solid State Physics
20,
37, (1967)2. M. Chandrasekhar, J.B. Renucci and M. Cardona, Phys. Rev. B
17,
1623 (1 978)3. D. OlegoandM. Cardona, Phys. Rev. 23, 6592(1981) 4. 0. Madelung, Landoldt-B8rnstein ~ a b l z , in press
5. C. Jacoboni and L. Reggiani, Adv. Phys. 28, 493 (1979) 6. D. Gl5tze1, O.K. Andersen, J.A. Verges aflZi M. Cardona,
to be published
7. F. Cerdeira and M. Cardona, Phys. Rev. @, 4723 (1973)
C6-636 JOURNAL DE PHYSIQUE
Fig. 1.
-
Selfenergy o f TA [I111 pho- nons i n n-Si.
Fig. 3.
-
Selfenergy of TA [lo01 pho--
nons In n-Si : i t remains c o n s t a n t frcmaq/211 = 0.5 t o edge of BZ.
Fig. 2.
-
Selfenergy of LA 11001 phonons i n n-Si: i t remains c o n s t a n t from aql211 = 0 . 5 t o edge o f BZ.Fig. 4 .
-
Selfenergy of TA [1111 pho-r
nons i n p-Si.
Fig. 5.
