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THE INFLUENCE OF THE DYNAMICAL
JAHN-TELLER EFFECT OF ACCEPTORS ONTO THE PHONON-TRANSPORT MECHANISM IN
CUBIC SEMICONDUCTORS
J. Maier, E. Sigmund
To cite this version:
J. Maier, E. Sigmund. THE INFLUENCE OF THE DYNAMICAL JAHN-TELLER EFFECT OF ACCEPTORS ONTO THE PHONON-TRANSPORT MECHANISM IN CUBIC SEMICONDUC- TORS. Journal de Physique Colloques, 1981, 42 (C6), pp.C6-232-C6-234. �10.1051/jphyscol:1981666�.
�jpa-00221602�
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZe'ment au n012, Tome 42, de'cembre 1981 page C 6 - 2 3 2
THE INFLUENCE OF THE D Y N A M I C A L JAHN-TELLER EFFECT OF ACCEPTORS ONTO T H E PHONON-TRANSPORT MECHANISM I N C U B I C SEMICONDUCTORS
J. Maier and E. Sigmund
I n s t i t u t f l r Theoretische Physik, Universit&t Stuttgart, 7000 Stuttgart, F. R. G.
Abstract.- Phonon scattering experiments o f various types in cu- b i c semiconductors doped with deep effective mass acceptors in- dicate a n extra resonance scattering a t some meV. W e show, using Green's function and transformation techniques that these re- sonances are d u e to a dynamic Jahn-Teller effect o f the r8- acceptor ground state. Their influence o n t o the thermal conduc- tivity is calculated in terms o f a specific single mode relaxa- tion time.
1. Introduction.- A series o f phonon e ~ ~ e r i m e n t s ~ ' ~ ' ~ and optical e x - periments4 indicate a resonance energy in the meV-range f o r the deeper acceptors GaAs(Mn), GaP(Zn), Si(1n) and Si(B). These energies a r e much larger than the splitting d u e to random internal fields, which may be o f the order o f 1 0 to 1 0 0 peV in these crystals. In the following w e s h o w that these additional resonances are d u e to the dynamical Jahn- Teller effect and w e discuss their influence onto the phonon transport mechanism.
2. T I - T h e theoretical analysis o f our
calculation is based o n the Boltzmann equation5 o f the phonon distri- bution function Nxq(r,t):
w h e r e Nxq(r,t)=Nxq+GNxq 0 and N O =lexp(h whq/k~)-1l-'.v is the group
q ?q
velocity o f the phonon and defined by v = V w The f i r s t term repre- xq q Xq'
sents the local time-derivation, the second one arises from the d r i f t motion o f the phonons i n a gradient field and the third term is the source term and describes the rate o f phonon production and annihila- tion by the heater and detector. The fourth term is called the colli- sion o r scattering term and it reflects the physics of all phonon inter- action processes arising in the studied sample. The treatment o f this scattering term is the root problem in solving the Boltzmann equation.
I t describes the change of the phonon distribution function due to the different scattering processes. In relaxation time approximation this term is given by
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981666
a N
Aq 6 N ~ q
( ) = - -
at c T
1 q
T ~ , is the single-mode relaxation time. If all relaxation rates are kniwn the thermal conductivity can be calculated.
3. T h e Acceptor Ground State in Cubic Semiconductors.- T h e ground state o f acceptors in cubic semiconductors is fourfold degenerate (r8) and the interaction with the lattice vibrations leads to the possibility of a Jahn-Teller effect, which may cause a dynamical splitting of the elec- tronic levels.
T h e acceptor-hole-lattice interaction Hamiltonian c a n be written as 6
where pi and o a r e Dirac's 4 x 4 matrices. T h e coupling functions rlq j
and are given by J
h w 1 - e ] etc.
r ~ " ( ~ ) 1 ' 2 2 M c A f(q) 5 12CZeAz - 3,eAx Ay
and biq are the annihilation and creatien operators f o r the phonon bhq A
Xq. q is the unit vector along ;, i s the polarization vector and M the mass o f the crystal. DE(=D:) and D~(=D:,) a r e the deformation po- tential constants f o r a [1,0,0] and [1,1,1] strain respectively. f (q) is the cut-off function reflecting the extended nature o f t h e defect state.
4. The Single-Mode Relaxation Time.- T h e scattering r a t e o r the inverse l i f e time o f a single phonon can be related to the imaginary part o f the T-matrix:
- 1
T-I = -LoAq I. T(w )
Aq Xq A q , X q
T h e T-matrix itself is defined by the phonon Green's functions
(G=Go-GOTGO) of the unperturbed (Go) and perturbed (G) crystal. For the calculations w e used a n isotropic model f o r the crystal, w e expanded the Green's function hierarchy u p to the fourth order and w e calculated the thermal expectation values by use o f a n exponential transformation.
As a final result the mean scattering rate can be written in the Lorentz- ian-like form 6
<r-l>(w) = P ( w )
(.-A(.)) 2 + r ( w ) 2
P , A and r are rather lengthy exoressions, therefore w e have omitted them. In Fig. 1 the Jahn-Teller induced relaxation r a t e is drawn.
C6-234 JOURNAL DE PHYSIQUE
5. The Thermal Conductivity.- Taking into account the boundary (B) and isotopic (I) scattering a s well as the phonon scattering by the a c - ceptor states (JT) and the Umklapp processes ( U ) the thermal conducti- vity K(T) can be calculated7. In Fig. 2. the thermal conductivity o f the system Si(1n) for different In-concentrations is drawn. Its be- haviour is in good agreement with t h e experimental results.
- -c+
set1
I
lo9
?08
.lo7
lo6
.lo5
.loL F i g u r e
:::1
1-
h w l rneV5 10
Fig.1: Relaxatio t h e h n - T e l l er by the boundary n=1 . 1 0 6 c m - ~ ) as
n rate of longitudinal phonons by the scattering by state ( l / ~ by the isotopic scattering ( 1 / ~ ) , and s c a t t e r i n g ~ ~ l ) r ) in in-doped Si ( ~ n - c o n c e n t r a E i o n a function of t/e energy h w . T h e calculated resonance energy is 3.8 meV.
Fig.2: Calculated thermal conductivity K in Si(1n) as a function o f temperature f o r different In-concentrations n.
References:
(1) d e Combarieu, A . , Lassmann, K., Phonon Scattering in Solids, p. 3 4 0 , Plenum Press (1976)
(2) Lassmann, K., Schad, Hp., Solid State Commun. 18, 449 (1976) (3) Schenk, H., Forkel, W., Eisenmenger, W . , Fruhjahrstagung DPG,
Freudenstadt 1978
(4) Sauer, R., Schmid, b!., Weber, I., Solid State Commun. 27, 7 0 5 (1978) (5) Klemens, P.G., Sol. S t a t e Phys. 7 (1958)
(6) Sigmund, E., Lassmann,K., Phonon Scattering i n Condensed Matter, p. 4 1 7 , Plenum Press (1979)
(7) Callaway, J., Phys. Rev. 113, 1046 (1959)
