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Non-radiative transition rates: Accepting and promoting phonon modes in an n-mode model
Journal of Luminescence 40&41 (1988) 609—610 North-Holland, Amsterdam
609
NON—RADIATIVE TRANSITION RATES: ACCEPTING AND PROMOTING PHONON MODES IN AN N—MODE MODEL
A. Schenk, D. Suisky, and R. Enderlein Humboldt—Universitdt zu Berlin, Sektion Physik, 1040 Berlin, Invalidenstrafle 110, GDR
The emission rate from deep levels due to multiphonon processes is calculated in the presence of an electric field adopting an N—mode model for the lattice vibrations. A closed analytical formula is obtained for any number of promoting and accepting modes. Experimental data on Si:Au are interpreted by means of this formula.
1. INTRODUCTION Transitions between localized and band states
2. field assisted multiphonon transitions (FAMT) Whereas in PAT the role of promoting modes is
induced by lattice vibrations are known to be
played by the non—diagonal part of the electric
the most important mechanism of radiationless
field interaction, both accepting and promoting
processes in semiconductors and insulators. The
modes are involved in FANT. PAT can be shown to
quantum mechanical transitions between the ml—
be much less effective than FANT for not too
tial and final electron states are caused by the
strong electric fields. In the present paper we
non—diagonal part of the electron—phonon inter—
concentrate ourselves on the latter process. We
action, whereas the diagonal part makes that the
use our theoretical results for an analysis of
transitions take place with the simultaneous ex—
experimental data on electric field dependent
citation and deexcitation of a certain number of
emission rates of Si:Au2.
phonons, balancing the electronic energy change. For a given phonon mode one of these parts may
2. THEORY
dominate over the other. If this happens one di—
We consider the emission rate e
of an elec—
stinguishes between promoting (non—diagonal part
tron captured at a deep centre of binding energy
essential) and accepting modes (diagonal part
EB into the conduction band due to multiphonon
essential). A theory of multiphonon transitions
and electric field induced processes. The fol—
is needed in this case which takes more than one
lowing general expression for en can be derived:
mode into account. Such a theory has been deve— loped by several authors’, restricting however
_
en
=
iT2
IdE IdE’ D(E, E’
)
~(E’, E)
.
(1)
to the minimum number of modes, i.e. two. In this paper we present a closed expression for
Here D(E,E’) means the electronic part of the
the transition rate which is valid for any num—
transition probability from a localized state
ber of promoting and accepting modes. The ex—
at energy E into a band state at E’, and ~(E’,E)
pression holds also in the presence of an exter—
is the corresponding lattice part. Closed ex—
nal electric field. It is known that the latter
pressions are known for ~E’,E) in the case of
can cause remarkable changes of the transition
one promoting and one accepting mode, including
rate via phonon assisted tunneling (PAT) and/or
the case that one and the same mode is both pro-
0022—2313/88/$O3.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A. Schenk et a!.
610
/ Non-radiative
transition rates
moting and accepting. A closed expression can also be obtained for an arbitrary number of mo—
~(E’,E)
exp(—~ r1 S r (2N r
=
+
1))*
des. It reads n
+~“
k
1,k 2’ ..k-°~m=1 n
2~
Si:Au
n
~
k (1+_~)~2
v’S
f ~
m
*
Sm
~
N 1 +1~~l _____ )2 N ‘Ik1~ (2S1/N1(N1-1- 1) 1
ó(E
—
)
*
0)-’
2
E’
+
~ hw k s1 a s
—
)
E
.
(2)
rel
4 Here the following symbols are used: hw.—phonon energy of mode i, S~Huang—Rhys factor, N~—pho—
3 2
-
I-
/,/
310K 280K —
modified Bessel function,
Erel~lattice relaxa—
element of electron—phonon interaction,In(z)_nth tion energy. 3. APPLICATIONS We apply expression (2) to a two—mode model,
‘S
1-
non occupation number, ft—non—diagonal matrix
•‘
i1,2, where mode 1 is of promoting type, i.e.
250
f1~0, S1=0, and mode 2 of accepting type, i.e.
0
220K /
—1
‘
I
0
/ /400K
f2=0, S2~0. The resulting expression data is used the2+—centre reinterpretation of so experimental on for the in Si which far2 have been anaAu lysed in a one—mode model. As can be seen from
/
Figure 1 good agreement between theory and expe—
2 4 6 Field strength / iü~Vcm —1
riment can be stated in the whole range of tern— peratures and field strengths.
In particular ,the
failure of the one—mode model at low temperatures and high fields is removed.
FIGURE 1 Field dependent emission rate of the Au 2+ acceptor in Si. Experimental points from’,theoretical curves for one—mode (dashed line) and two—mode models (full line). The following parameters are used for the one—mode model (i0):hw 00.O68eV, S 2.4, L(0.553eV-1.54 kT), and for the two— m~demode~hw 0.O7OeV, hw20.O68eV, E8(O.553eV —2.7 kT), S~ 3.8
REFERENCES 1. Yu.E. Perlin, and A.A. Kaminskii, phys. stat. sol.(b) 132 (1985) 11.
2. A.Schenk, K.Irmscher, D.Suisky, R.Enderlein, F.Bechstedt, and H.Klose, in:Proc.l7th Int. Conf.Physics of Sem., eds. J.D. Chadi and W.A. Harrison (Springer Verlag, New York, 613—616. 1985) pp.
609
NON—RADIATIVE TRANSITION RATES: ACCEPTING AND PROMOTING PHONON MODES IN AN N—MODE MODEL
A. Schenk, D. Suisky, and R. Enderlein Humboldt—Universitdt zu Berlin, Sektion Physik, 1040 Berlin, Invalidenstrafle 110, GDR
The emission rate from deep levels due to multiphonon processes is calculated in the presence of an electric field adopting an N—mode model for the lattice vibrations. A closed analytical formula is obtained for any number of promoting and accepting modes. Experimental data on Si:Au are interpreted by means of this formula.
1. INTRODUCTION Transitions between localized and band states
2. field assisted multiphonon transitions (FAMT) Whereas in PAT the role of promoting modes is
induced by lattice vibrations are known to be
played by the non—diagonal part of the electric
the most important mechanism of radiationless
field interaction, both accepting and promoting
processes in semiconductors and insulators. The
modes are involved in FANT. PAT can be shown to
quantum mechanical transitions between the ml—
be much less effective than FANT for not too
tial and final electron states are caused by the
strong electric fields. In the present paper we
non—diagonal part of the electron—phonon inter—
concentrate ourselves on the latter process. We
action, whereas the diagonal part makes that the
use our theoretical results for an analysis of
transitions take place with the simultaneous ex—
experimental data on electric field dependent
citation and deexcitation of a certain number of
emission rates of Si:Au2.
phonons, balancing the electronic energy change. For a given phonon mode one of these parts may
2. THEORY
dominate over the other. If this happens one di—
We consider the emission rate e
of an elec—
stinguishes between promoting (non—diagonal part
tron captured at a deep centre of binding energy
essential) and accepting modes (diagonal part
EB into the conduction band due to multiphonon
essential). A theory of multiphonon transitions
and electric field induced processes. The fol—
is needed in this case which takes more than one
lowing general expression for en can be derived:
mode into account. Such a theory has been deve— loped by several authors’, restricting however
_
en
=
iT2
IdE IdE’ D(E, E’
)
~(E’, E)
.
(1)
to the minimum number of modes, i.e. two. In this paper we present a closed expression for
Here D(E,E’) means the electronic part of the
the transition rate which is valid for any num—
transition probability from a localized state
ber of promoting and accepting modes. The ex—
at energy E into a band state at E’, and ~(E’,E)
pression holds also in the presence of an exter—
is the corresponding lattice part. Closed ex—
nal electric field. It is known that the latter
pressions are known for ~E’,E) in the case of
can cause remarkable changes of the transition
one promoting and one accepting mode, including
rate via phonon assisted tunneling (PAT) and/or
the case that one and the same mode is both pro-
0022—2313/88/$O3.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A. Schenk et a!.
610
/ Non-radiative
transition rates
moting and accepting. A closed expression can also be obtained for an arbitrary number of mo—
~(E’,E)
exp(—~ r1 S r (2N r
=
+
1))*
des. It reads n
+~“
k
1,k 2’ ..k-°~m=1 n
2~
Si:Au
n
~
k (1+_~)~2
v’S
f ~
m
*
Sm
~
N 1 +1~~l _____ )2 N ‘Ik1~ (2S1/N1(N1-1- 1) 1
ó(E
—
)
*
0)-’
2
E’
+
~ hw k s1 a s
—
)
E
.
(2)
rel
4 Here the following symbols are used: hw.—phonon energy of mode i, S~Huang—Rhys factor, N~—pho—
3 2
-
I-
/,/
310K 280K —
modified Bessel function,
Erel~lattice relaxa—
element of electron—phonon interaction,In(z)_nth tion energy. 3. APPLICATIONS We apply expression (2) to a two—mode model,
‘S
1-
non occupation number, ft—non—diagonal matrix
•‘
i1,2, where mode 1 is of promoting type, i.e.
250
f1~0, S1=0, and mode 2 of accepting type, i.e.
0
220K /
—1
‘
I
0
/ /400K
f2=0, S2~0. The resulting expression data is used the2+—centre reinterpretation of so experimental on for the in Si which far2 have been anaAu lysed in a one—mode model. As can be seen from
/
Figure 1 good agreement between theory and expe—
2 4 6 Field strength / iü~Vcm —1
riment can be stated in the whole range of tern— peratures and field strengths.
In particular ,the
failure of the one—mode model at low temperatures and high fields is removed.
FIGURE 1 Field dependent emission rate of the Au 2+ acceptor in Si. Experimental points from’,theoretical curves for one—mode (dashed line) and two—mode models (full line). The following parameters are used for the one—mode model (i0):hw 00.O68eV, S 2.4, L(0.553eV-1.54 kT), and for the two— m~demode~hw 0.O7OeV, hw20.O68eV, E8(O.553eV —2.7 kT), S~ 3.8
REFERENCES 1. Yu.E. Perlin, and A.A. Kaminskii, phys. stat. sol.(b) 132 (1985) 11.
2. A.Schenk, K.Irmscher, D.Suisky, R.Enderlein, F.Bechstedt, and H.Klose, in:Proc.l7th Int. Conf.Physics of Sem., eds. J.D. Chadi and W.A. Harrison (Springer Verlag, New York, 613—616. 1985) pp.