Influence of Excitonic Scattering on Charge Carrier Ambipolar ... - Imec

1 Introduction
<strong>Influence</strong> <strong>of</strong> <strong>Excitonic</strong> <strong>Scattering</strong> on Charge
Carrier Ambipolar Diffusion in Silicon
Enn Velmre and Andres Udal
Tallinn Technical University, Dept. <strong>of</strong> Electronics, Ehitajate tee 5, EE0026
Tallinn, ESTONIA
Tel.: (372) 6202 150, Fax: (372) 6202 151,
e-mail: evelmre@santra.va.ttu.ee
Abstract: Calculation <strong>of</strong> drift mobilities and the ambipolar diffusion coefficient
dependences versus free electron and hole concentrations in low and medium
doped silicon have been performed and compared to experimental results in
order to evaluate the influence <strong>of</strong> excitonic scattering on charge carrier
transport in silicon at room temperature. Our analysis shows that in contrast to
electron-hole scattering the excitonic scattering may be an important factor to
explane the experimentally observed ambipolar diffusion coefficient reduction at
higher free carrier densities.
The decrease <strong>of</strong> electron and hole drift mobilities in semiconductors due to electron-hole scattering
(EHS) at high free charge carrier densities above 10 16 cm –3 is a well known fact. Hall [1] was the
first who pointed out that power diode I/V characteristics are affected by the EHS at higher current
densities. Fletcher [2] suggested that the total mobility should be calculated via summarizing <strong>of</strong>
reciprocals from lattice, ionized impurity and EHS related mobility components, i.e. by use <strong>of</strong>
Matthiessen’s rule. At that diffusion coefficients were rather formally related to mobilities by
means <strong>of</strong> Einstein’s relations. This approach predicted a significant reduction <strong>of</strong> ambipolar
diffusion coefficient Da, following the mobilities decrease at high excess carrier concentrations.
Measurements <strong>of</strong> mobilities [3]-[5] and the ambipolar diffusion coefficient [6] supported the
Fletcher’s approach and it has been conventionally used in most <strong>of</strong> device simulators.
However, even a simple analysis indicates that in ambipolar diffusion conditions when
electrons and holes move in the same direction with the same velocity, the mutual collisions
should not decrease the diffusion velocities at higher carrier densities. This physical picture has
been supported by sophisticated analyses (see e.g., papers [7]-[9] and references therein) based on
an extended transport model containing terms with ∇ϕ , ∇n and ∇p in both electron and hole
current density equations. However, measurements [10], [11] still show that the Da is rather
decreasing than levelling at high carrier densities. Recently, an attempt has been made [12] to
resolve this conflict between theory and experiment within a three-carrier transport model
considering electrons, heavy and light holes. Despite <strong>of</strong> diminished gap between experimental and
theoretical data a conclusion was drawn that inter- and intracarrier scattering effects cannot be
responsible for so strong decrease <strong>of</strong> ambipolar diffusion coefficient Da.

Obviously some new physical effects should be implemented to explane the Da behaviour. In
this paper we examine electron and hole scattering on Wannier-type excitons as a additional
scattering mechanism to remove the above-mentioned discrepancy.
2 Model description and calculation results
Present analysis is based on our previously published [13] charge carrier low-field transport and
mobility model, developed by use <strong>of</strong> Kohler’s variational principle solution <strong>of</strong> Boltzmann transport
equation. In addition to conventional lattice, ionized impurity and inter- and intracarrier scattering,
the scattering <strong>of</strong> free carriers on Wannier-type excitons is taken into account assuming spherical
parabolic bands for both electrons and holes. Excitons are considered as positronium-like particles,
which can scatter free carriers similar to neutral impurity atoms [14]. The relevant scattering terms
e ex
drs − *
h ex
and drs − *
* ee * hh
contribute to the determinant G elements Γrs
and Γrs
[13] like ion
e i
scattering terms drs − *
h i
and drs − *
do.
For exciton density description we propose two sub-models. The first (see Fig.1) is based on
thermodynamics <strong>of</strong> a mixed gas <strong>of</strong> free electrons, holes, and excitons. In this "chemical picture"
the mass action law is written as nex = n ⋅ p K(
T,
γ ) , where the chemical mass equilibrium
constant for exciton formation and dissociation in silicon is
3 2
K( T,
γ ) = Ccv(
T 300K
) exp( −ΔEex
⋅γ
kBT
) . In the present calculations we used exciton
unscreened ground state binding energy E = 14.
7 meV and the state density coefficient
18 -3
cm
Δ ex
C cv = 1 . 43×
10 , according to the degeneracy factor f = 21 [15]. The screening parameter
γ ( n + p)
, which approaches 1 at low carrier concentration and turns to 0 at Mott transition, was
calculated by use <strong>of</strong> a method proposed in [16]. For exciton effective mass the value
m 0.
1545 was chosen in order to assure γ = 0 Mott concentration
* =
ex
17 -3
cm
n Mott = pMott
= 5 . 2×
10 [15] at T = 300 K.
The second exciton concentration sub-model (see Fig. 2) is based on a rigorously derived
quantum-statistical mass action law for two-component Fermi systems with statically screened
Coulomb interactions [17], [18]. In this theory the correlated particle density includes
contributions from bound (i.e. excitons) and scattering states: at Mott transition the dissociation <strong>of</strong>
bound states is fully compensated by the jump in the density <strong>of</strong> scattering states. In this work,
when calculating mobilities (Fig. 3) and Da (Fig. 4) above Mott transition concentration, we have
considered scattering state similarly to the excitonic states.
Conclusions
In this work, we developed a model <strong>of</strong> electron and hole scattering on free excitons and
implemented it to explane the problematic ambipolar diffusion coefficient reduction at high excess
carrier concentrations in silicon. Despite <strong>of</strong> some eclecticism <strong>of</strong> our preliminary model, calculation
<strong>of</strong> drift mobilities and the ambipolar diffusion coefficient are in a quite satisfactory agreement with
experiments. It is deepening the belief that excitons may have a remarkable effect on device
behavior even at the room temperatures.

Acknowledgement
This work has been supported by the Estonian Science Foundation Project 2828.
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