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Maximum-entropy principle for static and dynamic high-field transport in semiconductors
Full text linkBy introducing generalized kinetic fields we have developed
an extended HD approach based on the maximum entropy
principle, which includes an arbitrary number of moments
of the distribution function. Then, for the perturbation
of these moments a set of coupled balance equations is constructed
and analytical expressions for all the small signal
coefficients are obtained in the time and frequency domains.
In particular, the generalized expressions for the response
matrix , the perturbing forces −e E
t
E, the response
functions Kt, and the differential mobilities
=X
+iY
are directly calculated for any moment of
interest. By generalizing previous results, the theory provides
also some relations in integral form and in asymptotic form
that are used to describe the small signal analysis. From the
knowledge of the quantities K0, X0 and of the derivatives
dK / dt0+,
d2K / dt20+,
dY /d
0,
d2X /d
20 the
anatomy of all response functions and differentials mobilities
is inferred in the time and frequency domains. The power of
the present approach stems from the construction of an algebraic
in place of an integral formulation of the theory.
Thus, from the explicit knowledge of ,
E the small
signal coefficients are consistently obtained in algebraic
form.
The theory is formulated at a kinetic level, without the
need to introduce external parameters, and it has been carried
out within a total energy scheme, thus using an energy dispersion
of general form full-band approach. The physical
plausibility of the theory has been confirmed by analysing
the high field transport in n-Si. To this purpose, as generalized
kinetic fields we have taken the independent quantities
=pui1 ̄uis and as unique independent mean quantities
the traceless moments F =Fpi1 ̄is, where the indices p
and s are associated with the isotropic and deviatoric parts of
the tensors, respectively. The analysis of dc and ac numerical
results shows that the behavior of all moments is determined
essentially by the competition between the action of the electric
field and that of the dissipative scattering processes. In
particular, the action of the electric field prevails on the moments
which have an increasing isotropic part while the action
of dissipative processes are more evident in the moments
with a large deviatoric part. By studying the
eigenvalues and eigenvectors of the response matrix we have
analyzed the coupling among the different macrovariables
moments and we have found that this coupling leads to a
nonexponential decay of the corresponding response functions.
In particular, by considering the moments F0s, with
null isotropic part and increasing deviatoric part, the combined
action of the electric field and dissipative processes has
been quantitatively investigated. We have thus demonstrated
that, at high fields: i the vanishing dc differential mobility
of different moments, ii the presence of complex eigenvalues,
iii the negative values taken by the response functions,
iv the positive overshoot of differentials responses, and v
the maximum of the real and imaginary parts of the ac differentials
mobility, are all related to the efficiency of dissipative
scattering processes. Analogously, by considering the
moments Fps with increasing isotropic part p1 we
have established that a simple relaxation approach based on a
single time scale loses of validity, because both the response
functions and the corresponding differential responses evolve
with different time scales. In particular, the energy response
function KW ̃ evidences the streaming character of hot carriers
through a nonmonotonic behavior with a maximum which
separates different time scales. The limits of the concept of a
single relaxation time are also evident in the shape of the
corresponding ac differential mobilities which show a nonregular
behavior at increasing frequencies before reaching
the cutoff. The theory has been validated by comparing the
present results with those obtained from MC simulations and
with available experiments for the standard quantities of direct
physical interpretation v,W ̃ ,S ̃. Therefore, we believe
that the present approach represents a useful standard to obtain
a generalized modeling of the relevant static macrovariables
of interest and of the small-signal dynamics coefficients
in terms of rigorous analytical formulas associated
with microscopic peculiarities of the single carrier transport.
In addition to offering an approach complementary to existing
kinetic method based on Monte Carlo simulations and/or
iterative solutions of the Boltzmann equation, the theory has
the advantages of providing a systematic framework to investigate
transport phenomena under far from equilibrium
conditions and of operating within a contained computational
environment
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