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Calculating the density of electronic states of disordered materials
Journal of Non-Crystalline Solids 75 (1985 ) 311-312 North-Holland, Amsterdam
311
CALCULATING THE DENSITY OF ELECTRONIC STATES OF DISORDERED MATERIALS T.M. Hayes Xerox Palo Alto Research Center, Palo Alto, California 94304, USA J.L. Beeby Department of Physics, University of Leicester, Leicester LE1 7RH, UK The powerful descriptive and theoretical techniques developed to calculate the electronic properties of crystals cannot be used to study systems without long-range translational order. There is no equivalent to the Bloch theorem. Theoretical treatments have been restricted accordingly to simplified models for the potential, such as bonding or tight binding, and to cluster models of the structure. 1 We describe here a new method with enough power and flexibility to overcome both of these restrictions and yield accurate densities of electronic states for a wide variety of disordered systems. In our approach, 2 a multiple-scattering expression for the density of states n(E) is partitioned into contributions F a, one for each atom a in the system. Although associated with an individual atom, F a depends on the positions of many atoms and is in no sense a local density of states.
As defined, however, F a is expected to depend principally on the
positions of atoms near a, and to be insensitive to the exact positions of distant atoms. A set of parameters is chosen so as to describe those structural aspects of each atom's environment which are judged to be important in determining n(E).
The set of variable
parameters will include the directions of local coordinate axes and may also include the number and type of neighbors, bond lengths, and so forth. For a given system (with specified atom coordinates), the set of exact integral equations coupling each F a with those associated with other atoms is written in a form wherein each F is expressed as a function of its own set of values of these parameters. The underlying assumption is that configurations with identical parameter values will make identical contributions to n(E). 3
These equations are averaged over the ensemble of all
configurations of the disordered system characterized by identical parameter values.
In
addition to the original set of parameter values, the structural information which survives this partial ensemble average is a set of (conditional) probability distributions for finding neighboring atom environments characterized by particular parameter values given specified parameter values associated with a central atom.
The exact set of integral
equations has been reduced to a single approximate equation with its kernel determined by these conditional probabilities. This equation is solved and the result averaged over the remainder of the complete ensemble to yield n(E). 0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
312
T.M. Hayes, J.L. Beeby / Density o f electron& states o f disordered materials Structural information is accordingly incorporated in n(E) in two ways--through the
probability distribution of the chosen parameter set over the ensemble used to describe the disordered system, and additionally through conditional probabilities characterizing the correlations among parameter values on neighboring atoms.
This structural ansatz is
clearly justified if the parameter set used is so large that it can describe precisely the positions of as many neighbors of each atom as is necessary, but represents an important approximation in the usual situation of a small set of parameters.
Results presented
elsewhere 2 suggest that ours is a good approach even with a very small parameter set. The local environment of each atom is retained in this treatment. NO part of the system is omitted but distant atom positions in a disordered system are averaged more than near neighbor positions. AS a result, each local configuration of atoms is embedded in an infinite network, the properties of which are self.consistently determined and matched to that local configuration. This network of atoms is characterized by the chosen degree of short-range order but can be devoid of translational periodicity.
Unlike the situation in other theories,
this self-embedding is an implicit part of our procedure.
It is well beyond the capability of
previous treatments in which the properties of the extended matrix usually affect the calculation in an unphysical way.
From the perspective of those calculations based on
specific clusters of atoms, we have formulated a procedure by which a local structural unit can be terminated with a self-consistent and in some sense ideal electron propagator. Although addressing in some respects the same issues as does the coherent potential approximation, or CPA, our approach is more sophisticated in that it preserves crucial elements of short-range order in a system without translational periodicity.
Note also that
this theory can accommodate any degree of short.range order from complete disorder to crystalline order.
In fact, it treats correctly the limit of translational periodicity.
Its
computational requirements are surprisingly modest given these attributes. In addition to being able to incorporate complex short-range order, this method .can accommodate any potential expressible in non-overlapping muffin-tin form.
It is therefore
applicable to all manner of disordered systems of current interest, including silicate glasses, amorphous semiconductors, metallic glasses, and liquid metals.
The reader is referred
elsewhere for details. 2 REFERENCES 1) For reviews, see B. Kramer and D. Weaire, Theory of electronic states in amorphous semiconductors, in: Amorphous Semiconductors, Topics in Applied Physics 36, ed. M.H. Brodsky (Springer-Verlag, Berlin, 1979) pp. 9-40; F. Yonezawa and M.H. Cohen, Theory of electronic properties of amorphous semiconductors, in: FundamentalPhysics of Amorphous Semiconductors, Solid State Sciences 25, ed. F. Yonezawa (Springer-Verlag, Berlin, 1981)pp. 119-144. 2) J.L. Beeby and T.M. Hayes (to be published). 3) For a discussion, see J.L. Beeby, Phil. Mag. B 4 8 (1983) L23.
311
CALCULATING THE DENSITY OF ELECTRONIC STATES OF DISORDERED MATERIALS T.M. Hayes Xerox Palo Alto Research Center, Palo Alto, California 94304, USA J.L. Beeby Department of Physics, University of Leicester, Leicester LE1 7RH, UK The powerful descriptive and theoretical techniques developed to calculate the electronic properties of crystals cannot be used to study systems without long-range translational order. There is no equivalent to the Bloch theorem. Theoretical treatments have been restricted accordingly to simplified models for the potential, such as bonding or tight binding, and to cluster models of the structure. 1 We describe here a new method with enough power and flexibility to overcome both of these restrictions and yield accurate densities of electronic states for a wide variety of disordered systems. In our approach, 2 a multiple-scattering expression for the density of states n(E) is partitioned into contributions F a, one for each atom a in the system. Although associated with an individual atom, F a depends on the positions of many atoms and is in no sense a local density of states.
As defined, however, F a is expected to depend principally on the
positions of atoms near a, and to be insensitive to the exact positions of distant atoms. A set of parameters is chosen so as to describe those structural aspects of each atom's environment which are judged to be important in determining n(E).
The set of variable
parameters will include the directions of local coordinate axes and may also include the number and type of neighbors, bond lengths, and so forth. For a given system (with specified atom coordinates), the set of exact integral equations coupling each F a with those associated with other atoms is written in a form wherein each F is expressed as a function of its own set of values of these parameters. The underlying assumption is that configurations with identical parameter values will make identical contributions to n(E). 3
These equations are averaged over the ensemble of all
configurations of the disordered system characterized by identical parameter values.
In
addition to the original set of parameter values, the structural information which survives this partial ensemble average is a set of (conditional) probability distributions for finding neighboring atom environments characterized by particular parameter values given specified parameter values associated with a central atom.
The exact set of integral
equations has been reduced to a single approximate equation with its kernel determined by these conditional probabilities. This equation is solved and the result averaged over the remainder of the complete ensemble to yield n(E). 0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
312
T.M. Hayes, J.L. Beeby / Density o f electron& states o f disordered materials Structural information is accordingly incorporated in n(E) in two ways--through the
probability distribution of the chosen parameter set over the ensemble used to describe the disordered system, and additionally through conditional probabilities characterizing the correlations among parameter values on neighboring atoms.
This structural ansatz is
clearly justified if the parameter set used is so large that it can describe precisely the positions of as many neighbors of each atom as is necessary, but represents an important approximation in the usual situation of a small set of parameters.
Results presented
elsewhere 2 suggest that ours is a good approach even with a very small parameter set. The local environment of each atom is retained in this treatment. NO part of the system is omitted but distant atom positions in a disordered system are averaged more than near neighbor positions. AS a result, each local configuration of atoms is embedded in an infinite network, the properties of which are self.consistently determined and matched to that local configuration. This network of atoms is characterized by the chosen degree of short-range order but can be devoid of translational periodicity.
Unlike the situation in other theories,
this self-embedding is an implicit part of our procedure.
It is well beyond the capability of
previous treatments in which the properties of the extended matrix usually affect the calculation in an unphysical way.
From the perspective of those calculations based on
specific clusters of atoms, we have formulated a procedure by which a local structural unit can be terminated with a self-consistent and in some sense ideal electron propagator. Although addressing in some respects the same issues as does the coherent potential approximation, or CPA, our approach is more sophisticated in that it preserves crucial elements of short-range order in a system without translational periodicity.
Note also that
this theory can accommodate any degree of short.range order from complete disorder to crystalline order.
In fact, it treats correctly the limit of translational periodicity.
Its
computational requirements are surprisingly modest given these attributes. In addition to being able to incorporate complex short-range order, this method .can accommodate any potential expressible in non-overlapping muffin-tin form.
It is therefore
applicable to all manner of disordered systems of current interest, including silicate glasses, amorphous semiconductors, metallic glasses, and liquid metals.
The reader is referred
elsewhere for details. 2 REFERENCES 1) For reviews, see B. Kramer and D. Weaire, Theory of electronic states in amorphous semiconductors, in: Amorphous Semiconductors, Topics in Applied Physics 36, ed. M.H. Brodsky (Springer-Verlag, Berlin, 1979) pp. 9-40; F. Yonezawa and M.H. Cohen, Theory of electronic properties of amorphous semiconductors, in: FundamentalPhysics of Amorphous Semiconductors, Solid State Sciences 25, ed. F. Yonezawa (Springer-Verlag, Berlin, 1981)pp. 119-144. 2) J.L. Beeby and T.M. Hayes (to be published). 3) For a discussion, see J.L. Beeby, Phil. Mag. B 4 8 (1983) L23.