A disorder induced electronic filter is implemented using molecular systems such as semiconductor heterostructures and polymers. Two or more different materials are arranged to form a large number of layers with the arrangement being random and disordered so that no long range periodicity is created in the layers.

TECHNICAL FIELD 
This invention is in the field of solid-state physics. 
BACKGROUND ART 
Numerous applications in engineering and physics require the use of filters 
to selectively pass and reject various components of an incoming signal. 
For example, electromagnetic signals such as electrical and optical 
signals are filtered using filters implemented by electronic circuitry and 
optical lenses, respectively. Similarly, acoustic signals are filtered 
using mechanical means. Finally, charge-carrying particles, such as 
electrons, are filtered by electronic filters. 
Filters in each of these regimes invariably suffer from one main problem: 
the passband of the filter--and, in particular, its center frequency and 
width--are difficult to control precisely. This is especially true in the 
case of filters that are intended to be extremely selective, i e., those 
having extremely narrow passbands. Also, the response time of such filters 
while extremely fast, is limited by the transit time of the filter medium. 
A need exists, therefore, for a high-speed filter which has a very narrow 
passband. 
SUMMARY OF THE INVENTION 
The present invention discloses a class of disorder-induced electronic 
filters characterized by their extreme frequency selectivity. These 
filters are implemented using molecular systems such as semiconductor 
heterostructures and polymers. In the former case, two (or more) different 
materials are arranged to form a large number of layers, with the 
arrangement being random once certain constraints have been met. That is 
the layers associated with the heterostructure are disordered, in that 
there is no long range periodicity associated with these layers. The case 
of polymers is similar, with two (or more) molecular groups occurring 
repeatedly in a long polymeric chain, again with the arrangement of these 
groups being constrained in a specific sense but otherwise random. In the 
general case of molecular systems, two (or more) different molecules 
occupy lattice sites in a one, two or three dimensional lattice, again 
random with certain constraints. 
These arrangement constraints, which uniquely characterize the invention, 
are of two basic types. The first is embodied in the Random Dimer Model 
(RDM), in which at least one of the materials (in the semiconductor 
heterostructure case) is assigned to pairs of successive layers in the 
heterostructure. A generalization of the RDM exists and can be expressed 
mathematically. The second is the Random Binary Alloy (RBA), in which one 
of the materials cannot appear consecutively (i.e., a layer of this 
material must have layers of the other material on either side). A 
specific example of the RBA arrangement is the quasiperiodic Fibonacci 
arrangement of layers. 
In order to be used as an electronic device, the semiconductor 
heterostructure is equipped with an electrical contact at at least one 
end. Electrons are supplied at one of these contacts either by the 
radiation of an electron beam from an external source of electrons or by 
physically coupling the metal contact to external circuitry. For the layer 
arrangements just described, under certain conditions, only those 
electrons in a narrow energy range (a narrow band of electronic states) 
transport superdiffusively through the heterostructure without being 
attenuated, i.e., without being scattered. Hence, the heterostructure 
functions as a bandpass filter. The center frequency and width of the 
filter are determined by details of the heterostructure and by the number 
of layers in the heterostructure, respectively. In particular, these two 
filter parameters are easily predetermined to achieve extremely precise 
selectivity. In addition, it should be noted that the heterostructures of 
the invention can be easily fabricated using current deposition 
techniques, such as molecular beam epitaxy (MBE), chemical vapor 
deposition (CVD), and organometallic CVD (OMCVD), or even by depositing 
molecules one-by-one with the use of scanning tunneling microscopes (STM) 
to make extremely small heterostructures. 
Finally, the RDM and RBA molecular systems of the invention have 
application as high-speed solid-state switches (with the switching being 
controlled by a bias voltage) and in laser technology. In the switch 
application, electrical contacts are at both ends of the semiconductor 
heterostructure, and a bias voltage is applied across these contacts. When 
the bias voltage exceeds some threshold value, the heterostructure 
experiences a sudden dramatic drop in its resistivity, and electronic 
states in the heterostructure that lie in the heterostructure's passband 
transport through the device superdiffusively. The extremely short 
propagation time associated with these electronic states result in 
switching times on the order of picoseconds. 
In addition, the semiconductor heterostructures of the invention may be 
used in laser applications. Specifically, the heterostructure may be used 
to optically pump a lasing medium in a narrow frequency band.

DETAILED DESCRIPTION OF THE INVENTION 
A semiconductor heterostructure embodiment of the invention is shown in 
FIG. 1. The heterostructure 100 comprises layers of two different 
materials. Materials A and B may comprise semiconductor compounds 
consisting of elements from different columns of the periodic table. For 
example, material A may be a III-IV compound, such as GaAs, and material B 
may be a different compound, such as InP or GaP. Alternatively, material A 
may be a III-IV compound, as above, and material B may be a II-VI 
compound, such as CdS or ZnO, or a IV-VI compound, such as PbS. In fact, 
material A or B may be a quaternary alloy, such as InAlAsP. 
A metal contact 102 at one end of the heterostructure provides an 
electrical connection between the heterostructure and an external source 
of electrons. Specifically, electrons are applied at contact 102 either by 
the radiation of an electron beam from an external source of electrons or 
by physically coupling the metal contact to external circuitry. The 
arrangement of the layers of materials A and B result in a novel 
superdiffusive property. Specifically, certain electrons propagate through 
the heterostructure with a mean-square displacement that grows as 
t.sup.3/2, thereby giving rise to infinite conductivity in contrast to 
electrons being transported by standard diffusion in which displacement 
grows as t and conductivity is finite. 
There are two distinct layer arrangements that can yield superdiffusion. 
The first is called a Random Dimer Model (RDM), shown in FIG. 1. In the 
RDM, layers of at least one of the materials (i.e., either A or B or both) 
are present in pairs in the heterostructure. That is, for at least one of 
the materials. e.g., material B, an odd number of layers of material B may 
not lie between layers of material A. 
The second arrangement is that of the Random Binary Alloy (RBA). shown in 
FIG. 2. In the RBA. at least one of the materials A or B may not occur in 
adjacent layers but instead must always appear singly with the other 
material on either side. 
One example of an RBA is the Fibonacci lattice arrangement, an arrangement 
of considerable experimental interest due to its quasiperiodicity. The 
Fibonacci arrangement, shown in FIG. 3, is constructed in the following 
recursive manner. Fibonacci arrangements having 1 and 2 layers are given 
by A and AB. To construct the 3 layer Fibonacci arrangement, the 1 layer 
arrangement is appended to the end of the 2 layer arrangement, yielding 
(AB)(A)=ABA. Similarly, the 5 layer arrangement is constructed by 
appending the 3 layer arrangement to the end of the 2 layer arrangement, 
and so on. As is evident, the B's will always occur singly, and so the 
Fibonacci arrangement is a special case of the RBA. 
I. NARROWBAND ELECTRONIC FILTERS 
Under certain conditions, each of the two basic layer arrangements, RDM and 
RBA, exhibits the superdiffusive property that can be exploited to 
construct narrowband electronic filters. That is, electrons are donated at 
the heterostructure's metal contact (i.e., contact 102 of FIG. 1, contact 
202 of FIG. 2, or contact 302 of FIG. 3). If certain conditions are 
satisfied, then the electrons having energies in some range will transport 
superdiffusively through the heterostructure and emerge at the other end 
of the heterostructure. These conditions are stated below, with the more 
theoretical discussion being relegated to Section III. 
I.A. RANDOM DIMER MODEL 
Consider a layered heterostructure consisting of N layers formed from 
materials A and B according to the RDM. Let .epsilon..sub.A and 
.epsilon..sub.B denote the site energies (i.e.. binding energies in the 
conduction band) for materials A and B, respectively wherein the site 
energies are determined by the well depths in the semiconductor 
heterostructure. Also denote by V the constant nearest neighbor matrix 
element connecting successive layers. 
Then, if 
EQU .vertline..epsilon..sub.A -.epsilon..sub.B .vertline.&lt;2V, (1) 
a quantity proportional to 1/.sqroot.N of the electronic states in the 
conduction band will transport superdiffusively through the 
heterostructure in a narrow energy range. It can be shown (see Section 
III.B.) that the band of unscattered states corresponds to electrons whose 
momenta are within 1/.sqroot.N of 
##EQU1## 
where q is the fraction of layers of material B (i.e., the probability of 
choosing material B in the random arrangement) and W=.epsilon..sub.A 
-.epsilon..sub.B. Such electronic states can be thought of as being 
transparent to the disorder. All the other electronic states are 
localized, and hence are unable to contribute to transport. 
From this discussion, it is apparent that if Equation 1 is satisfied by 
appropriately choosing the heterostructure well depths (i.e., site 
energies) and well spacings (i.e.. separations), then the device 
consisting of the heterostructure equipped with metal end contacts can be 
used as an electronic filter. The center frequency (See Equation 2) of the 
passband of this filter is determined by the matrix element V, the site 
energies .epsilon..sub.A and .epsilon..sub.B, and the relative frequencies 
of the dimer defects. The width of the passband is determined by the 
number of layers in the heterostructure. Hence, the longer the 
heterostructure, the more selective the filter. 
The filter operates as follows. Referring to FIG. 1, electrons are donated 
at metal contact 102. As shown in the plot of electron density versus 
wavevector K. the inputted electronic signal at contact 102 is broadband. 
Only those electrons with momenta within the range .theta.-1/.sqroot.N to 
.theta.+1/.sqroot.N transport superdiffusively through the heterostructure 
100 and emerge at the output end 104 of the heterostructure. The 
electronic states lying outside this range remain localized inside the 
heterostructure. Hence, the outputted electronic signal is narrowband, and 
the heterostructure operates as a bandpass filter. 
I.B. RANDOM BINARY ALLOY 
Now consider a semiconductor heterostructure with materials A and B being 
arranged according to the RBA, with .epsilon..sub.A and .epsilon..sub.B 
denoting the same quantities as before. Here, however, two nearest 
neighbor matrix elements are needed, and accordingly let V.sub.A denote 
the matrix element corresponding to two successive type-A defects (i e., 
layers) and let V.sub.C denote the element for adjacent type-A and type-B 
defects. Note that two type-B defects may not occur successively, as must 
be true in the case of the RBA arrangement. 
As in the RDM case, superdiffusion is exhibited by a quantity of the 
electronic states proportional to 1/.sqroot.N, if a certain inequality is 
satisfied. The inequality here is slightly different: 
EQU V.sub.A .vertline..epsilon..sub.A -.epsilon..sub.B 
.vertline.&lt;2.vertline.V.sub.A.sup.2 -V.sub.C.sup.2 .vertline.. (3) 
As before, all the other electronic states are localized and are hence are 
unable to contribute to transport. It can be shown (see section III.C.) 
that the band of unscattered states corresponds to electrons whose momenta 
are within 1/.sqroot.N of 
##EQU2## 
Hence, if the condition of Equation 3 is satisfied by appropriately 
choosing the heterostructure well depths and well spacings, then the 
device consisting of the heterostructure equipped with a metal end contact 
can function as an electronic filter. The center frequency is given by 
Equation 4 and the width of the passband is given by 1/.sqroot.N. 
II. HIGH-SPEED SWITCH AND OPTICAL PUMPING APPLICATIONS 
Thus far, the use of RDM and RBA heterostructures in the context of 
bandpass filtering of electronic states has been described. In fact, these 
heterostructures exhibit behavior that allows them to function as 
high-speed switches. 
Consider the system of FIG. 4, which illustrates a semiconductor 
heterostructure 400 having input metal contact 402 and output metal 
contact 404. A nominal bias voltage E is applied across the contacts 402 
and 404 such that an electric field F.sub.E is produced in the 
heterostructure, with field lines pointing from right to left for a 
positive value of E. Electrons present in the heterostructure drift in 
response to the electric field. 
However, as is indicated by the FIG. 5 plot of resistivity .rho. versus 
1/E. as the magnitude of the applied voltage increases from its nominal 
value, the resistivity abruptly decreases near some threshold value. Hence 
for sufficiently high values of E, electrons in the heterostructure that 
lie in the energy band corresponding to the range of momenta .theta..sub.0 
-1/.sqroot.N to .theta..sub.0 +1/.sqroot.N transport superdiffusively 
through the heterostructure. The threshold behavior which causes the 
abrupt transition to superdiffusive transport for a band of electronic 
states permits the system to function as an electronic switch. Moreover, 
since the transit time for an electron within the superdiffusion passband 
to pass through the heterostructure is on the order of picoseconds, so too 
is the switching time, thereby yielding an extremely fast switching 
device. 
The RDM and RBA heterostructures can also be used in laser applications. 
Referring to FIG. 6, the heterostructure 600 is used to optically pump a 
lasing medium 602 with extremely narrowband radiation. 
III. THEORY AND DERIVATIONS 
In section I above, two types of disorder, namely RDM and RBA, that cause 
certain electronic states to undergo superdiffusion were described. It is 
this selective character of the heterostructure which permits the device 
to function as an electronic filter. In fact, RDM is one model in a 
general class of models that can be described mathematically. In this 
section, this general class of disorder models is introduced. Moreover, 
theoretical details that were omitted in Section I are included. 
III. A. GENERALIZATIONS OF RANDOM DIMER MODEL 
Assume that n is the direct lattice vector for a particular site an 
operator a.sub.n.sup.+ creates an electron at this site, and .mu. is a 
positive unit vector originating at n and pointing to the nearest neighbor 
sites along the .mu..sup.th direction of the crystal. Also, assume that 
V.sub..mu. is the bare bandwidth along the .mu..sup.th direction, and 
G.sub..mu.;n,n+.mu. is a random bond variable connecting sites n and 
n+.mu.. In the context of structurally induced disorder, 
G.sub..mu.;n,n+.mu. can be shown to be a function of the relative 
displacement between ions located at lattice sites n nd n+.mu.. Then, any 
system described by a Hamiltonian of the form 
##EQU3## 
will exhibit transmission resonances which give rise to superdiffusive 
transport if the site energies .epsilon..sub.n and transfer matrix 
elements V.sub..mu.;n,n+.mu. can be written as 
##EQU4## 
RDM arrangement is an example of such systems in that its well depths and 
barrier spacings are formed so as to satisfy Equations 6 and 7. 
III.B. DERIVATION OF RDM FILTER AMETERS 
To determine the center frequency (given by Equation 2) and width of the 
passband of the RDM filter, consider an otherwise ordered lattice with a 
single dimer defect (i.e , all but two adjacent layers are of material A). 
Place the dimer on sites 0 and 1 and assign the energy .epsilon..sub.A to 
all sites except these two sites which are assigned energy 
.epsilon..sub.B. A constant nearest neighbor matrix element V mediates 
transport between the sites. First it will be shown that 1/.sqroot.N of 
the electronic states are unscattered by the dimer impurity. Then, the 
center frequency of the passband for an arrangement containing randomly 
placed dimers (as opposed to a single dimer defect) is derived. 
To proceed, the reflection and transmission coefficients through the dimer 
impurity are calculated as follows: Consider the eigenvalue equation 
EQU EC.sub.n =.epsilon..sub.n C.sub.n +V (C.sub.n+1 +C.sub.n-1) (8) 
for the site amplitudes, where C.sub.n is the amplitude that the electron 
is at site n. Expressing the site amplitudes as C.sub.n =e.sup.ikn 
+.beta.e.sup.-ikn for n.ltoreq.-1 and C.sub.n =.gamma.e.sup.ikn for 
n.gtoreq.1 where .beta. and .gamma. are the reflection and transmission 
amplitudes, respectively, it follows from Equation 8 for sites -1 and 1 
that C.sub.0 =1+.beta.=.gamma.(We.sup.-ik +V)/V with W=.epsilon..sub.A 
-.epsilon..sub.B. Substitution of this result into Equation 8 for site 0 
results in the closed-form expression: 
##EQU5## 
for the reflection probability. Note that the reflection coefficient 
vanishes when .epsilon..sub.A -.epsilon..sub.B =2Vcosk or equivalently 
when -2V&lt;.epsilon..sub.A -.epsilon..sub.B &lt;2V. The location in the parent 
ordered band of the perfectly transmitted electronic state corresponds to 
the wavevector k.sub.o =cos.sup.-1 [(.epsilon..sub.B 
-.epsilon..sub.A)/2V]. To determine the total number of states that behave 
in this fashion, .beta. is expanded around k.sub.o. To lowest order, in 
the vicinity of k.sub.o, .vertline..beta..vertline..sup.2 
.about.(.DELTA.k).sup.2 where .DELTA.k=k-k.sub.o. 
Consider now a crystal containing a certain fraction of randomly-placed 
dimer impurities (i.e., not just a single dimer defect). Electronic states 
in the vicinity of k.sub.o will be reflected with a probability 
proportional to (.DELTA.k).sup.2. The time between scattering events .tau. 
is inversely proportional to the reflection probability. As a result, in 
the random system, the mean free path 
.lambda.=&lt;velocity&gt;.tau..about.1/(.DELTA.k).sup.2 in the vicinity of 
k.sub.o. Let .DELTA.k=(1/2.pi.)(.DELTA.N/N). Upon equating the mean free 
path to the length of the system (N) the total number (.DELTA.N) of states 
whose mean free path is equal to the system size is found to scale as 
.DELTA.N=.sqroot.N. Because the mean free path.about.localization length 
is in 1-dimension, the total number of states whose localization lengths 
diverge is .sqroot.N. Consequently, in the random dimer model, .sqroot.N 
of the electronic states remain extended over the total length of the 
system. Such states move through the crystal ballistically with a constant 
group velocity .nu.(k) except when they are located at the bottom or the 
top of the band where the velocity vanishes. Because all the other 
electronic states are localized, the diffusion constant is determined 
simply by integrating .nu.(k).lambda.(k) over the width of k-states that 
participate in the transport. The upper limit of the integration is then 
proportional to the total fraction of unscattered states or 1/.sqroot.N 
and .lambda.(k).about.N. In the case when the velocity is a nonzero 
constant, D.about..sqroot.N. Because the states which contribute to 
transport traverse the length of the system with a constant velocity, t 
and N can be interchanged so that D.about.t.sup.1/2. Consequently the 
mean-square displacement grows as t.sup.3/2. At the bottom or the top of 
the band where the group velocity vanishes, .nu.(k).about.k and 
D.about.-0(1). 
Finally, the exact location of the set of perfectly transmitted electronic 
states in the energy band of the disordered system (in other words, the 
passband center frequency) must be determined. To do so consider the 
correlated disorder model of Equations 5-7. Having shown above that the 
fraction of states in the vicinity of k=.theta..sub.0 that remains 
unscattered scales as 1/.sqroot.N, the value of .theta..sub.0 must be 
determined. To apply the model of Equations 5-7 to the problem at hand, 
note that the site energies in the Random Dimer Model can be constructed 
from a constrained bi-valued distribution of G's, i.e., from a 
distribution of the form G.sub.n,n.+-.1 =G.sub.a and G.sub.n,n.+-.1 
=G.sub.B with probabilities P and 1-P, respectively. Because the site 
energies are of the form .epsilon..sub.n =G.sub.n,n+1 +G.sub.n,n-1' 
G.sub.B cannot occur consecutively in the lattice. The resultant site 
energies will be .epsilon..sub.A =2G.sub.A and .epsilon..sub.B =G.sub.B 
+G.sub.A with the .epsilon..sub.B 's occuring in pairs. The matrix 
elements that are generated by G.sub.A and G.sub.B must be equal in the 
RDM. Solving the two simultaneous equations that result from Equation 7 
for the value of cos.theta..sub.o that makes the matrix elements equal 
yields the general condition for the location of the unscattered state: 
##EQU6## 
where q is the concentration of .epsilon..sub.A. Note that the location of 
the unscattered state is then a function of the concentration, as well as 
the relative disorder V/W with W=.epsilon..sub.a -.epsilon..sub.b. 
Substitution of Equation 10 into the restriction 
-1.ltoreq.cos.theta..sub.0 .ltoreq.1 yields the general result that 
-1.ltoreq.W/2V.ltoreq.1 for an unscattered state to exist. Note that when 
.epsilon..sub.A -.epsilon..sub.B =.+-.2V, cos.theta..sub.o =.+-.1, 
regardless of the concentration q. In this case, the unscattered states 
have zero velocity and diffusion is obtained. For all other values of W, 
provided that -1.ltoreq.W/2V.ltoreq.1, the location of the unscattered 
states depends on q and will have a non-zero velocity. 
III.C. DERIVATION OF RBA FILTER AMETERS 
To determine the center frequency (given by Equation 4) and width of the 
passband of the RBA filter, the reflection and transmission coefficients 
on an otherwise ordered infinite 1-dimensional lattice with a single 
type-B defect are computed. Place the defect with energy .epsilon..sub.B 
on site 0. All sites other than site 0 are assigned the energy 
.epsilon..sub.A. The matrix element V.sub.A connects all nearest-neighbor 
site pairs except site pairs (-1.0) and (0.1), which are connected by 
V.sub.C. It is now shown that .sqroot.N of the electronic states have unit 
transmission coefficients through the single defect. 
The eigenvalue equation for the site amplitudes is: 
EQU EC.sub.n =.epsilon..sub.n C.sub.n +V.sub.n,n+1 C.sub.n+1 +V.sub.n,n-1 
C.sub.n-1' (11) 
with site amplitudes C.sub.n expressed as C.sub.n =e.sup.ikn +Re.sup.-ikn 
for n.ltoreq.-1 and C.sub.n =Te.sup.ikn for n.gtoreq.1, where R and T are 
the reflection and transmission amplitudes, respectively. From the 
eigenvalue equation for sites -1 and 1, it follows that C.sub.O =V.sub.A 
/V.sub.B (1+R)=TV.sub.A /V.sub.B. Substitution of this result into the 
eigenvalue equation for site 0 results in the closed-form expression: 
##EQU7## 
for the reflection probability, where W=.epsilon..sub.A =.epsilon..sub.B. 
The reflection probability .vertline.R.vertline..sup.2 vanishes when: 
EQU V.sub.A .vertline..epsilon..sub.A -.epsilon..sub.B 
.vertline..ltoreq.2.vertline.V.sub.A.sup.2 -V.sub.C.sup.2 .vertline., 
and solving for the corresponding value of k.sub.o yields 
EQU k.sub.o =cos.sup.-1 (WV.sub.A /2(V.sub.A.sup.2 -V.sub.C.sup.2)) 
as the wavevector of the perfectly transmitted wave. 
To determine the width of the states in the vicinity of k.sub.o that remain 
unscattered. R is expanded about k.sub.o. To lowest order in 
.DELTA.k=k-k.sub.o, .vertline.R.vertline..sup.2 .about.(.DELTA.k).sup.2. 
Because the scattering time .tau. is inversely proportional to the 
reflection coefficient, the mean free path in the RBA heterostructure in 
the vicinity of k.sub.o scales as 
.lambda.=&lt;velocity&gt;.tau..about.1(.DELTA.k).sup.2. Equating .lambda. with 
the system size N yields the total fraction of states having mean free 
paths or equivalently, localization lengths (for d=1) equal to or longer 
than the system size as being given by .DELTA.k.about.1/.sqroot.N.