Fundamentals of Reliability Calculations for Molecular Devices and

6

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Fundamentals of Reliability Calculations for Molecular Devices and Photochromic Memories 1

Albert F. Lawrence and Robert R. Birge Department of Chemistry and W. M . Keck Center for Molecular Electronics, Syracuse University, Syracuse, NY 13244

Although molecular electronics offers significant promise, the issue of reliability remains a critical component to the analysis of potential commercialization. This chapter examines the mathematical foundations and the methods and procedures associated with calculating the reliability of a molecular-based device. The principal sources of error and their effects on the probability of correctly setting or reading the state of a binary device are defined. These results are used to derive formulas to determine the minimum number of molecules that must be used to define a binary state. In addition we examine the reliability of selected molecular devices currently proposed or under development for application as optically coupled gates and memories. In addition to accounting for sources of error such as quantum uncertainty, imperfect absorption, or measurement error, we also identify error sources arising from the dynamics of the read and write operations. These latter types of error can be described by using the mathematics of chaos in the statistics of ensembles.

THEPATHTOMOLECULARELECTRONICpresents some formidable technical problems, although systems based on molecular transformations offer great potential for achieving the ultimate limits in speed and density of processing components. Because the relevant physics on the nanometer scale is fundamentally different than the physics on the micrometer scale, it is probably unlikely that we will build exact molecular analogs to the devices that are familiar to the electronics engineer. Thus, 1

Alternative address: Biological Components Corporation, Menlo Park, CA 94025 0065-2393/94/0240-0131$09.80/0 © 1994 American Chemical Society

Birge; Molecular and Biomolecular Electronics Advances in Chemistry; American Chemical Society: Washington, DC, 1994.


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M O L E C U L A R AND BIOMOLECULAR ELECTRONICS

such devices as transmission lines, amplifiers, gates, and switches will probably not be available to a hypothetical circuit designer who works with structures composed of a few atoms. Nevertheless, it should still be desirable to build devices like digital computers, if only because of the rich and powerful computational theory that a Turing machine embodies. As a result, we require devices that embody basic logical functions such as " a n d , " "not," and " o r " and that perform their functions with adequate and well-defined reliability. Numerous molecular systems found in nature exhibit the transmission and control of signals through modulation of an effective potential. Often, the fundamental mechanics also include (1) a conformational switch from one metastable state to another, (2) the motion of a charged particle via a combination of tunneling and escape processes along a periodic potential, and (3) the direct coupling of the conformational switch to the transport process (1-4). A binding event or absorption of a photon of energy represents common mechanisms for initiating this chain of events. In both natural and synthetic molecular systems, movement of charge or mass in multiple potential wells represent the dominant mode of state change. This chapter describes the calculations that must be performed to estimate the reliability of a device. We will consider the barest of essentials: setting a switch and reading the state of a switch. In the physical world, both of these operations require the input of energy to change the state of a system of atoms. This is the primary or basic event. Whether the switch is a single molecule or a semiconductor structure containing more than 10 atoms, both of these operations are subject to error. We will discuss the theory that applies to a single read-write event and to the average of many read-write events. 12

Dynamics of a Molecular Switch The underlying mechanics of a molecular switch can be quite complex and subtle. Our present investigation emphasizes photochromic molecular switches. One of the most well-studied examples is bacteriorhodopsin, the light-harvesting component in the purple membrane of the halophilic microorganism Halobacterium halobium (4, 5). The photophysical properties of this protein are discussed in detail in Chapters 20-23 in this book. Thus, our discussion will be limited to a very brief overview relevant to the bacteriorhodopsin-based devices that will be examined at the end of this chapter. The light-adapted form of bacteriorhodopsin undergoes a photochemical cycle that is responsible for transporting protons from the inside (cytoplasm) to the outside (extracellular) of the membrane. The primary molecular event is a trans to cis photoisomerization around the C i = C i 3


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Reliability Calculations

double bond. The initial phototransduction event provides the motive force for the proton transport. At liquid nitrogen temperature, the native protein photochemically interconverts between a green absorbing state (bR, X = 577 nm) and a red absorbing state (K, X = 620 nm). At near ambient temperature, the native protein photochemical interconverts between a green absorbing state (bR, X = 568 nm) and a blue absorbing state (M, Xmax = 412 nm). These photochromic transformations occur with highquantum efficiency (4,5) and at liquid nitrogen temperatures, picosecond reaction times (6-8). The retinyl chromophore exhibits multiple metastable configurations in both the excited and ground states. The primary event involves a change in the shape of the conformational potential energy surface (due to the excitation of the electrons) followed by a conformational change and a nonradiative decay to the ground state (9). Because the barrier to conformational change in the excited state is small, or negative, absorption of light leads to the exceedingly rapid photoisomerization. In a sense, conformational motion in the excited state acts to effectively gate conformational changes in the ground state. The dynamics of the initial phototransduction in bacteriorhodopsin exhibit a number of features that complicate the mathematical analysis. First, the rapidity of the conformational change eliminates the assumption of adiabaticity. The nuclear motion must be analyzed as occurring in a changing potential. Many of the standard analyses of tunneling in a double well potential assume weak coupling between the tunneling motion and the other degrees of freedom of the system. This is not the case with bacteriorhodopsin. Finally, as the potential wells become more shallow, states other than those occupying the lowest level in each well must be taken into account. Because of these difficulties, a rigorous mathematical analysis is not yet available. Nevertheless, attempts to solve various special cases of this problem have led to a number of mathematical developments (9-11). At one time or another, analyses involving several areas of pure mathematics have entered the picture (12-21). The difficulty of the mathematical analysis of the double well potential underscores the need for a simplified picture of the dynamics. For this reason and for reasons of quantum uncertainty that will be described in the next section, we must limit ourselves to a probabilistic analysis. max

max


max

Origins of Error Although the dynamics of a molecule are deterministic in principle, statistics play a major role in determining the interaction of a molecule with an external perturbation. In the case of photonic events, this result follows from the collapse of the wave function that occurs upon the



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M O L E C U L A R A N D BIOMOLECULAR ELECTRONICS

absorption of a quantum of light. Previously, we described an example of a switch that is activated via the absorption of a photon of light by a chromophore from a flux of photons. This event has a certain quantum amplitude. That amplitude corresponds to a probability of observation that is never unity, and thus, a highly reliable device based on optical coupling will usually require more than one molecule to participate in the transformation of interest. Another important example is the transfer of an electron from one site to another. The observation of electron transfer from one part of a molecule to another is also governed by quantum statistics. Accordingly, single electron devices, which exploit the motion of a single charge, must also exploit the properties of an ensemble, but in a more subtle way (12-14). Exceptions to this statement are only possible when the write operation is carried out with a stimulus and a barrier to interconversion that far exceeds fcT, and when the read operation is nondestructive. The former requirement has typically re quired extremely low temperatures to be realized ((


n-*-oo

We can write similar expressions for the absorptions. If we define hl\ η

η

M,•0+1, a,n

(27)

hi η

then t/o photons of wavelength λ and t/i photons of wavelength λ are absorbed. In particular: 0

M

= N lim

\yi)

( Σ

0

ΜΟ+Χ,,^ΜΟ,Μ,,)^^) -

^\k=i

M

0

+

J* ) 0

1

(28)

W

J\xil

where Ν is a suitable normalization constant, (x , %i) is the state of the ensemble, and M i , is defined in a similar way to M . The matrices M and M +i, can be written in terms of matrix exponentials and integrals of matrix exponentials. Specifically, 0

0 +

i > n

M

0+ltt

M

0

+

M

0

i » = Exp[u- (M ,t + M 0

0 +

i, = a

N

a

U

-

(29a)

21)]

(29b)

M M (u)du 0Xa

Jo

0+ht

where u is the "time" parameter. Equations 20 through 22 and equations 24 hold with some of the M replaced by M or M , i , replaced by ijt

M

0 +

0+ht

0

a

i, . a

The next step in our analysis of probabilities is to consider errors in the measurement apparatus. We will assume that measurement errors are Gaussian distributed. Then by equation 13, Exp

P [(yo, t/i)l(*o, *i)] m

X

1

Expl

yo -

*o

_ (yi -

zi\

2


(30)

6.

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LAWRENCE & BIRGE

Furthermore, from equation 9 PAiyo, yi)\(xo, *i)] f

l

1

Γ

/y -zo\ 0

2

Exp 1


X

( 2 ' σ,„

ff V2i .

I

1

m

1

ExP

2

Expj

1

Zi - μ

&

2 'W

àz

x

(31)

l

Each integral is the convolution of Gaussians, so the joint conditional probability is the product of Gaussian distributions P r K y o , yi)\(xo, χι)] 1

Exp

\2

V / .

V

_ (tjl ~

Exp

Mr

(32)

where the parameters σ ° and μ ° are derived via equation 24 and the rules for convolutions. In particular, Γ

Γ

(33a) σ° = V( τ

Similar equations hold for σ

ffm

and

ι

τ

°) + 2

(33b)

(σ Ύ α

μ. ι

τ

Ensembles with Unknown Starting States. The final step in our analysis is to consider the effect of uncertainty as to the initial state (x , Xi) of the ensemble. We take this into account by considering the probabilities on the molecular level. For an ensemble set to state 0 we assume that an individual molecule has probability p of being in state A and probability p of being in state B, where we can take p = μο, , and so on. The terms p and p denote the corresponding probabilities for an ensemble in state 1. Then the photon absorption probabilities are as follows 0

0A

0B

0A

i A

Po,

5

1 B

PoA^A.a

£

Ρθ,ζ

=

0

+

ΡθΒ&Β,α°

POAk^aVoB^Ba

(34a) (34b)

1

and similarly for ensembles in state 1. Then we can approximate the absorption probability for an ensemble in state 0: PÂ(yo, y ύ\ο]

ι ο,

σ

Exp

0\2 οκ2

(yo - no,Q,a \ '

Exp

,a°/


MOLECULAR AND BIOMOLECULAR ELECTRONICS

148 where

Vpo, °(l-Po,a )

n

0

a

(36a)

VN

(36b)

MO,a° = Po,a°

and


(37a) M0,a

= PO,*

X

(37b)

1

Similar formulas hold for an ensemble in state 1. Read probabilities can be calculated as in equations 12, 13, and 31: Pr[(yo,yi)\i] = f

f P [(yo,yi)\(z ,z )]P [(z z )\i]dz dz m

0

1

a

0i

l

0

and so on

l

(38)

J o JO

Recall that the values E defined by equation 14 constitute the en tries of the confusion matrix for reading the state of the sample after writing either a 0 or a 1. In particular Ε and E i give the error rates for reading ensembles in the 0 and 1 states, respectively. The error terms may be calculated by substitution of equations 30, 36, 37, and 38 into equations 14. To make definite calculations we need a parametric description of the regions R and R i . If an optimal decision criteria is desired, that is, a threshold curve that minimizes E , i and E then the regions R and R will be separated by a curve which is a conic section. This is because the minimal error is associated with the threshold curve. This can be seen by observing that moving the threshold curve away from C increases the error. In particular, C is the curve on which f J

0Λ

) 0

0

0

0

(39)

C = {(»i, y )\P [(yi, ife)|0] = PA(yi, «/ )U]} 2

Exp 0,r

ff

ït0

x

r

(yi ~ M0,r ο,Λ ] \2 σ , 2

1/2

Exp

2

ι

0

,r° /

J σο,/1

Exp

t/2 ~ M0,r l/2 "Ό,ι

1\2

2

Exp

(40) «ι/

I .

2 r

Specific Devices These calculations are applied to some specific examples by Birge et al. (20,21). The parameters used in the exposition in this section are derived


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149



from references 20 and 21 and the references cited therein. We sum marize some of the results of the second paper and add a new discussion of the reliability of three-dimensional optical memories. In reference 21 Birge et al. make the basic assumption that the ensemble state as signments for the " 0 " and " 1 " states are given in terms of a probability for the correct assignment of the state of a single switch. This probability is denoted P. To simplify the calculations, quantum yield ratios are arbitrarily assigned to be 0.25 for the 0 state and 0.75 for the 1 state, so that the expected ensemble yields MO and μι are taken to be Mo = N(0.75 - 0.5P )

(41a)

Mi = N(0.25 + 0.5Ρ )

(41b)

a>

(1>

Here, Ν is the number of switches in the ensemble. Because of the uncertainties in prior state, the two extreme values for the standard deviation are given as follows:
7

0

max

1

max

s

-1

χ

-1

0

1

h

h

Three-Dimensional Optical Memories. Two-photon threedimensional (3D) optical addressing architectures offer significant promise for the development of a new generation of ultra-high density RAMs (see reference 5 and Chapter 7 by Dvornik and Rentzepis in this volume). These memories read and write information by using two or thogonal laser beams to address an irradiated volume (1-50 μπι ) within a much larger volume of a nonlinear photochromic material. A twophoton process is used to initiate the photochemistry, and this process involves the unusual capability of some molecules to capture two photons simultaneously to populate an energy level within the molecule with an energy equal to the sum of the energies of the two photons absorbed. 3


152


Because the probability to a two-photon absorption-process scales as the square of the intensity, photochemical activation is limited to a first approximation to regions within the irradiated volume. (Methods to cor rect for photochemistry outside the irradiated volume are described in reference 5.) The 3D addressing capability derives from the ability to adjust the location of the irradiated volume in three dimensions. Twodimensional (2D) optical memories have a storage capacity that is limited to — l / λ , where λ is the wavelength, which yields approximately 10 bits/cm . In contrast, 3D memories can approach storage densities of 1/ λ , which yields densities in the range 10 to 1 0 bits/cm . The design described subsequently is designed to store 18 GBytes (1 Gbyte = 10 bytes) within a data storage cuvette with dimensions of 1 . 6 X 1 . 6 X 2 cm. Our current storage capacity is well below the maximum theoretical limit of —512 GBytes for the same —5 c m volume. Bacteriorhodopsin has four characteristics that combine to enhance its comparative advantage as a nonlinear optical photochromic medium (4, 5). First, it has a large two-photon absorptivity due to the highly polar environment of the protein binding site, and the large change in dipole moment that accompanies excitation. Second, bacteriorhodopsin exhibits large quantum efficiencies in both the forward and reverse di rection. Third, the protein gives off a fast electrical signal when lightactivated that is diagnostic of its state (see Chapter 22 in this volume). Fourth, the protein can be oriented in optically clear polymer matrices permitting photoelectric state interrogation. The two-photon induced photochromic behavior is summarized in the scheme below: 2

8


2

3

11

13

3

9

3

bR(state 0)(X

max

s 1140 nm) ^ ^ 9 5 M(state l)(X

max

s 820 nm)

Photochemistry is initiated by using a two-photon process and laser wavelengths twice those used to initiate normal (i.e., one-photon) pho tochemistry. We arbitrarily assign bR to binary state 0 and M to binary state 1. The chromophore in bR has an unusually large two-photon ab sorptivity (b = 290 X 1 0 " c m s molecule" photon" ), a value that is approximately 10 times larger than absorptivities observed for other polyene chromophores (4). This permits the use of much lower intensity laser excitation to induce the forward photochemistry. The above wave lengths are correct to only ± 4 0 nm, because the two-photon absorption maxima shift as a function of temperature and polymer matrix water content. The optical design of the two-photon 3D optical memory is shown in Figure 3. The bacteriorhodopsin is contained in a cuvette and is ori ented by using electric fields prior to polymerizing the polyacrylamide gel matrix. This orientation is required to use the photoelectric signal 50

4

1

1



6.

LAWRENCE & BERGE

153


Figure 3. Schematic diagram of the principal optical components of a twophoton three-dimensional optical memory based on bacteriorhodopsin. The write operation involves the simultaneous activation ofLDj and LD (0 1) or LD and LD (1 0) to induce two-photon absorption within the irradiated volume and partially convert either bR ίο Μ (0 1) or M to bR (1-+0). The write operation uses a 10-ns pulse and a pulse simultaneously ofl ns. The protein is oriented within the cuvette by using an electric field prior to polymerization of the polyacrylamide gel. A polymer sealant is then used to maintain the correct polymer humidity. The SMA connector is at tached to the indium-tin-oxide conducting surfaces on opposing sides of the cuvette and is used to transfer the photoelectric signal to the external amplifiers and box-car integrators. Symbols and letter codes are as follows: (a) sealing polymer, (b) indium-tin-oxide conductive coating, (c) BK7 op tical glass; (d) SMA or OS50 connector; (e) Peltier temperature controlled base plate (0-20 °C); AT (achromatic focusing triplet); bs (beam stop); DBS (dichroic beam splitter); LD (laser diode); FL (adjustable focusing lens). Computer simulations of the probability of two-photon-induced photochem istry (vertical axis) as a function of location relative to the center of the irradiated volume (AX/ and AYf ) in microns are shown in (f) and (g). The top right contour plot (f) shows the probability after two 1140-nm laser beams have been simultaneously directed along orthogonal axes cross ing at the center of the irradiated volume. The bottom right contour plot (g) shows the probability after two 820-nm "cleaning pulses' have been independently directed along the same axes. The maximum conversion probability at χ = 0, y = 0 is normalized to unity for both contour plots. 3

2

4

ocus

ocus

9



154


to monitor the state of the proteins occupying the irradiated volume. A write operation is carried out by firing simultaneously the two 1140nm lasers (to write a 1) or the two 820-nm lasers (to write a 0). To eliminate unwanted photochemistry along the laser axes, nonsimultaneous firing of the lasers not used in the original write operation is carried out immediately following the write operation. The methods and procedures of this "cleaning" operation are described in reference 5, and the results are shown in the contour plots shown in Figure 3. The position of the cuvette is controlled in three dimensions by using a series of actuators that independently drive the cuvette in the x, y, or ζ direction. For slower speed maximum density applications, piezo electric-stepper motor micrometers are used. For higher speed, lower density applications, voice-coil actuators are used. Parallel addressing of large data blocks can also be accomplished by using holographic lenses or other optical architectures. A key requirement of the two-photon memory is to generate an irradiated volume that is reproducible in terms of xyz location over lengths as large as 2 cm. In the present case, our memory cuvettes are typically —1.6 cm in the χ and y dimension and —2 cm in the ζ direction (see Figure 3). These dimensions are variable up to 2 cm on all sides and can be as small as 1 cm on a side depending upon the desired storage capacity of the device. By using a set of fixed lasers and lenses and by moving the cuvette by using orthogonal translation stages, excellent reproducibility can be achieved (±1 μπι for piezoelectric micropositioners and ± 3 um for voice-coil actuators). Refractive inhomogeneities that develop within the protein-polymer cuvette as a function of write cycles adversely affect the ability to position the irradiated volume with reproducibility. This problem is due to the change in refractive index associated with the photochemical transformation. The problem is min imized by operating with a relatively large irradiated volume (—30 μτη ) and by limiting the photochemical transformation to 60:40 versus 40: 60 in terms of relative b R : M percentages. A detailed examination of the reliability of 3D memories has not been published, in part because each design has unique aspects that mediate the analysis. A reliability analysis based on a preliminary analysis of the experimental results obtained for the optical 3D memory shown in Figure 3 suggests that this memory has a unit component reliability of P = 0.512, which is one of the lowest unit reliabilities encountered in our studies to date. The relatively low value is due to a number of characteristics unique to volumetric memories, which must be designed to measure the state of an ensemble within a larger volumetric medium (5). The curve shown in Figure 2 indicates that 10 (—501,000) mol ecules must be included in the ensemble (irradiated volume) to generate a reliability of ξ ^ 10. The following equation estimates the concentra3

57


6.

LAWRENCE & BIRGE

155


tion, C, of protein necessary to achieve reliable operation as a function of linear bit separation, S: c ^ - t L e e x i o - ^ ^

1

^ ^

(47)

For example, a 3D memory operating with a linear bit separation of 3 μτη, requires a concentration of 3.1 X 10~ mol/1 of protein to achieve reliability (£> 10;N = 5 Χ 10 ). This concentration is difficult to achieve for oriented memory media, and the largest concentration we have achieved to date while maintaining high homogeneity and orientation is 1.3 X 10" mol/1. This result suggests one of two scenarios. A reliable 3D memory based on bacteriorhodopsin will either require larger linear bit separations (S > 4 μτη) or error-tolerant designs. The error-tolerant approach has been adopted in our prototypes, where we store 10-bitsper-8-bit character (data byte), thereby providing single bit error cor rection and double bit error detection. The above observations should not be used to argue that 3D optical memories do not offer high density and high reliability, simultaneously. A very conservative design based on S = 6 μτη will still provide a projected 200-fold improvement in storage density based on overall package size. 5


5

5

Monoelectronic and Monomolecular Devices. The possibility of constructing monomolecular (21, 23, 24, 42-44) and single electron devices (12-14, 45, 46) is the subject of active experimental and theo retical study. For convenience, we will refer to any device that relies on a single molecule or single charge carrier as a single quantized entity (SQE) device. One might conclude from the previous discussion that such devices are unrealistic, because the device having an ensemble of only one quantized entity will require an unrealistically high value of F (—1) to yield reliable operation. The probability of correctly as signing the state of an SQE, Ρ, is never exactly unity. Based on equation 44, we can write the best-case probability of an error in state assignment as follows: P n mi

error

(N, P ) s - E r f

(2P -l)iN

+ 1)νΝ

, is the probability of determining the state of the device if only one molecule (or information carrier) is present. The sec ond variable, N , is the number of molecules (or information carriers) operating to assign the state of the device (i.e., the number of quantized entities within the ensemble). The third variable, ξ, is the limiting log arithmic reliability factor, which is a function of the entire system and the environment in which it must operate. The first variable, is often difficult to assign experimentally. If we can measure the reliability of the device for a known ensemble size, our procedures can be used to define reliability as a function of smaller and larger ensembles. Sample reliability analyses are carried out on four devices to deter mine the minimum number of molecules, N, that must be included in the ensemble to achieve an error probability of less than 1 0 " . The devices studied include a molecular electrostatic gate (N = 1600), an optically coupled molecular N A N D gate (N = 130,000), a 2D optical RAM based on bacteriorhodopsin (Ν ^ 320,000), and a 3D (volumetric) R A M based on bacteriorhodopsin (N £ 500,000). Although optical coupling will typically require larger ensembles, optical coupling provides the most convenient method of generating the ensemble averaging that is often required for device reliability while maintaining the speed advantages inherent in molecular or nanoscale ballistic devices. χ

10




158

Heisenberg's uncertainty principle places constraints on the speed with which the state of a device can be assigned and for typical molecular devices an observable diminution in the probability of correct state as signment occurs for aperture (state assignment) times less than 1 ps. For virtually all systems that involve ensemble averaging, "Heisenberg energy broadening" is of little physical consequence until one ap proaches aperture times in the subpicosecond regime. Monomolecular or one-electron devices will require a P value of 0.9985 to achieve reliability in the absence of assignment averaging. Averaging can be used, however, to achieve reliability with a concom itant reduction in speed. For example, if 1000 assignment averages are taken, a P value of —0.7 is sufficient. Quantum effects limit the max imum frequency of operation of monomolecular and one-electron de vices to approximately f (GHz) « 0.96 v /N where v is the energy separation of the two states of the device in wavenumbers, and Ν is the number of state assignments that must be averaged to achieve reliable state assignment. max

s

s

Acknowledgments This work was supported in part by grants from the W . M . Keck Foun dation, The U.S. Air Force Rome Laboratory (F30602-91-C-0084), the National Institutes of Health (GM-34548), the Office of Naval Re search (N00014-88-K-0359), and the Industrial Affiliates Program of the W. M . Keck Center for Molecular Electronics at Syracuse University. The authors thank A. Aviram, P. A. Dowben, R. W . Keyes, Κ. K. Likharev, L . A . Nafie, M . Reed, and B. Ware for interesting and helpful discussions.

References 1. Birge, R. R.; Findsen, L . Α.; Pierce, Β. M . J. Am. Chem. Soc. 1987, 109, 5041-5043. 2. Kobilka, B. W.; Kobilka, T. S.; Daniel, K.; Regan, J. W.; Caron, M . G.; Lefkowitz, R. J. Science (Washington, D.C.) 1988, 240, 1310-1318. 3. Pnevmatikos, S. Phys. Rev. Lett. 1988, 60, 1534-1537. 4. Birge, R. R. Annu. Rev. Phys. Chem. 1990, 41, 683-733. 5. Birge, R. R. Computer 1992, 25, 56-67. 6. Polland, H. J.; Franz, Μ. Α.; Zinth, W.; Kaiser, W.; Kolling, E.; Oesterhelt, D. Biophys. J. 1986, 49, 651-662. 7. Mathies, R. Α.; Brito Cruz, C. H.; Pollard, W. T.; Shank, C. V. Science (Washington D.C.) 1988, 240, 777-779. 8. Mathies, R. Α.; Lugtenburg, J.; Shank, C. V. In Biomolecular Spectroscopy; Birge, R. R.; Mantsch, H . H . , Ed.; The International Society for Optical Engineering: Bellingham, WA, 1989; pp 138-145. 9. Birge, R. R. Biochim. Biophys. Acta 1990, 1016, 293-327. 10. Miller, W. H.; George, T. F. J. Chem. Phys., 1972, 56, 5637.



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RECEIVED for review March 12, 1992. 1993.

ACCEPTED revised manuscript March 24,


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