A Fowler-Nordheim Integrator can Track the Density of Prime Numbers

11/24/2017 ∙ by Liang Zhou, et al. ∙ Washington University in St Louis 0

"Does there exist a naturally occurring counting device that might elucidate the hidden structure of prime numbers ?" is a question that has fascinated computer scientists and mathematical physicists for decades. While most recent research in this area have explored the role of the Riemann zeta-function in different formulations of statistical mechanics, condensed matter physics and quantum chaotic systems, the resulting devices (quantum or classical) have only existed in theory or the fabrication of the device has been found to be not scalable to large prime numbers. Here we report for the first time that any hypothetical prime number generator, to our knowledge, has to be a special case of a dynamical system that is governed by the physics of Fowler-Nordheim (FN) quantum-tunneling. In this paper we report how such a dynamical system can be implemented using a counting process that naturally arises from sequential FN tunneling and integration of electrons on a floating-gate (FG) device. The self-compensating physics of the FG device makes the operation reliable and repeatable even when tunneling-currents approach levels below 1 attoamperes. We report measured results from different variants of fabricated prototypes, each of which shows an excellent match with the asymptotic prime number statistics. We also report similarities between the spectral signatures produced by the FN device and the spectral statistics of a hypothetical prime number sequence generator. We believe that the proposed floating-gate device could have future implications in understanding the process that generates prime numbers with applications in security and authentication.

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1 Prime number density and FN tunneling

Consider the hypothetical prime number generator shown in Fig. 1(a), the output of the filter is given by

(1)

where denotes the set of prime numbers, is a constant time factor and denotes a convolution operation. Then, as a consequence of the prime number theorem [22, 23]

(2)

where the parameter is determined by the gain of the filter . Denoting the equation 2 can be rewritten as,

(3)

where is a solution of a first-order dynamical system given by

(4)

The right-hand side of equation 4 has a similar form as the FN tunneling current density  [21] given by:

(5)

where is the electric-field across a tunneling barrier and are material and temperature dependent parameters. Assuming an uniform electric-field across the barrier of thickness the first-order dynamical system given by 4 can be implemented using FN current as

(6)

where denotes the integration capacitance of the floating-gate and is the voltage across the barrier during the process of tunneling. The similarity between the two dynamics serves as a motivation to compare and and illustrates a fundamental connection between the density of primes and the proposed FG device.

2 Asymptotic tunneling current distribution

Equation 6 captures the change in floating-gate potential for an uniform oxide or FN barrier thickness. In practice, the oxide thickness would spatially vary as illustrated in Fig. 1

(b) and can be modeled as a two-dimensional random variable

, with and being the two spatial dimensions. Then cumulative dynamics in equation 6 can be expressed as the sum of tunneling current over the whole cross-sectional area where is the total floating-gate capacitance and is the unit area. The first order dynamics guarantees that . It is also highly unlikely that oxide barriers at two different spatial locations will have the same thickness, in which case we can assume that there exists a location such that . In this case the ratio of the tunneling current

(7)

approaches a Dirac-delta function as decreases asymptotically. This is what has been verified using Monte-Carlo simulations in Fig. 1(c) and (d). For each instance of the simulation, the two-dimensional oxide thickness was generated from a uniform random distribution. We used a 100100 tile array to model the distribution of oxide thickness across a 66

cross-sectional area. The thickness within each tile is uniform and selected from the uniform distribution with mean value of 13 nm and relative standard deviation of 5

. To capture the temporal response, we used equation 6 as the dynamics to do numerical integration over time with step size of 1 second. The capacitance is chosen as 2 pF and , are calculated from physics constants and calibrated using measurement data.

3 Device implementation

We fabricated eight different variants of the device on a standard double-poly CMOS process with a nm gate-oxide thickness. Four of these devices have different integration capacitors but identical tunneling junction area, while other four devices have different tunneling junction areas but identical integration capacitors. The micro-photograph of the fabricated die is shown in supplementary Fig. S1(a), and the form factors including the capacitance and tunneling junction areas are all summarized in the table shown in the supplementary Fig. S1(b). Multiple-sections of micrograph were captured using a high resolution microscope and were stitched together as shown in the supplementary Fig. S1(a). The temporal behaviors of all the fabricated devices were measured and also shown in the figure. Supplementary Fig. S1(c) shows the evolution of the tunneling rate with respect to time for each device, where we can observe that the effect of self-compensation (SC) that leads to a converging tunneling response for different devices. Supplementary Fig. S1(d) shows the measured temporal response of all the devices, each of which asymptotically converges to a response.

4 Device programming and characterization

The device can be initialized to operate in different modes by programming the charge on the floating-gate. The common method for programming FG transistor is by using FN tunneling or by using hot-electron injection [26]. FN tunneling removes the electrons from FG node by applying a high-voltage ( 15 V for 13nm oxide thickness) across a parasitic nMOS capacitor acting as a programming junction. Hot-electron injection, however, requires lower voltage ( 4.2 V in 0.5-m CMOS process) than tunneling and hence is the primary mechanism for accurate programming of floating-gates. The hot-electron programming procedure involves applying a voltage larger than 4.2 V across the source and drain terminals. The large electric field near the drain of the pMOS transistor creates impact-ionized hot-electrons whose energy when exceeds the gate-oxide potential barrier (3.2 eV) can get injected onto the floating-gate. A combination of FN tunneling and hot-electron injection can program the FG voltage to target value.

While the FG voltage () can be easily programmed to the FN tunneling region where electrons can continuously tunnel through the gate oxide, the readout circuits need to be carefully designed to avoid interference with the FG potential. Initially, we programmed the FG node voltage to a value around 3V. If we want to activate the FN tunneling integration process, we connect the terminal of the integration capacitor () to a fixed voltage such as 6V. Since the pMOS transistor is biased at accumulation mode, all the capacitors can be assumed to be constant and FG voltage depends linearly with . To measure the FG potential, we connect to ground, which pulls the FG voltage below 3V and hence can be interrogated using a standard unity-gain buffer. The buffer input is directly connected to the FG node using the same polysilicon layer which avoids the use of metallic vias that might introduce trap states in the surrounding oxide.

5 Matching and Alignment

To time-align the response of the FG device (T) and the prime number generator (P) as shown in Fig. 1(a), we used the closed-form expression of the FG device model which is obtained from the first-order differential equation 6 reported in [27] and given by:

(8)

- are model coefficients determined by process parameters, form factors, physics constants and initial conditions. After measuring the temporal response of the device, we used the proposed model to fit the data and extract the coefficient . A similar process was conducted on the prime number density using the same form as:

(9)

where - are model parameters determined by prime number distribution. The scalar mapping between the domains (absolute time and prime number generator) was derived using the parameters (, ). The process is illustrated in supplementary Fig. S2, where we first obtained the model parameters for the device and the prime number density respectively. A linear mapping from absolute time to integer was calculated using those model parameters, which is then used to translate the timer output to prime number density. For alignment, we used the first 50 million prime numbers and table shown in supplementary Fig. S2 summarizes the corresponding model parameters.

6 Time-stitching procedure

To verify the long-term response of the FG device, we programmed the device in three different regions, as shown in Fig. 3(a). We estimated the device model equation in 8 using the measured data from Region 1. We then extended the model to Region 2 and Region 3 as illustrated by Fig. 3(a). The alignment was achieved by mapping the mid-point of Region 2 and Region 3 to the model value. As verified by the results, the model can capture the behavior for a long-term operation equivalent of 1.5 years. The model was then used to map the three regions to the prime number density distribution, as shown in Fig. 3(b) using the mapping method discussed in the previous paragraph.

7 Spectral analysis

A spectrogram was used to compare the quasi-stationary fluctuations in the output of the FG device and the output of the hypothetical prime-number generator. We used a 100000 tap finite-impulse-response (FIR) filter to smooth out the impulse train in Fig. 1(a). This ensures that the output retains some of the high-frequency signatures in the prime-number data without over-smoothing. A time-alignment procedure was used to synchronize and and a rectangular window (of 100 samples) was used to generate the spectrograms corresponding to the prime number generator and FG device output.

References

  • [1] van Dam, W. Quantum computing and zeroes of zeta functions. arXiv preprint quant-ph/0405081 (2004).
  • [2] Latorre, J. I. & Sierra, G. There is entanglement in the primes. arXiv preprint arXiv:1403.4765 (2014).
  • [3] Riemann, B. On the number of primes less than a given magnitude. Monthly Reports of the Berlin Academy (1859).
  • [4] Dyson, F. Birds and frogs. Notices of the AMS 56, 212–223 (2009).
  • [5] Planat, M., Solé, P. & Omar, S. Riemann hypothesis and quantum mechanics. Journal of Physics A: Mathematical and Theoretical 44, 145203 (2011).
  • [6] Schumayer, D. & Hutchinson, D. A. Colloquium: Physics of the riemann hypothesis. Reviews of Modern Physics 83, 307 (2011).
  • [7] Bunimovich, L. & Dettmann, C. Open circular billiards and the riemann hypothesis. Physical review letters 94, 100201 (2005).
  • [8] Schaden, M. Sign and other aspects of semiclassical casimir energies. Physical Review A 73, 042102 (2006).
  • [9] Rivest, R. L., Shamir, A. & Adleman, L. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21, 120–126 (1978).
  • [10] Dickson, L. History of the theory of numbers, vol. 1, carnegie institution of washington, washington, dc, 1919. Dickson1History of the Theory of Numbers1919 (2005).
  • [11] Maynard, J., Tao, T. & Ford, K. Chains of large gaps between primes (2016).
  • [12] Oliver, R. J. L. & Soundararajan, K. Unexpected biases in the distribution of consecutive primes. Proceedings of the National Academy of Sciences 113, E4446–E4454 (2016).
  • [13] Daniel, R., Rubens, J. R., Sarpeshkar, R. & Lu, T. K. Synthetic analog computation in living cells. Nature 497, 619 (2013).
  • [14] Bounds, D. G. New optimization methods from physics and biology. Nature 329, 215–219 (1987).
  • [15] van der Pol, B. An electro-mechanical investigation of the riemann zeta function in the critical strip. Bulletin of the American Mathematical Society 53, 976–981 (1947).
  • [16] Firk, F. W. & Miller, S. J.

    Nuclei, primes and the random matrix connection.

    Symmetry 1, 64–105 (2009).
  • [17] Connes, A. Trace formula in noncommutative geometry and the zeros of the riemann zeta function. Selecta Mathematica 5, 29 (1999).
  • [18] Sekatskii, S. K. On the hamiltonian whose spectrum coincides with the set of primes. arXiv preprint arXiv:0709.0364 (2007).
  • [19] Bender, C. M., Brody, D. C. & Müller, M. P. Hamiltonian for the zeros of the riemann zeta function. Physical Review Letters 118, 130201 (2017).
  • [20] Schumayer, D., van Zyl, B. P. & Hutchinson, D. A. Quantum mechanical potentials related to the prime numbers and riemann zeros. Physical Review E 78, 056215 (2008).
  • [21] Lenzlinger, M. & Snow, E. Fowler-nordheim tunneling into thermally grown sio2. Journal of Applied physics 40, 278–283 (1969).
  • [22] Dusart, P. Estimates of some functions over primes without rh. arXiv preprint arXiv:1002.0442 (2010).
  • [23] Kotnik, T. The prime-counting function and its analytic approximations. Advances in Computational Mathematics 29, 55–70 (2008).
  • [24] Berry, M. V. & Keating, J. P. The riemann zeros and eigenvalue asymptotics. SIAM review 41, 236–266 (1999).
  • [25] Maynard, J. Small gaps between primes. arXiv preprint arXiv:1311.4600 (2013).
  • [26] Huang, C., Sarkar, P. & Chakrabartty, S. Rail-to-rail, linear hot-electron injection programming of floating-gate voltage bias generators at 13-bit resolution. IEEE Journal of Solid-State Circuits 46, 2685–2692 (2011).
  • [27] Zhou, L. & Chakrabartty, S. Self-powered timekeeping and synchronization using fowler–nordheim tunneling-based floating-gate integrators. IEEE Transactions on Electron Devices 64, 1254–1260 (2017).