Statistical device simulations of III-V nanowire resonant tunneling diodes as physical unclonable functions source

In this paper, utilising the non-equilibrium Green ’ s function (NEGF) formalism within the new device simulator NESS (Nano-Electronic Software Simulator) developed at the University of Glasgow ’ s Device Modelling Group, we present quantum mechanical simulations of current flow in double-barrier III-V GaAs-AlGaAs nanowire resonant tunneling diodes (RTDs). NESS is a fast and modular Technology Computer Aided Design (TCAD) tool with flexible architecture which can take into account various sources of statistical variability in nanodevices. The aim of this work is to show that, in the RTD devices with nano-scale dimensions, there is a direct correlation between the position and the numbers of random dopants and the key device parameters, e.g., position of the resonant-peak (V R ) variations as well as the shape and number of peaks in the output current – voltage (I-V) characteristics. Such V R variability can be used as a quantum fingerprint which can provide robust security and hence can be used to deliver Physical Unclonable Functions (PUFs).


Introduction
The RTDs are semiconductor quantum well (QW) structure nanodevices with finite rectangular barriers which have negative differential resistance (NDR) in their I-V characteristics. Owing to the presence of this unique NDR property and their relatively simple structure and fabrication process, the RTDs are competitive candidates for interesting digital and analog circuits applications. For example, terahertz (THz) communications is an emerging area related to 5-6G technology development. Due to the ultra-high frequency capabilities, the RTDs are also a very promising choice for wireless and optical resonant THz frequency generation and detection [1][2][3][4]. Moreover, the so-called PUFs [5], which are nano-sized devices, have drawn an enormous amount of interest for hardware authentication in cybersecurity applications [6][7][8]. PUFs can be added to an already existing integrated circuit design to generate a unique output that is determined by material parameters and the structural characteristic, which cannot be duplicated or cloned. Recently, the RTDs have been proposed as the main building blocks for quantum-confinement-based physically unclonable functions (QC-PUFs) [9,10].
Here, we present a numerical investigation of the important role of the random dopant fluctuations (RDFs) variability in the RTDs, showing that the locations of the resonance peaks for each RTD devices are unique, random, and extremely difficult to predict, which indeed makes them suitable for PUFs generation and cryptographic purposes.

Simulation methodology
Time and cost-effective computer simulation tools pave the way to experimental electronic device fabrication as they provide deep insight into the physics of device operation. The simulations presented in this work are performed self-consistently utilizing the NEGF solver of our Nano-Electronic Simulation Software (NESS) framework [11][12][13][14][15]. NESS is designed to simulate quantum charge transport in ultra-scaled nanoelectronic devices using various integrated solvers such as NEGF, drift-diffusion, and Kubo-Greenwood formalism where the corresponding transport modules are solved self-consistently within the Poisson equation. The NEGF solver implemented in NESS allows us to capture the detailed physical picture of the quantum mechanical effects such as quantum confinement, coherence, particle-particle interactions, and resonant tunneling of electrons through barriers in structures under investigation. The charge density, potential profile, and the current flow in the system are obtained by performing a self-consistent solution of the Poisson equation and the NEGF transport equations in the coupled mode-space representation. We can either study the transport in (i) pure ballistic limit, (ii) include acoustic or g-type optical electron-phonon scattering mechanisms with the self-consistent Born approximation approach, or (iii) incorporate surface roughness scattering within our NEGF. Moreover, NESS can introduce various sources of statistical variability, e.g., RDF, Line Edge Roughness (LER), or Metal Gate Granularity (MGG).
The NEGF formalism is a widely applied framework for analyzing the quantum transport in mesoscopic and nanoscale devices. Having the system in steady state, one has to solve first at each energy E the relevant components of the retarded G R , advanced G A and the lesser Green's function G < , which are governed by the following system of equations [16] Where E, 1, η, and Σ R (Σ ≶ ) are the energy, identity matrix, an infinitesimal positive real number, and retarded (lesser/greater) selfenergies, respectively. The above-mentioned self-energies take into account scattering of the electrons and their interaction with the contacts. The electrons are described in this solver by an effective mass approximation. The one-particle Hamiltonian, h(r), reads where, U(r) is the external applied electrostatic potential at position r = (x, y, z). The semiconductor band-structure is captured by the effective masses along the x, y, and z directions -namely m j with j = x, y, and z.
Employing the retarded Green's function component, we can obtain the charge [17,18] and the current through the l th layer as where h l+1,l (G < l,l+1 ) is representing the matrix elements of the Hamiltonian (G < ) between the basis states on layer l +1 (l) and l (l + 1). The inset depicts its schematic illustration where L S =L D =19 nm, L i1 =L i2 =3 nm, L B1 =L B2 =3 nm, and L W =5 nm. The device has a square cross-section of 10 nm × 10 nm, and the total length is 55 nm.

Results and discussions
The inset of Fig. 1 shows the simulated double barrier III-V RTD. At the central region of the device, we consider two 3 nm thick regions (L B1, B2 ) made of Al 0.3 Ga 0.7 As as the tunneling barriers, enclosing a 5 nm GaAs section (L W ) which serves as the QW. These are separated from the source and drain (L S,D ) by 3 nm buffer regions (L i1,i2 ). The source and drain regions are n-type doped with a high doping concentration of 2 × 10 18 cm − 3 , whereas the rest of the device has intrinsic doping.
As depicted in Fig. 1, the current through the RTD rises steadily to reach the first resonant-peak at a bias of 0.29 V, followed by a sharp drop in current down to the first valley at 0.30 V. Such current profile is know as the negative differential resistance (NDR). By applying bias, three distinct regions are formed within the I-V characteristic. For low biases, the first confined state (resonant state) between potential barriers gets closer to the source Fermi level as bias increases, therefore, the current it carries increases. Further increasing the bias makes the first confined state lower in energy, gradually going into the energy range of the bandgap, hence the current it carries drops abruptly. The second confined state is still too high in energy to conduct any significant current at this moment. As the last step, since the second confined state becomes closer to the source Fermi level by further increasing the bias, it can carry more current, therefore, the current will increase similarly to the first positive resistance region.
Moreover, we have also investigated the transmission spectra T(E) in a 2D cut along the transport direction illustrated in Fig. 2(a). The figure shows the presence of sharp resonance-peaks at V R =0.29 V, which are significantly lower at a bias of 0.30 V, yielding the above-mentioned NDR region. Moreover, as depicted in Fig. 2(b), the charge distribution in the device also shows a significant discharging of the QW in the post-resonance condition, which is consistent with the T(E) profiles.
From the local density of states (LDOS) shown in Fig. 3(a), the corresponding alignment of the discrete states on the source, well, and drain side at the bias of 0.29 V is clear, thus concurring with the I-V characteristics and the sharp transmission peaks discussed previously. The T(E), which is shown superimposed on the LDOS with the red solid line, indicates that these peaks are specifically located at the energies at which there is an alignment of the states at − 0.059, 0.096, and 0.355 eV for the bias condition V R =0.29 V. This alignment is very sensitive to the applied drain bias and vanishes when the bias is increased to the postresonance voltage, see Fig. 3(c). The corresponding current spectra at Fig. 3(b)-(d) also reveal the resonant tunneling phenomena by the two  Fig. 1 (inset) very prominent lines at two such aligned energy states and an overall significant and more discrete tunneling current contribution. The tunneling across the potential barriers is also prominent in these simulation results. When the device is driven beyond the resonance condition, it is clearly visible that the current spectra become smaller in magnitude and rather smeared in nature. Furthermore, we have carried out simulations considering a sample of 10 devices with the unique position of the RDFs to investigate their impact on the V R (see Fig. 4). In Fig. 5, we look closely at four of those samples and confirm in the insets that devices No. 4, 6, 7, and 9 contain   Fig. 4). Accordingly, the bottom row presents the corresponding energy-resolved current spectra of the aforementioned devices. respectively 7, 4, 5, and 6 dopants each but with a different and unique 3D spatial distribution located within the RDF region. Comparing their I-V characteristics with the smooth (without RDFs) device presented in Fig. 1, we see that there is a direct link between the number and positions of the RDFs and the position of the resonant-peak, position of the valley, and magnitude of the simulated current. Therefore, making them distinguishable from one another in terms of the output I-V characteristics. Hence, these unique I-V characteristics of each device can be used as a unique fingerprint which is determined by characteristics and the material parameters, and thereby cannot be cloned or duplicated for counterfeiting purposes.
Lastly, Fig. 6 shows the energy sub-band structure, transmission spectra, LDOS, and current spectra of two of the aforementioned devices. In terms of the sub-band structures (white dashed line) and the average potential (solid black line), the changes due to the number and position of dopants are very much visible. We also see that the T(E), plotted as a solid red line, spikes at different energy levels, and also the current conduction through the channels has different weights for each case. The significant differences in terms of the potential, band alignment condition, and the transmission spectra all reflect the variations in the I-V characteristics of these devices and therefore display how the quantum nature of the resonance condition can be significantly altered with RDFs variability in these III-V nanowire RTD structures.

Conclusion
In summary, we have reported quantum mechanical simulation, utilizing ballistic Non-equilibrium Green's Function (NEGF) approach, of double-barrier III-V resonant tunneling diodes (RTDs) based on nanowire structures. focusing on the influence of random dopant fluctuations (RDFs), we have simulated an ensemble of 10 devices, each with a different and specific RDF distribution. we have shown that the quantum nature of the resonance condition can be significantly altered due to the RDFs. Our software NESS can indeed be used to predict and tailor RTDs behaviour for various future applications such as tunable THz communications, 5G/6G technology, and PUFs.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Asen Asenov received the Ph.D. degree in solid-state physics from the Bulgarian Academy of Sciences, Sofia, Bulgaria, in 1989. He was a Chief Executive Officer with Gold Standard Simulations, Ltd., Glasgow, U.K. Currently, he is a James Watt Professor of Electrical Engineering with the University of Glasgow.
Vihar P. Georgiev received his Ph.D. degree from the University of Oxford, Oxford, U.K., in 2011. In 2011, he joined the Device Modelling Group, School of Engineering, University of Glasgow, where he was a Research Associate until 2015 and currently a Senior Lecturer in electronics and nanoscale engineering. He is also UKRI EPSRC innovation fellow.