Simulation Of Ultra-Small Electronic Devices: The .

Simulation of Ultra-Small Electronic Devices: The Classical-Quantum Transition Region2: Quantum Corrected Transport Modelsof quantum effects which are increasing in significance inthis device (see Figure 1). Other projects in the NAS Semiconductor Device Modeling (SDM) Program [6] focus onlonger term quantum simulation approaches and devices.One goal of the NAS SDM Program is cover the entirerange from classical devices and physics to purely quantumcomputing. This spectrum is described in terms of particular (proposed or demonstrated) electronic devices in Table1. Note from Table 1 that as quantum (wave transport)effects become more fundamental to device operation,classical effects (due to inelastic scattering) become moredetrimental to proper device operation, and vice-versa.The preceding paragraphs describe the motivation forthis study of quantum effects in near-future electronicdevices and the general approach that will be taken. Theremainder of this document develops the specific tasks andplans for this project in more detail. The two specific taskseach pursue one of the approaches being taken by the electronic device modeling community to answer the industry’squestions about quantum effects in these electronicdevices: adding quantum corrections to classical electronicdevice models (Section 2), and using a fully quantummodel (Section 3). Finally, Section 4 contains a discussionof issues which continue to shape our approach in thiseffort at NAS to develop a widely useful electronic devicesimulation capability in general, and a quantum effect andquantum device simulation capability in particular.As stated above, quantum effects such as those depictedin Figure 1 will increasingly affect electronic device operation as devices are scaled to smaller and smaller dimensions. As a result, electronic device models based onclassical mechanics, such as drift-diffusion (DD), hydrodynamic (HD), and Boltzmann transport equation (BTE), arebecoming progressively less accurate. At the same time,the increasing cost of experimental R&D with smalldevices makes it imperative to use device modeling to agreater extent in the advancement of electronics into thefuture. One way to reconcile these incompatible trends(maintain accuracy of device models in the face of increasing quantum effects) is to add some form of quantum correction to the classical models. The main strength of thisapproach is that it retains all of the accumulated experienceand refinement that has made classical device models efficient, robust, and acceptably accurate for past and currentelectronic devices such as the MOSFET. The main weakness is that an independent approach is required to determine when the quantum correction is accurate, and underwhat conditions it too breaks down. This weakness isaddressed by the task discussed in Section 3.Many forms of quantum corrections to classical electronic device models have been proposed or implemented.These include MOSFET-specific quantum corrections [7,8, 9, 10, 11, 12] and generic quantum corrections to thedrift-diffusion [13], hydrodynamic [14, 15, 16], and Boltzmann transport equation [17] models.It is impossible for a researcher to single-handedlyimplement and adequately investigate more than one ortwo of these quantum correction approaches. [This fact isfurther discussed in Section 4.] Unfortunately, choosingone or a few approaches from the wide array is not straightforward. The main trade-off between models is computational efficiency versus accuracy and generality. Differentquantum correction approaches are undoubtedly preferablefor various device sizes, device types, or simulation tasks.Given that the strength of NAS in large numerical computations is to be exploited in this project, the choice in thiscase will favor accuracy and generality, at the probableexpense of higher computational cost. The two methodsthat appear to best fit this description are a 3-D density-gradient quantum correction to the DD model, and a 3-Dquantum-corrected HD model. These will be the first quantum correction models implemented in this task.Even having chosen just two quantum correction modelsto investigate, these models should be implemented asexpeditiously as possible. This task will therefore use ageneral PDE solver called PROPHET [18] (and possiblysimilar tools) to quickly implement and investigate thesemodels. Rapid model implementation is further discussedTable 1: Classical to Quantum Electronic DevicesClassical (inelastic scattering) effects become moredetrimental, and quantum (wave transport) effectsmore essential, as devices transition from classical toquantum. Note: HET hot electron transistor, QWLD quantum well laser diode, QUIT quantum interference transistor, SQUID superconducting quantuminterference device, SET single electron EffectsMOSFET, BJTDominantParasiticMODFET, HET,QWLDDominantUsefulRTD, RTTSignificantSignificantQUIT, SQUIDParasiticDominantQuantum Computation-killerExclusiveBryan A. Biegel3

Simulation of Ultra-Small Electronic Devices: The Classical-Quantum Transition Regionin Section 4.The density-gradient quantum correction to the DDmodel will be investigated first in this task, since it shouldrequire the less time to implement and fewer computationalresources. Indeed, implementation of the 3-D DG model inPROPHET is already underway. The DG model formalizesthe quantum mechanical requirement that wavefunctions,and thus carrier densities, can not change abruptly versusposition. For example, classically, carriers can reach veryhigh densities directly against the MOS gate oxide in theinversion layer, and drop to zero just inside the oxide.Quantum mechanically, carrier wavefunctions are nearzero in the oxide, and they must decrease smoothly towardszero in the neighboring inversion layer. The result is thatthe inversion charge is smoothed out and forced away fromthe gate oxide by some (classically unknown) distance,decreasing gate capacitance and thus MOSFET transconductance. The DG model can also model quantum tunneling.The mathematical description of the DG model is a simple extension of the classical DD model. The classical DDmodel can be written: ( ε ψ ) – ρ – q ( p – n ND–tronic devices.After implementing and investigating the density-gradient model, the quantum hydrodynamic (QHD) model willbe implemented in 3-D using PROPHET. The idea of theHD transport model (classical or quantum) is that, ratherthan resolve the momentum distribution of carriers exactly,the momentum distribution is assumed to be a mathematically simple modification of the equilibrium. Under thisassumption, the distribution can be described by a fewcharacteristic values. The equations for these values arederived by taking one or more moments of the Boltzmann(classical) or Wigner function (quantum) transport equation. The standard HD electronic device model uses threemoments, with the resulting characteristic values beingdensity, average velocity, and average energy.We now describe the QHD model mathematically. Several forms of the HD and QHD transport equations havebeen proposed. For illustration purposes, we present onewhich has been written in both classical and quantum corrected form. In classical form, for a spatially-independenteffective mass, and in 1-D, this HD model is [17]: ρ ρp ------ 0------ t x m N A) Jn n----- -------------- ( – n µ n ψ D n n ) ,q t Jp p------ – --------------- ( pµ p ψ D p p ) tq2 ( ρp ) U ( ρp ) ρ p ( ρkT ) -------------- -------------- --------- ρ t c x x t x m (1) W W pW ρpkT ρp U--------- ------------- ------------- -------- t c t x m x m m xwhich equations are solved for electrostatic potential ψ ,electron density n , and hole density p . Mathematically,the DG model [13] modifies the two continuity equationsby adding a “quantum potential” (the Bohm potential [19])to ψ : n----- ( – n µ n ψ*n D n n ) t,(2) p------ ( pµ p ψ *p D p p ) t2h 2 pψ*p ψ – -------------- -------------- 6m *p q p , (4)23ρkT ρ pW ------------- --------2m2which are solved for carrier density ρ , average momentump , and average energy W . A full mathematical HDdescription including three moments each for electrons andholes requires seven equations (including the Poissonequation).The quantum-corrected system of HD transport equations corresponding to (4), as derived from the Wignerfunction-corrected BTE [17], are:where2 2 nhψ*n ψ -------------- ------------- 6m*n qn Bryan A. Biegel ρ ρp ------ 0------ t x m .(3)2 Q ( ρp ) ρ p --------- ρ U ---- ( ρkT )-------------- x 3 x t x m ( ρp ) -------------- t cThe DG model has only been implemented in 1-D, usually assuming an infinite oxide band gap. This task willstudy the DG model in 3-D, with a physically correct(finite) oxide band gap. In this way, gate oxide tunnel current can be investigated, along with other quantum effectsin MOSFETs (see Figure 1) and other “classical” elec-Q W pW ρpkT ρp ------------- -------------- U ---- -------- 3 t x m x m m x 2ρh 1 ρ p W– ---------- --- ---- -------- 12m x ρ x x m t c4,(5)

Simulation of Ultra-Small Electronic Devices: The Classical-Quantum Transition Region(6)Schrödinger EquationQuantum Transportwhere Q is the Bohm quantum potential. This (or a similar) QHD model should give more accurate electronicdevice simulations than purely classical models, but theexpense of the computation may be very high, and numerical robustness may suffer. Analysis of such expectationswill be an important aspect of the investigation of quantum-corrected classical models in this task.EquationsTransfer MatrixScattering MatrixGreen’s Functions Density MatrixWigner FunctionPath IntegralFigure 3: Partial Quantum Mechanics Family TreeRelationships between quantum mechanics formulations relevant to the simulation of electronic devicesare shown. Models implemented in existing NAS 1-Dsimulation software tools are shown in bold.3: Quantum Models with ScatteringEven with the near-term device modeling focus of thisproject, more accurate models of quantum effects in electronic systems are needed to complement quantum correction models such as those described above. These moreaccurate models will be used to determine the accuracy andlimitations of the quantum correction models, and possiblyto derive more accurate and computationally efficientquantum correction models. The second specific task inthis project, described in this section, will use a fully quantum model to accomplish these goals.Figure 3 shows many of the quantum formulations thathave been used for electronic device modeling. As with thequantum correction models, it is only possible for a singleresearcher to implement and adequately investigate one ortwo of these models, so a choice must be made amongthese formulations. Because classical devices inherentlyexhibit significant inelastic scatteri

longer term quantum simulation approaches and devices. One goal of the NAS SDM Program is cover the entire range from classical devices and physics to purely quantum computing. This spectrum is described in terms of particu-lar (proposed or demonstrated) electronic devices in Table 1. Note from Table 1 that as quantum (wave transport)