Quantum. Effects in MOS Devices. presented by. Dipl. Phys. Universität Karlsruhe. accepted ANDREAS WETTSTEIN - PDF

Diss. ETH No Quantum Effects in MOS Devices A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doktor der technischen Wissenschaften presented by ANDREAS

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Diss. ETH No Quantum Effects in MOS Devices A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doktor der technischen Wissenschaften presented by ANDREAS WETTSTEIN Dipl. Phys. Universität Karlsruhe born 21st June 1969 citizen of Germany accepted on the recommendation of Prof. Wolfgang Ficht er, examiner Dr. Wim Schoenmaker, coexaminer 2000 Acknowledgements I wish to thank Prof. Wolfgang Fichtner for giving me the opportunity to do work which is strongly related to theoretical physics, but has still some links to the 'real world', i. e. to questions relevant for the semiconductor industry. I enjoyed many interesting (and sometimes even useful) discussions with the members of the TCAD group of the Institut für Integrierte Systeme (IIS), Andreas Schenk, Andreas Scholze, Fabian Bufier, Frank Geelhaar, Bernhard Schmithüsen, Bernd Witzigmann, especially indebted to my and others. I am advisor Andreas Schenk for his continuous stim ulation and encouragement, and for keeping me free of tedious adminis trative stuff. I found working at the IIS particularly agreeable due to the ideal working conditions provided by the computer system adminis trators and the kind and unbureaucratic support by the secretaries and technicians. This work would have been hardly possible without support by former and present employees of ISE AG. Ulrich Krumbein and James Litsios helped me to get started with Dessis.isb. Eugeny Lymukis, Oleg Penzin, and Boris Polski spent quite some time in helping me to make my Schrödinger solver a useful part of the commercial version of DESSIS_ISB. I acknowledge financial support by the ETH Zürich, the Schweiz erischer Nationalfond (SNF ), the QuCuiSS project European Union (ESPRITIV 29795), gie und Innovation (KTI ). of the and the Kommission für Technolo ljkmg^ Ln **%& /re.» VsasJT / f Contents Zusammenfassung xiv Abstract xvî 1. Introduction 1 2. ID Schrödinger Equation Current equation in 2+1 dimensions Physical models Basic form of the Hamiltonian Boundary conditions and current in quantisation direction Effective masses Problems with position dependent mass Carrier density of discrete levels Continuum contributions to the density Numerics and implementation Solving the initial value problem Iterating the energy Integration into the simulator Selfconsistency iterations Discussion Results and applications A look inside the device 28 V Lowfrequency CVcharacteristics of a MOS diode Doublegated SOI MOSFET Conclusions Direct Tunnelling Physical modelling Bardeen's perturbation method How can Bardeen's method work at all? Application to tunnelling Hole tunnelling Open questions Implementation Applications Is selfconsistency important? The influence of quantisation Stacked insulators A 2D example Channel Mobility Linearised Boltzmann equation Relaxation time Ansatz Phonon scattering Adaption of bulk matrix elements Scattering rates Relaxation time Interface roughness scattering Model Hamiltonian Matrix elements Screening Relaxation time Results for the effective mobility Results for transistors Discussion The Quantum DriftDiffusion Equation Quantum Liouville and Wigner equation Method of moments Equilibrium density matrix 80 VI Integral equation for the equilibrium density matrix Application to the model Hamiltonian Numerical aspects A note on dimensionality Simple examples The limit of slow variation of potential and mass A numerically practical form of the quantum term Implementation Results MOS diode MOSFET Doublegated SOI MOSFET Conclusions 102 Appendix 103 A.l. Mobility computation using Kohler 's method 103 A.2. Derivative of matrix exponential 106 A.3. Integrals for the equilibrium density matrix 106 A.4. Quantum correction for large potential step 107 Bibliography 110 Author Index 119 Curriculum Vitae 123 List of Figures 2.1. Energy correction and number of zeroes as a function of trial energy Tensor grid in a MOSFET channel and insulator Electron density in a MOSdiode, with and without quan tisation Wave functions Eigenenergies and potential profile CVcharacteristics of a MOSdiode, dependence of quanti sation on substrate orientation CVcharacteristics of MOSdiodes, doping dependence of quantisation Sketch of doublegated SOI MOSFET Potential profile and charge density in doublegated SOI MOSFET Dependence of currentvolt age characteristics for doublegated SOI MOSFETs on silicon layer thickness Selfconsistent vs. fixed Fermi level computation of tunnel current Comparison of tunnelling currents obtained with numeri cal and analytical model Comparison of gate currents for single oxide, ON and ONO dielectrics Wave functions in ONO structure 52 Vlll 3.5. Sketch of an oxide and equivalent highe material barrier Band edge in ON and ONO Gate and drain currents in a MOSFET Fit of the mobility model (no screening) to the universal curve for 300 K Mobility computed with screening compared to universal curve Comparison of computed mobility to experiments for 77K Drain current vs. gate voltage of a MOSFET using the mobility model Effective mobility in a doublegated SOI MOSFET Level occupation in doublegated SOI MOSFET, effective field dependence adependence of effective mobility Potential step example: comparison of various quantum corrections Comparison of CVcurves for MOS diode obtained with different quantisation models Comparison of CVcurves for two MOS diodes for two tem peratures obtained with different quantisation models Comparison of density distribution in MOSFET channels. 98 for different quantisation models Drain current vs. gate voltage for a MOSFET, obtained with different quantisation models Drain current vs. gate voltage for doublegated SOI MOS FET obtained for different quantisation models Density profiles in doublegated SOI MOSFET channel ob tained for different quantisation models 101 List of Tables 2.1. Parameter values for electron mass in Si Parameters values for hole mass in Si Parameters used for tunnelling simulations Parameters for the mobility model Fit constants for the quantum drift diffusion model... 96 List of Symbols [A,B]_ di Bi dt _ a b ab commutator AB BA iih component of V ith component of V time derivative * scalar product of vectors a and b dyadic product of vectors a and b a* complex conjugate of a a\ix...in j^l 'In dildi_na limffoö»! dina [...] function arguments a nonparabolicity factor ß reciprocal carrier thermal energy l/k&t. ßL reciprocal lattice thermal energy 1//cb2l. c\ sound velocity d d Dac dimensionality (degrees of freedom) of the physical syste dimensionality in which calculations are done acoustic deformation potential constant Uvn.t,vv' deformation potential constant for intervalley {y phonon scattering S[x] Dirac's delta function Sij Kronecker symbol Ao amplitude of interface roughness XI e E elementary charge (positive) energy t^iit^iv energy of ith. subband (in valley v) Ê electric field Ee& Ej? effective electric field Fermi energy e material dependent permittivity fj 2D position vector (x, y) ÎB BoseEinstein distribution function & n nx n2 Fermi integral of order i Hamiltonian Hamiltonian for zdirection Hamiltonian for xydirections n Planck's constant divided by 27T i imaginary unit, i2 Id * drain current 1 3 particle current density fcb k Boltzmann constant 3D wave vector ^xy K K Lx LG t^corr 2D vector: xy components of k 2D wave vector in the xyplane multi index to abbreviate uvok system size in xdirection (Ly, Lz likewise) thickness of the gate correlation length of interface roughness Hj vector from grid point i to grid point j 'qm A As Ahp m characteristic quantum length scale vh2ß/2m potentiallike quantum correction simplified form of A approximation for A for high potential step effective mass tensor me mx TflXy free electron mass xcomponent of effective mass tensor (my, mz likewise) 2DDOS mass in xyplane ( yjmxmy) m\ iaxis effective mass component for electron in valley v mh heavy hole mass m1 light hole mass Xll m\ big mass component of electrons in silicon rat p, mobility small mass component of electrons in silicon //eff effective mobility n particle density or subband index N multi index to abbreviate nva Nb band edge density of states (conduction or valence band) V derivative with respect to 3D (centre of mass) coordinate V derivative with respect to 3D position difference coordinate index labelling conduction band valley or hole bands frequency of intervalley phonon Q,i volume of box i v uovvi 4 electrostatic potential $ potential (electrostatic and band edge contributions) $1 ^dependent part of the potential $2 arydependent part of the potential 3 c part of the potential that couples xy $b band edge contribution to the potential 3?m q R r p pq Pcmr pcr Pij Pqm pw g a 'mass' driving potential dkqtlogm/2 particle charge 3D (centre of mass) position coordinate 3D position difference coordinate and ^direction charge density, density matrix Classical limit of the Wigner function (Boltzmann distribution function) density matrix in centre of mass representation density matrix in position coordinate representation elements of the density matrix elements Pcmr [R, T, ß] exp [^171^ /2ß%2] Wigner function crystal mass density spin index (Tij area of face between box i and box j ç conductivity sgn [x] signum function 20 [x] T carrier temperature Tl lattice temperature 1 Xlll r kinetic energy operator r e Tr UB ug v relaxation time unit step function trace drain voltage, relative to source gate voltage, relative to substrate (for diodes) or source (for MOSFETs) (group) velocity scattering rate W x, y, z coordinate axes c 2D position vector (x, y) Zusammenfassung Die industrielle Simulation von Halbleiterbauelementen fußt derzeit auf den DriftDiffusionsGleichungen bzw. hydrodynamischen Gleichungen. Die Ladungsträger werden darin wie eine klassische Flüssigkeit behan delt. Mit fortschreitender Miniaturisierung elektronischer Bauelemente wird die Gültigkeit dieser Beschreibung mehr und mehr in Frage gestellt. Schon heute sind die GateOxide in MOSFETs nur wenige Nanometer dick; die Kanaltiefe ist ähnlich klein. Damit sind quantenmechanische Längenskalen erreicht. Die lateralen Dimensionen sind erheblich größer: Bei MOSFETs ist eine Kanallänge von 180 nm (in Produktion) möglich, für das Jahr 2011 sind 50 nm projektiert [1], was aus quantenmechani scher Sicht bei Zimmertemperatur immer noch makroskopisch ist. Es liegt also nahe, die Richtung senkrecht zum Kanal gesondert zu behandeln und den Transport längs des Kanals weiterhin klassisch zu be schreiben. Kapitel 2 erklärt, wie man die eindimensionale Schrödingergleichung in einen DriftDiffusionsSimulator einbauen kann. Dabei wird im Kanal und im GateOxid ein TensorGitter benutzt, auf dessen zur Si Si02Grenzfläche senkrechten Linien jeweils die ldschrödingergleichung gelöst wird. Das Ergebnis wird benutzt, um die Zustandsdichte so zu ver ändern, dass die klassischen DriftDiffusionsGleichungen den korrekten quantenmechanischen Dichteverlauf im Kanal liefern. Da im Vergleich zur klassischen Rechnung die Ladungsträger von der Grenzfläche wegge drängt werden, reduziert sich die GateKapazität. Die Schwellspannung von MOSFETs wird durch die Nullpunktsenergie im engen Kanalpoten zial erhöht. XV In Kapitel 3 werden die Eigenlösungen dazu verwendet, speziellen Form zeitabhängiger Störungstheorie [2] mithilfe einer Tunnelströme durchs GateOxid zu berechnen. Das Modell liefert fast dasselbe liefert wie ein einfacheres, analytisches. Es ist aber auch geeignet, um Tunneln durch aus mehreren Schichten aufgebaute GateIsolatoren zu untersuchen, was mit einfachen Modellen nicht möglich ist. Kapitel 4 beschreibt ein Beweglichkeitsmodell, Lösung der ldschrödingergleichung gewonnenen das auf den aus der Informationen basiert. Das Modell beinhaltet Fononenstreuung und abgeschirmte Grenzflächen streuung. Die Störstellenstreuung wurde weggelassen, da sie nur bei ge ringer Dichte freier Ladungsträger im Kanal wichtig ist, das Augenmerk aber vorwiegend auf dem Bereich der Inversion und hoher effektiver Felder liegt. Ziel war es, die universellen Beweglichkeitskurve zu reproduzieren. Obwohl das Modell nicht in der Lage ist, die rapide weglichkeit bei hohen Feldern vor herzusagen, liefert Abnahme der Be es qualitativ die in Experimenten beobachtete Abnahme des Drainstroms mit steigender Ga tespannung im Bereich tiefer Inversion, was lokale Beweglichkeitsmodelle nicht leisten. Die Modellierung von Quanteneffekten mit der Schrödingergleichung ist sehr zeitaufwändig. Die Implementierung Kanal voraus, was sich in realen (prozesssimulierten) erreichen lässt. Hinzu kommt das physikalische Problem, dass Nichtgleichgewicht in Quantisierungsrichtung setzt ein TensorGitter im Strukturen kaum nicht beschrieben werden kann. Die QuantenDriftDiffusions (oder DichteGradient) Methode [3] ver spricht Abhilfe bei diesen Problemen. Sie besteht darin, die klassische DriftDiffusionsGleichung um eine quantenmechanische treibende Kraft, die sich durch Ableitungen der Ladungsträgerdichte ausdrücken lässt, zu erweitern. Kapitel 5 beschreibt die Herleitung, Implementierung und Ve rifizierung des Modells. Eine Besonderheit der Herleitung ist die Berück sichtigung der Ortsabhängigkeit der effektiven Masse. Es zeigt sich, dass die QuantenDriftDiffusionsGleichungen Quanteneffekte erstaunlich gut beschreiben. Hoffnungen auf höhere Rechengeschwindigkeit tät des Verfahrens im Vergleich zur IDSchrödingerMethode bisher nicht erfüllt. und Stabili haben sich Abstract Contemporary industrial device simulation relies on the driftdiffusion or on the hydrodynamical equations. a classical fluid. These equations treat the carriers like With ongoing miniaturisation of electronic devices, the validity of this description becomes more and more questionable. Today's MOSFETs have gate oxides that are only a few nanometers thick; channel depth is of the same order of magnitude. This means that quan tum mechanical length scales have already been reached. For the lateral dimensions, the length scales are significantly larger. the Currently, MOS According FETs with 180 nm channel length are being mass produced. to Ref. [1], for the year 2011 the channel length will be reduced to 50 nm, which is from a quantum mechanical point of view still macroscopical at room temperature. This suggests to describe the direction perpendicular to the channel separately, while still treating the transport along the channel classically. Chapter 2 explains how the onedimensional Schrödinger equation can be integrated into a driftdiffusion simulator. oxide, a tensor grid is used. On interface, the 1DSchrödinger equation is solved. In the channel and the gate each grid line perpendicular to the SiSi02 The results are used to modify the density of states in such a way that the classical driftdiffusion equations yield the correct quantum mechanical density distribution in the channel. As the charge carriers are moved away from the interface in comparison to a classical calculation, the gate capacity is reduced. Due to the zero point energy in the narrow channel potential voltage of MOSFETs increases. the threshold XV11 In chapter 3, tunnelling currents through the gate oxide are computed using a special form of timedependent perturbation theory [2] the bound eigenstates. This expensive based on model leads to almost the same results as a simpler, analytical model. However, in contrast to simple models, it can also be used to investigate gate insulators made up of multiple layers. Chapter 4 describes a mobility model that is based on the information obtained by the solution of the IDSchrödinger equation. The model includes phonon scattering and screened interface roughness scattering. Impurity scattering has been neglected, as it is important only charge carrier concentration, and the focus of this investigation inversion conditions and large effective electric fields. was to reproduce the universal mobility curve. at low is on The aim of the work While the model cannot predict the rapid degradation of the mobility at large fields, in contrast to local mobility models, it can qualitatively reproduce the experimentally observed decrease of the drain current with increasing gate voltage regime of deep inversion. Modelling quantum effects by the Schrödinger equation is in the very timeconsuming. The implementation requires a tensor grid in the channel, which is hardly possible in realistic (processsimulated) structures. An additional, physical restriction is that nonequilibrium in quantisation di rection cannot be described. The quantumdrift diffusion (or densitygradient) method [3] attempts to solve these problems. In this method, the classical driftdiffusion equations are extended by a quantum me chanical force term, which can be expressed in terms of derivatives of the charge carrier density. Chapter 5 describes the derivation, implementa tion and verification of the model. The special feature of the derivation is that it takes into account the position dependence The quantumdrift diffusion equations effects amazingly well. of the effective mass. are found to describe quantum The hopes regarding increased speed of compu tation and better numerical stability as compared to the 1DSchrödinger approach did not come true yet, however. Chapter Introduction The rapid progress in technologj^ keeps the simulation of semiconduc tor devices exiting. With new designs, new materials and processes, and smaller length scales, physical effects that were not important be fore start to need careful attention. Novel device concepts like Quantum Cellular Automata [4] or Quantum Dots require entirely new approaches to device simulation [5]. However, ultrasmall CMOS devices are gener structures for the ally expected to stay the economically most important coming 1015 years, despite the foreseeable technological challenges [1,6]. Accurate simulation of MOS devices therefore remains very important. From the device simulation point of view, miniaturisation poses new problems of different type. The first category is related to statistics. The smaller the device, the smaller is the number of microscopic vari ables that determine its are the relative fluctuations. average, macroscopic behaviour, and the larger A wellknown example are random dopant fluctuations (see, for example Ref. [7] and references therein). Another category of problems is related to ballistic transport. Electrons can pass through small devices with only a few scattering events. Due to the strong electric fields, they can reach very high energies. Therefore, ex treme nonequilibrium carrier distributions occur, transport description choice to deal with this aspect. and full bandstructure becomes a must. Monte Carlo is the method of The third category of problems the TCAD physicist faces are quantum effects. The thermal wavelength of an electron of mass ra is given by Trhv2/mkBT, which is about 8nm at room temperature when ra equals 2 the free electron mass. While this is still significantly smaller than the channel length of MOSFETs in the foreseeable future [1], the channel depth and the oxide thickness for current devices are already smaller than the thermal wavelength. Quantum effects therefore play today's MOS devices. of them on a solid physical base, yet applicable a role even in This work attempts to model the most important to industrial device simulation. in a manner that makes the models In principle, transport in semiconductor devices should be computed with a manyparticle Hamiltonian for the electrons and atomic cores of the device material. In practice, simplifications are unavoidable, and manyparticle effects are typically ignored (or hidden in material param eters) and the effective mass approximation [8] is used. Note that the latter approximation is not strictly valid in the cases of interest, as its derivation assumes that potentials change
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