A. Arun Goud Network for Computational Nanotechnology (NCN) Electrical and Computer Engineering

1. NEGF Simulation of Electron Transport in Resonant Tunneling and Resonant Interband Tunneling Diodes A. Arun Goud Network for Computational Nanotechnology (NCN) Electrical and Computer Engineering 11/28/2011

2. Beyond CMOS For the last 3 decades CMOS scaling driven by Moore’s law has been the norm • Scaling challenges • Leakage effects – High k dielectrics • Gate control – Non-planar structures • Variability – Process improvement • Mobility – Strain, III-V • Another line of thought… • Quantum mechanical effects • Tunneling • Interference • Quantization, etc. Emerging devices will have to utilize these effects while delivering high performance (high speed, low power consumption) ITRS 2009 - Emerging Research Devices

3. Outline • Example of a quantum device – • Resonant tunneling diode (RTD) • Characteristics • Applications…Why show interest in RTDs? • Shortcomings...Why RTDs are not common? • Simulation tool using NEMO5…To understand Physics behind RTDs • NEGF formalism…A quantum formalism to calculate charge and current • Resonant interband tunneling diode (RITD) • Alternative to RTDs • Overcomes some drawbacks with RTDs • Modeling of RITDs • Two other simulation tools – • 1dhetero • Brillouin zone viewer

4. Quantum device – RTD (GaAs/AlGaAs) First demonstrated by Chang, Esaki and Tsu (1974) Grown using MBE Vertical devices  current flows along growth direction IV characteristics showing NDR I n+ GaAs (c) Ip (b) n GaAs z Iv AlxGa1-xAs L < Phase coherence length (a) GaAs AlxGa1-xAs Vv Vp V n GaAs Peak to Valley Current Ratio = Ip/Iv (figure of merit) Requirements  Large Ip, low Iv. n+ GaAs (a) (b) (c)

5. Motivation - RTDs for digital applications • RTDs have been used for microwave circuits such as oscillators due to NDR. • Oscillations as high as 2.5 THz! (TCLG Sollner, Applied Physics Letters 43: 588) • (a) Ultra-high switching speeds (b) Not transit time limited • (c) Low voltage Digital circuit applications? YES! Peak current should be larger than leakage currents of read/write FETs Else there is unwanted state transition Multi-Functional devices - Simulation models needed. Should be Physics driven instead of compact model 6T SRAM memory cell RTD latch

6. So why are RTDs not widespread • 2 terminal  No isolation • Low drive capabilities. Peak current, PVR must be increased • Device variations from die to die More importantly, Compatibility with mainstream Si technology? • AlGaAs/GaAs, InGaAs/InAlAs, etc are popular choices but not compatible with Si technology and are expensive • Si/SiGe RTDs have been demonstrated. Tend to have poor PVRs at 300K… So is the emphasis laid on RTDs totally unfounded? Advances in MBE, integration techniques  Viable way to integrate RTDs with mainstream processes is likely (InP based RTD/HEMTs already exist) Perfect Lab for studying quantum phenomena - Physics involved and Simulation techniques devised will be useful for analyzing other devices too

7. Contribution - RTD NEGF tool • Features - • Coherent simulation of GaAs/AlGaAs RTDs • - Charge density 1. Semiclassically (Thomas-Fermi) • 2. Quantum self-consistent (Hartree) • - Effective mass Hamiltonian • - NEGF formalism for transport • Scattering/Relaxation in emitter reservoir • NEMO5 driven • Output - • Energy band diagram, Resonance levels • Transmission coefficient • Well, Emitter quasi-bound |Ψ|2 • Current density • IV • Charge & sheet density profiles • Resonances vs voltage • Energy resolved charge profiles a) Charge - 1. Thomas-Fermi method 2. Hartree method (NEGF) b) Transport - NEGF

8. RTD modeling – Thomas-Fermi The converged potential is used by NEGF solver to calculate current Free charge density non-zero only in reservoirs Thomas-Fermi expression Solved iteratively with Poisson’s equation. BCs are φ(z=L)=V and φ(z=0)=0

9. RTD modeling - Hartree Charge treated semiclassically in terminals Quantum charge calculated in Quantum region Current calculated only in Non-equilibrium region

10. NEGF - Quantum Charge and Current EQ  NEQ 1. RGF method 2. Dyson’s equation 3. iη relaxation model (Right contact will be ignored in this explanation ) gN,N = GN,N Mimics broadening just as imaginary part of Only 1st and Nth column of G are needed

11. Simulation flow – Thomas-Fermi Described in previous slide

12. Simulation flow - Hartree

13. Thomas-Fermi vs Hartree IV CB profile PVR = 2 Well charge vs Bias Quantization  Low charge density => Low potential energy Well charge  CB raises to block further flow of charges into well Hartree Resonance drops below Ec slower w.r.t bias in Hartree method than in Thomas-Fermi method Thomas-Fermi HartreeVp > TF Vp

14. Approximations made • Parabolic transverse dispersion • Higher order subband minima are overestimated • => 2nd and further turn-on voltages are overestimated J. Appl. Phys. 81 (7), 1997 • Transverse energy and momentum are separable • T(E,k||)  T(Ez) • => Current calculation involves integration over only Ez Full transverse dispersion and integration over k|| for exact analysis of coherent RTDs Scattering self-energies also for incoherent simulation

15. Recap • Resonant tunneling diode (RTD) • Characteristics…NDR  • Applications…Memory • Shortcomings...Low PVR at 300K  • Simulation tool…To understand Physics behind RTDs  • NEGF formalism…To calculate current  Is there a way to increase PVR?... We can draw inspiration from the Esaki diode

16. From Esaki diodes to RTDs to RITDs Esaki diode operation - I V 1) High peak to valley current ratio due to drastic reduction in valley current 2) Major drawback  - Heavily doped junctions difficult to produce - High capacitance which degrades speed of operation We need a mix In the case of RTD’s, Barriers are not effective in reducing valley current – low PVR  Barriers and well are undoped – low capacitance 

17. Esaki diode + RTD = RITDs InAs/AlSb/GaSb RITD Exhibit larger PVR at 300K than RTDs by reducing valley current. - Type II broken gap - Interband like Esaki diode InAs non-parabolicity Mixing of CB, VB states Multiband model is needed for proper description.

18. Tight binding Hamiltonian Form Bloch sum of localized orbitals in the transverse plane σ1σ2 v … … α Cation or anion orbitals (10 for sp3s*) σ  Layer index || Wavefunction is expressed in terms of planar orbitals in each layer Δ=a0/2 Cation Anion z Real space Schroedinger equation can be transformed to this basis using Open boundary conditions using NEGF

19. RITD multiband simulation IV 1. Thomas-Fermi charge model 2. sp3s* TB model with spin orbit coupling 3. Numerical k|| integration to compute current PVR = 50 Valley region is broad because effectively electrons see bandgap of AlSb+GaSb+AlSb layers

20. J(kx) at Vp and Vv ky kx,ky grid  (0.15,0.15) * 2π/a a = 0.6058 nm 2π/a = 10.37 /nm kx 0 1 2 3 4 Majority of the current is due to tunneling through Г state

21. Energy resolved electron density At peak voltage At valley voltage

22. 1dhetero Application Design and study of electrostatics within HEMTs • Features • Schroedinger-Poisson solver • 3 options for Hamiltonian • Single band • TB sp3s* with spin-orbit coupling • TB sp3d5s* with spin-orbit coupling • Semiclassical density-Poisson option • Choice of substrates Schroedinger domain Outputs 1. Energy band diagram 2. Potential 3. Resonances 4. Wavefunction magnitude squared 5. Sheet density, doping density 6. Resonance vs voltage Gate Bulk Poisson domain • Sheet charge density • Analytical method – Parabolic transverse dispersion • Numerical – Transverse dispersion from TB calculation used http://nanohub.org/1dhetero/usage

23. Brillouin Zone viewer • Application • Visualization of 1st Brillouin zones for lattice system • Cubic (SC,BCC,FCC) • Hexagonal (Wurtzite) • Honeycomb (Graphene) • Rhombohedral (Bi2Te3) Input Translational vectors Lattice constant Output 1st Brillouin zone Real space unit cell http://nanohub.org/brillouin/usage

24. Summary • RTD NEGF • Coherent simulation of GaAs/AlGaAs RTDs using effective mass model and NEGF for transport • Relaxation in equilibrium reservoir modeled using imaginary optical potential term iη • Future work – Implementation of self energy expressions for various scattering mechanisms, (111) wafer orientation RITD multiband simulation A coherent InAs/AlSb/GaSb RITD was simulated using NEMO5 with sp3s* SO model 1dhetero tool Simulation tool for the study and design of 1D heterostructures using a choice of substrates Brillouin zone viewer Simulation tool for visualizing the 1st Brillouin zones of cubic, hexagonal, honeycomb and rhombohedral lattice systems.

25. Acknowledgements Advisory committee Prof. Gerhard Klimeck Profs. Mark Lundstrom, Vladimir Shalaev NEMO5 developers Sebastian Steiger – 1dhetero, Brillouin and for answering other questions Hong-Hyun & Zhengping Jiang– RTD NEGF, NEGF simulation technqiues Tillmann Kubis & Michael Povolotskyi – NEMO5 simulation issues All other members of the Nanoelectronic modeling group…Presentation skills Xufeng Wang, JM Sellier – For code that went into 1dhetero Steven Clark – Tool installation Derrick Kearney George Howlett Cheryl Haines Vicky Johnson Funding agencies – NSF, SRC, NRI Rappture support Scheduling appointments, handling paperwork

26. Thank You!

27. Coherent tunneling Coherent tunneling – Translational periodicity in the transverse direction Two rules should be satisfied – 1) Total energy is conserved 2) Transverse momentum is conserved In Emitter In Well (Bulk like) (2D subband) Shaded disk in Fermi sphere indicates kx, ky states in emitter that take part in tunneling for a particular subband min. Eo in the well

28. IV at 0K Under equilibrium Relative position of Well subband & E-kx dispersion in emitter CB Profile, resonance position No overlap between well suband level & emitter bulk level => No tunneling channel kx, ky that take part in tunneling Contribution to current

29. IV at 0K V < Peak voltage Vp Relative position of Well subband & E-kx dispersion in emitter CB Profile, resonance position Some well suband levels & emitter bulk levels overlap => Tunneling channel kx, ky that take part in tunneling Contribution to current

30. IV at 0K V = Peak voltage Vp Relative position of Well subband & E-kx dispersion in emitter CB Profile, resonance position Maximum overlap of well suband levels & emitter bulk levels => Current is at its max. kx, ky that take part in tunneling Contribution to current