1. Network for Computational Nanotechnology (NCN) Purdue, Norfolk State, Northwestern, MIT, Molecular Foundry, UC Berkeley, Univ. of Illinois, UTEP 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. A.Arun Goud Beyond CMOS Scaling challenges  Leakage effects – High k dielectrics  Gate control – Non-planar structures  Variability – Process improvement  Mobility – Strain, III-V For the last 3 decades CMOS scaling driven by Moore’s law has been the norm ITRS 2009 - Emerging Research Devices 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)

3. A.Arun Goud 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. A.Arun Goud Quantum device – RTD (GaAs/AlGaAs) First demonstrated by Chang, Esaki and Tsu (1974) Grown using MBE Vertical devices  current flows along growth direction n GaAs Al x Ga 1-x As GaAs Al x Ga 1-x As n+ GaAs n GaAs n+ GaAs L < Phase coherence length z V I (a)(b)(c) (a) (b) (c) Vv Vp Ip Iv Peak to Valley Current Ratio = Ip/Iv (figure of merit) Requirements  Large Ip, low Iv. IV characteristics showing NDR

5. A.Arun Goud 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) 6T SRAM memory cell RTD latch  (a) Ultra-high switching speeds (b) Not transit time limited (c) Low voltage Digital circuit applications? Multi-Functional devices - YES! Peak current should be larger than leakage currents of read/write FETs Else there is unwanted state transition Simulation models needed. Should be Physics driven instead of compact model

6. A.Arun Goud So why are RTDs not widespread Compatibility with mainstream Si technology?  2 terminal  No isolation  Low drive capabilities. Peak current, PVR must be increased More importantly,  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… Advances in MBE, integration techniques  Viable way to integrate RTDs with mainstream processes is likely (InP based RTD/HEMTs already exist)  Device variations from die to die Perfect Lab for studying quantum phenomena - Physics involved and Simulation techniques devised will be useful for analyzing other devices too So is the emphasis laid on RTDs totally unfounded?

7. A.Arun Goud 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. A.Arun Goud RTD modeling – Thomas-Fermi 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 The converged potential is used by NEGF solver to calculate current

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

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

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

12. A.Arun Goud Simulation flow - Hartree

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

14. A.Arun Goud Approximations made Parabolic transverse dispersion Higher order subband minima are overestimated => 2 nd and further turn-on voltages are overestimated 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 J. Appl. Phys. 81 (7), 1997

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

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

18. A.Arun Goud Tight binding Hamiltonian Form Bloch sum of localized orbitals in the transverse plane z || α  Cation or anion orbitals (10 for sp3s*) σ  Layer index Δ=a0/2 σ1 σ2 v … … Wavefunction is expressed in terms of planar orbitals in each layer Real space Schroedinger equation can be transformed to this basis using Cation Anion Open boundary conditions using NEGF

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

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

21. A.Arun Goud Energy resolved electron density At peak voltage At valley voltage

22. A.Arun Goud 1dhetero 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 Application Design and study of electrostatics within HEMTs Sheet charge density  Analytical method – Parabolic transverse dispersion  Numerical – Transverse dispersion from TB calculation used Outputs 1. Energy band diagram 2. Potential 3. Resonances 4. Wavefunction magnitude squared 5. Sheet density, doping density 6. Resonance vs voltage GateBulk Schroedinger domain Poisson domain Users281 Simulation Sessions 1421 (WCT– 104 days) http://nanohub.org/1dhetero/usage

23. A.Arun Goud Brillouin Zone viewer Application Visualization of 1 st Brillouin zones for lattice system  Cubic (SC,BCC,FCC)  Hexagonal (Wurtzite)  Honeycomb (Graphene)  Rhombohedral (Bi2Te3) Input Translational vectors Lattice constant Output 1 st Brillouin zone Real space unit cell Users61 Simulation Sessions 157 (WCT – 5 days) http://nanohub.org/brillouin/usage

24. A.Arun Goud 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 1 st Brillouin zones of cubic, hexagonal, honeycomb and rhombohedral lattice systems.

25. A.Arun Goud 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. A.Arun Goud Thank You!

27. A.Arun Goud 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. A.Arun Goud IV at 0K Under equilibrium CB Profile, resonance position k x, k y that take part in tunneling Contribution to current Relative position of Well subband & E-k x dispersion in emitter No overlap between well suband level & emitter bulk level => No tunneling channel

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

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