Quantum Transport Simula0on: A few case studies where it is necessary Sayeef Salahuddin Laboratory for Emerging and Exploratory Devices (LEED) EECS, UC Berkeley sayeef@eecs.berkeley.edu
The celebrated Moore s Law Low power Architectures;Lecture #1:IntroducKon, Dr. Avi Mendelson Source: New Microarchitecture Challenges in the Coming GeneraKons o fcmos Process Technologies Fred Pollack, Intel Corp. Micro32 conference key note - 1999.
The celebrated Moore s Law Quantum Transport Boltzmann Transport Equa0on Dri<- Diffusion Hydrodynamic model
A few numbers to remember M: number of current carrying modes l: length of the conductor λ: mean free path λ: 10-20 nm in Si 20-30 nm in III- V materials We are approaching an era where quantum transport simulakon is necessary!
A Few Examples Conven0onal Device geometry source drain Direct Source to Drain tunneling source drain Off- State leakage through band- to- band tunneling Novel Devices Non- Semiconductor quantum devices Band- to- band tunneling transistor (presentakon by Dr. Luisier this morning)
Spin Transfer Torque Devices Top Electrode! CoFeB (3)! MgO (0.85)! Insulator! CoFeB (3)! Ru (0.85)! CoFe (2.5)! Bottom Electrode! So< layer Pinned layer Slonczewski, JMMM, 96 Berger, PRB,96 KaKne, PRL 2000 R 0 Pinned layer Oxide So< layer Kubota et. al., JJAP, 44, 0 40, 1237,2005 Current
Why STT Devices?
Key challenges for device simulakon Kubota et. al., JJAP, 44, 40, 1237,2005 Can we explain the (i) Amplitude of the switching current (ii)resistance With the same set of device parameters?
Self Consistent solu0on of the transport and magne0za0on dynamics NEGF (Non Equilibrium Green s Function) Voltage Electron Transport: NEGF Equations Current Magnetization Spin Dynamics Torque LLG S. Salahuddin and S. Daga: IEDM (2006)
Torque from conserva0on of angular momentum Fixed magnet oxide So< ferromagnet The difference of spin currents is absorbed by the so< magnet
Effec0ve mass treatment of the transport Band structure dependent parameters Fixed magnet oxide So< ferromagnet Band spliing Fermi level Barrier Height EffecKve mass in barrier E f K up K down W. H. Butler et al, PRB, 2001 S0les Group, PRL, 2008, Macdonald Group, PRL, 2008
Incorpora0ng transverse modes/k sampling Oxide z x y Cross seckon is typically larger than 50 nm X 50 nm Fixed magnet So< ferromagnet E K x >0 k z >0 k x =0 k z =0 E f k y K x >0 k z =0
Effect of the transverse modes Pure 1D 10 modes Pure 1D 100 modes
Physics of Spin Torque Top Electrode! CoFeB (3)! MgO (0.85)! Insulator! CoFeB (3)! Ru (0.85)! CoFe (2.5)! Bottom Electrode! Fixed magnet Oxide So< ferromagnet z x y Non Equilibrium distribu0on for the right magnet + Torque Torque felt by the selectrons = Torque felt by the delectrons Calculated from NEGF CalculaKons Known from band spliing MagneKzaKon of The magnet Can be calculated from the other three
Experimental Benchmark Experiment Theory 32 Fuchs et. al., PRL 96, 186603, 2006 TMR (%) 30 28 26 24 22-0.4-0.2 0 0.2 0.4 Current (ma)
Experimental Benchmark Experiment Theory: Salahuddin group and Daea Group 120 100 TMR (%) Current( µ A) 80 60 40 20 0 0 0.2 0.4 0.6 0.8 Voltage (V) T. Kawahara et. al., ISSCC, 2007
Experimental Benchmark PRL 99, 226602 (2007) Theory: UCB and Purdue http://arxiv.org/abs/ 0910.2489 Ef = 2.25 V = 2.15 ev m* = 0.2 m0 m* FM = m0 U b = 1.4 V
A typical hysteresis from self consistent NEGF- LLG simulakon
Typical contour of voltage induced switching
Experimental Benchmark Nat. Phys. 4, 67-71 (2008) Theory: UCB and Purdue http://arxiv.org/abs/ 0910.2489
Experimental Benchmark PRB 79, 224416 (2009) Theory: UCB and Purdue http://arxiv.org/abs/ 0910.2489 Perpendicular Component Perpendicular Component E f = 2.25 V = 2.15 V m * = 0.32 m0 m FM* = 0.73 m0 U b = 0.9 V
Experimental Benchmark Nature Physics, 4, 37, 2008 Theory: UCB and Purdue http://arxiv.org/abs/ 0910.2489
Nature Physics, 4, 37, 2008 Experimental Benchmark Theory: UCB and Purdue http://arxiv.org/abs/ 0910.2489 E f = 2.25 V = 2.15 V m * = 0.18 m0 m FM* = 0.73 m0 U b = 0.77 V
Band spliing Fermi level Barrier Height E f EffecKve mass in barrier K up K down http://arxiv.org/abs/0910.2489
Integrated system V H Quantum Device Simulator I R Circuit Simulator
Puing it with circuit First Device Circuit Simula0on Device and magnekzakon dynamics Circuit Variability Op0mal opera0ng point from device circuit co- simula0on Li, Augus0ne, SS, Roy, DAC, 2008, pp. 278-283.
Conclusion Quantum Transport SimulaKon is going to be necessary for many devices in the nano scale regime! Acknowledgement: NSF/NRI