Quantum Transport Simula0on: A few case studies where it is necessary
Sayeef Salahuddin Laboratory for Emerging and Exploratory Devices (LEED)
EECS, UC Berkeley
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
The celebrated Moore’s Law
Hydrodynamic model
Dri<-‐Diffusion
Boltzmann Transport Equa0on
Quantum Transport
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
Direct Source to Drain tunneling
source
drain
source
drain
Off-‐State leakage through band-‐to-‐band tunneling
Conven0onal Device geometry
Novel Devices
Band-‐to-‐band tunneling transistor
Non-‐Semiconductor quantum devices
(presentaKon by Dr. Luisier this morning)
Spin Transfer Torque Devices
Bottom Electrode!
CoFe (2.5)!
Ru (0.85)!
Insulator!
Top Electrode!
CoFeB (3)!
CoFeB (3)!
MgO (0.85)!
Pinned layer
So< layer
Oxide 0
0 R
Current Pinned layer So< layer
Kubota et. al., JJAP, 44, 40, 1237,2005
Slonczewski, JMMM, 96 Berger, PRB,96 KaKne, PRL 2000
Why STT Devices?
Can we explain the (i) Amplitude of the switching current (ii) Resistance
With the same set of device parameters?
Key challenges for device simulaKon
Kubota et. al., JJAP, 44, 40, 1237,2005
Electron Transport: NEGF
Equations Voltage Current
LLG
Spin Dynamics
Torque Magnet- ization
NEGF (Non Equilibrium Green’s Function)
S. Salahuddin and S. Daga: IEDM (2006)
Self Consistent solu0on of the transport and magne0za0on dynamics
The difference of spin currents is absorbed by the so< magnet
Fixed magnet oxide So< ferromagnet
Torque from conserva0on of angular momentum
Ef
• Band spliing
• Fermi level
• Barrier Height
• EffecKve mass in barrier
• Kup
• Kdown
Band structure dependent parameters
Fixed magnet oxide So< ferromagnet
Effec0ve mass treatment of the transport
W. H. Butler et al, PRB, 2001
S0les Group, PRL, 2008, Macdonald Group, PRL, 2008
Fixed magnet So< ferromagnet
Oxide
Ef
E
ky
kx=0 kz=0
Kx>0 kz=0
Kx>0 kz>0
Incorpora0ng transverse modes/k sampling
x
y
z
Cross secKon is typically larger than 50 nm X 50 nm
Effect of the transverse modes
Pure 1D 100 modes
Pure 1D 10 modes
Fixed magnet So< ferromagnet
Oxide
x
y
z
Non Equilibrium distribu0on for the right magnet
+
Torque felt by the delectrons Torque felt by the selectrons
Torque
Physics of Spin Torque
=
Known from band spliing
Calculated from NEGF CalculaKons
MagneKzaKon of The magnet
Can be calculated from the other three
Bottom Electrode!CoFe (2.5)!Ru (0.85)!
Insulator!
Top Electrode!
CoFeB (3)!
CoFeB (3)!MgO (0.85)!
Fuchs et. al. , PRL 96, 186603,
2006
Theory Experiment
-0.4 -0.2 0 0.2 0.4 22
24
26
28
30
32
Current (mA) T
MR
(%)
Experimental Benchmark
Experimental Benchmark
0 0.2 0.4 0.6 0.8 0
20
40
60
80
100
120
Voltage (V)
TMR
(%)
Cur
rent
( µ A
)
Experiment
T. Kawahara et. al., ISSCC, 2007
Theory: Salahuddin group and Daea Group
Ef = 2.25 V ∆ = 2.15 eV m* = 0.2 m0 m*FM= m0 Ub = 1.4 V
PRL 99, 226602 (2007)
Theory: UCB and Purdue http://arxiv.org/abs/
0910.2489
Experimental Benchmark
A typical hysteresis from self consistent NEGF-‐LLG simulaKon
Typical contour of voltage induced switching
Nat. Phys. 4, 67-‐71 (2008)
Theory: UCB and Purdue http://arxiv.org/abs/
0910.2489
Experimental Benchmark
Perpendicular Component
Perpendicular Component
Ef = 2.25 V ∆ = 2.15 V m*= 0.32 m0 mFM
*= 0.73 m0 Ub = 0.9 V
PRB 79, 224416 (2009)
Theory: UCB and Purdue http://arxiv.org/abs/
0910.2489
Experimental Benchmark
Nature Physics, 4, 37, 2008 Theory: UCB and Purdue
http://arxiv.org/abs/0910.2489
Experimental Benchmark
Nature Physics, 4, 37, 2008
Ef = 2.25 V ∆ = 2.15 V m*= 0.18 m0 mFM
*= 0.73 m0 Ub = 0.77 V
Theory: UCB and Purdue http://arxiv.org/abs/
0910.2489
Experimental Benchmark
Ef
• Band spliing
• Fermi level
• Barrier Height
• EffecKve mass in barrier
• Kup
• Kdown
http://arxiv.org/abs/0910.2489
Integrated system
Quantum Device Simulator
Circuit Simulator
V
H
I
R
Puing it with circuit
Device and magneKzaKon dynamics
Circuit Variability
Li, Augus0ne, SS, Roy, DAC, 2008, pp. 278-‐283.
Op0mal opera0ng point from device
circuit co-‐simula0on
First Device Circuit Simula0on
Conclusion
Quantum Transport SimulaKon is going to be necessary for many devices in the nano scale regime!
Acknowledgement:
NSF/NRI
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