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![N-type behavior in current-driven devices is typically accompanied by the formation
of filaments [Ridley, Pr. Phys. Soc. 82, 954 (1963)].
However, in the present case the cross-sectional area is so small that filamentation is
not likely to occur.
The device is current driven, so the
experiments yield, N-shaped (one
valued) V(I) curves.
Our investigation keeps the idea of filaments, but makes them to occur only in
ModelingModeling: qualitative analysis: qualitative analysis
Massimo Rudan
University of Bologna
12
Our investigation keeps the idea of filaments, but makes them to occur only in
energy, not in space.
The main transport mechanisms are:
o At low current → hopping processes through localized states [Mott
& Davis (1961); Buscemi et al., JAP 106 (2009)].
o At high current → conduction due to electrons occupying extended
states (here termed “band electrons”).](https://image.slidesharecdn.com/education-rudan-physicalmodelingdesignphasechangememories-130716014028-phpapp01/85/Physical-Modeling-and-Design-for-Phase-Change-Memories-12-320.jpg)
![The balance of the generation and recombination phenomena yields
(with I, IK > 0, neq ≤ n < N)
24
2
I
I
I n
Nn
n
n −+=
( )[ ].1/exp)(),( −=+= KCBI IIIrIrnnn
Models of the conductanceModels of the conductance —— IIII
Massimo Rudan
University of Bologna
19
The model is derived by combining the phonon-assisted net
recombination with the ME-emission generation in steady state, with:
nB
Ratio between the trap-emission and trap-capture rates for the phonon-
assisted transitions.
nC Threshold parameter for the electron emission.
( )[ ].1/exp)(),( −=+= KCBI IIIrIrnnn](https://image.slidesharecdn.com/education-rudan-physicalmodelingdesignphasechangememories-130716014028-phpapp01/85/Physical-Modeling-and-Design-for-Phase-Change-Memories-19-320.jpg)
![The lower branch at low currents can be considered near to
equilibrium. If T becomes larger the concentration n of the band
electrons becomes dominant. Assuming for simplicity a non-
degeneracy condition, the above yields:
( )[ ] 0,exp10 >−=−+≈ FCaBaC EEETkECCG
with k the Boltzmann constant and C , C , E parameters. Note that
TT--dependencedependence of the lower branchof the lower branch —— II
Massimo Rudan
University of Bologna
25
with kB the Boltzmann constant and C0, C1, Ea parameters. Note that
this result does not contradict the low-temperature conductance
expression (Mott’s law of variable-range hopping):
( )
1 4
0expCG T T ∝ −
](https://image.slidesharecdn.com/education-rudan-physicalmodelingdesignphasechangememories-130716014028-phpapp01/85/Physical-Modeling-and-Design-for-Phase-Change-Memories-25-320.jpg)
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Physical Modeling and Design for Phase Change Memories
The document presents a tutorial on physical modeling and design for phase change memories (PCM) at the 30th IEEE VLSI Test Symposium, discussing advancements in memory technology, specifically focusing on chalcogenide materials. It covers the cost-performance gap, macroscopic and microscopic modeling techniques, including Monte Carlo analysis, device architecture, and measurement results. The findings demonstrate the scalability and efficiency of PCM compared to traditional memory technologies.
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Physical Modeling and Design for Phase Change Memories
- 1. Special Session 8B:Embedded Tutorial Tue April 25, 2012 30th IEEE VLSI Test Symposium (VTS 2012) Physical Modeling and Design for Phase Change Memories Massimo Rudan University of Bologna 1 Massimo Rudan ARCES-University of Bologna, Italy Phase Change Memories Hyatt Maui, HI — April 23–26, 2012
- 2. Thanks…Thanks… F.F. GiovanardiGiovanardi “E. deCastro” Advanced Research Center on Electronic Systems (ARCES) University of Bologna, Italy University of Bologna F. BuscemiF. Buscemi E.E. PiccininiPiccinini Massimo Rudan University of Bologna 2 CNR Institute of Nanosciences (S3) University of Modena and Reggio Emilia, Italy R. BrunettiR. Brunetti C.C. JacoboniJacoboni A.A. CappelliCappelli
- 3. … more thanks……more thanks… Eric PopEric Pop University of Illinois at Urbana Champaign Massimo Rudan University of Bologna 3 Part of this work has been carried out under the contract 347713/2011 of the Intel Corporation, whose support is gratefully acknowledged.
- 4. OutlineOutline • Cost andperformance gap in memory technology. • Chalcogenide materials and Phase-Change Memories. • Macroscopic modeling and examples. Scaling issues. • Microscopic modeling: Monte Carlo analysis and Massimo Rudan University of Bologna 4 • Microscopic modeling: Monte Carlo analysis and detrapping mechanism. • Measurements. • Conclusions.
- 5. CPU vs HardDrive PerformanceCPU vs Hard Drive Performance —— II In 13 years (1996-2008) the media access time for 20k read has improved 175x for multicore CPU, 70x for CPU, 1.3x for Hard Disk. Cost & performance gap Massimo Rudan University of Bologna 5 CPU with Embedded Memory (SRAM cache) Main Memory DIMMS on MB (DRAM) Storage Hard Disk Drive (HDD) Latency = 1x $$/bit = 1x Latency = 100,000x $$/bit = 0,01x
- 6. CPU vs HardDrive PerformanceCPU vs Hard Drive Performance —— IIII CPU with Main StorageNVM Storage Massimo Rudan University of Bologna 6 CPU with Embedded Memory (SRAM cache) Main Memory DIMMS on MB (DRAM) Hard Disk Drive (HDD) Latency = 1x $$/bit = 1x NVM Storage Solid State Disk (Managed NAND) Latency = 100,000x $$/bit = 0,01x Latency = 1000x $$/bit = 0,1x
- 7. Possible implementationsPossible implementations FamilyExample Mechanism Selector Phase Change Chalcogenide alloys e.g.: GST 225 Energy (heat) converts material between crystalline (low resistance) and amorphous (high resistance) phases Unipolar Magnetic Tunnel Junction Spin Torque Transfer RAM Switching of magnetic resistive layer by spin-polarized electrons Bipolar Massimo Rudan University of Bologna 7 Junction Electro- chemical e.g.: CuSiO2 Formation / dissolution of “nano- bridge” by electrochemistry Bipolar Binary Oxide (filaments) e.g.: NiO Reversible filament formation by oxidation-reduction Unipolar Interfacial Resistance Memristors e.g.: doped STO, PCMO Oxygen vacancy drift diffusion induced barrier modulation Bipolar
- 8. What is achalcogenide?What is a chalcogenide? Chalcos (ore) + Gen (formation) → Chalcogen (ore formation) Electro-positive Element or + Chalcogen → Massimo Rudan University of Bologna 8 Chalcogenide: (GeTe)1+x(Sb2Te3)x x = 1 → Ge2Sb2Te5
- 9. Why chalcogenides?Why chalcogenides? H e a t CrystallineAmorphous Heat reversibly switches chalcogenides from crystalline to amorphous. The ratio ρa/ρc ~102 is the principle behind phase- change memories . Massimo Rudan University of Bologna 9 Charge Storage Based Memory (Flash) Phase Change Storage Based Memory (PCM) Endurance 105 1012 Read Time 25 ns 10 ns Write time 300 µs/page 50 µs/page
- 10. PhasePhase--Change Memory (PCM)ChangeMemory (PCM) Sub-threshold behavior SET current READ voltage RESET current Repr. from Wutting and Yamada, Nature Mater. 6, 824 (2007) Massimo Rudan University of Bologna 10 exponential Ohmic Sub-threshold behavior (courtesy: D. Ielmini) THRESHOLD voltage
- 11. Metal 1 M etal 2 Row Column Poly Si-Substrate Metal1 M etal 2 Row Column Poly Si-Substrate PCM architecturePCM architecture Massimo Rudan University of Bologna 11 DerChang Kau et al., “A stackable cross point phase change memory”, Proc. IEDM 2009. Due to their simplicity, PCM memories have a strong scalability advantage.
- 12. N-type behavior incurrent-driven devices is typically accompanied by the formation of filaments [Ridley, Pr. Phys. Soc. 82, 954 (1963)]. However, in the present case the cross-sectional area is so small that filamentation is not likely to occur. The device is current driven, so the experiments yield, N-shaped (one valued) V(I) curves. Our investigation keeps the idea of filaments, but makes them to occur only in ModelingModeling: qualitative analysis: qualitative analysis Massimo Rudan University of Bologna 12 Our investigation keeps the idea of filaments, but makes them to occur only in energy, not in space. The main transport mechanisms are: o At low current → hopping processes through localized states [Mott & Davis (1961); Buscemi et al., JAP 106 (2009)]. o At high current → conduction due to electrons occupying extended states (here termed “band electrons”).
- 13. Trap conduction Low current MEemission High current Scheme of the energy transitionsScheme of the energy transitions Massimo Rudan University of Bologna 13 The electron tunnels between traps and remains a low- mobility trap electron The electron becomes a high- mobility band electron
- 14. GC RS The wholedevice is described as a one-dimensional structure made of the series of the amorphous GST material of area A, length L, with conductance ( ) ( )nnqLAG nTTC µµ += and of a constant resistance RS due to the heater, crystalline cap, and upper contact. Macroscopic model of the snap backMacroscopic model of the snap back Massimo Rudan University of Bologna 14 GCT GCB RS Letting GCT , GCB be the conductance of the trap and band electrons the total resistance can be written as: CBCT S C S GG R G RR + +=+= 11
- 15. V Always low conductance Always saturated conductance Largetransition intervalLarge transition interval Massimo Rudan University of Bologna 15 II’ I’’ If IK = I’’ – I’ is large, the V(I) curve is not N- shaped.
- 16. V Always low conductance Always saturated conductance Smalltransition intervalSmall transition interval Massimo Rudan University of Bologna 16 II’ I’’ If IK = I’’ – I’ is small, the V(I) curve is N-shaped hence the I(V) curve is S-shaped.
- 17. Low conductance High conductance VV Low conductance High conductance Limiting casesLimiting cases The second case is important for measuring purposes Massimo Rudan University of Bologna 17 I’=I’’ I I’ I’’ I
- 18. Given the generalexpression of the conductance, ( ) ( ) nnNnnqLAG TnTTC +=+= ,µµ − += N n Nq L A G T Tn TC µ µµ µ 1 GC IK Models of the conductanceModels of the conductance —— II Massimo Rudan University of Bologna 18 I GC n << nT ≈ N µT << µn nT ≈ 0 µn >> µT I’ I’’
- 19. The balance ofthe generation and recombination phenomena yields (with I, IK > 0, neq ≤ n < N) 24 2 I I I n Nn n n −+= ( )[ ].1/exp)(),( −=+= KCBI IIIrIrnnn Models of the conductanceModels of the conductance —— IIII Massimo Rudan University of Bologna 19 The model is derived by combining the phonon-assisted net recombination with the ME-emission generation in steady state, with: nB Ratio between the trap-emission and trap-capture rates for the phonon- assisted transitions. nC Threshold parameter for the electron emission. ( )[ ].1/exp)(),( −=+= KCBI IIIrIrnnn
- 20. Note: At low currentsthere is only one branch for all values of b = µn /µT . The low-current branch is linear. However, it is found from VoltageVoltage--current relationcurrent relation —— II Massimo Rudan University of Bologna 20 However, it is found from experiments that the linear part is followed by an exponential part. Thus, improvements are necessary.
- 21. Field-enhanced mobility Note: At lowcurrents there is still only one branch for all values of b. The low-current branch is VoltageVoltage--current relationcurrent relation —— IIII Massimo Rudan University of Bologna 21 exponential. The region where the conductance saturates is not affected by the trap mobility.
- 22. The comparison with experimentshas been carried out on devices like the one on the left, shown in F. Xiong, A. Liao, E. Pop, APL 95, 243103 (2009). Example of devices under investigationExample of devices under investigation Massimo Rudan University of Bologna 22 Electrodes are made of a broken carbon nanotube, coated with a 10- nm GST layer.
- 23. The geometrical factorA/L in the above devices ranges from 10-8 to 10-7 cm. Note: In the experiments used here the upper branch is related to the crystalline phase. Fitting experimental dataFitting experimental data —— II Massimo Rudan University of Bologna 23 branch is related to the crystalline phase. However, the subthreshold behavior and the switching current are fairly reproduced. The trap concentration favorably compares with the data reported in Ielmini and Zhang, JAP 102, 054517 (2007).
- 24. Fitting experimental dataFittingexperimental data —— IIII Massimo Rudan University of Bologna 24
- 25. The lower branchat low currents can be considered near to equilibrium. If T becomes larger the concentration n of the band electrons becomes dominant. Assuming for simplicity a non- degeneracy condition, the above yields: ( )[ ] 0,exp10 >−=−+≈ FCaBaC EEETkECCG with k the Boltzmann constant and C , C , E parameters. Note that TT--dependencedependence of the lower branchof the lower branch —— II Massimo Rudan University of Bologna 25 with kB the Boltzmann constant and C0, C1, Ea parameters. Note that this result does not contradict the low-temperature conductance expression (Mott’s law of variable-range hopping): ( ) 1 4 0expCG T T ∝ −
- 26. The above expressionhas been fitted to the experimental data by Ielmini and Zhang (left) and Pop (right): It is interesting to note that Ea= 0.33 ± 0.01 eV in both cases. Parameter C0 is directly TT--dependencedependence of the lower branchof the lower branch —— IIII Massimo Rudan University of Bologna 26 Parameter C0 is directly related to physical quantities of interest: 0 T A C q N L µ=
- 27. One starts fromthe V(I) characteristic, where GC incorporates also the dependence of the trap mobility on I : ( ) 1 S C V R I G I = + The snap-back point is found by making the derivative dV/dI to vanish. This yields: dG G Scaling factorsScaling factors —— II Massimo Rudan University of Bologna 27 d 1 d C C S C G G R G I I + = The right hand side is invariant with respect to the physical and geometrical scaling factors of GC and (independently) of I. The result shows that in this model the snap-back current (obviously not necessarily the voltage!) is invariant with respect to such scaling factors.
- 28. Scaling factorsScaling factors—— IIII The scaling of the V(I) characteristic under variation in the geometrical factors (device length and cross-sectional area) is shown. Comparison of the model with the experimental curve at T = 295 K. The device is a 10-nm GST layer deposited over a 110- Massimo Rudan University of Bologna 28 nm gap opened within a 4.3-nm-diameter carbon nanotube. The symbols show the experiments, wheras the continuous line has been calculated by Sentaurus T-CAD.
- 29. The conduction processin the high-resistivity state of PCD is well interpreted assuming a transport model based on hoppinghopping throughthrough localizedlocalized statesstates via direct or thermally-assisted tunneling and/or Poole-Frenkel (thermionic) effect. Ielmini and Zhang, J. Appl. Phys. 102, 0545172 (2007) Poole-Frenkel emission Hopping ofHopping of trap electronstrap electrons Massimo Rudan University of Bologna 29 Thermally-assisted tunneling Tunneling Below threshold the I(V) curve is linear at low fields and exponential at increasing fields: the forward current is strongly enhanced and dominates the conduction. Band transport is activated at higher fields.
- 30. The numerical approach:the Monte Carlo methodThe numerical approach: the Monte Carlo method Massimo Rudan University of Bologna 30 a-GST layer Metal contactMetal contact Compensating charge Filled donor trap Empty donor trap
- 31. Generate traps Move acarrier from one contact to the other one Solve Poisson equation Calculate probabilities Initialize simulation (t) and injection (tI) times tI ← tI + e /I The numerical approach: currentThe numerical approach: current--driven Monte Carlodriven Monte Carlo Massimo Rudan University of Bologna 31 Calculate probabilities for all scattering events Calculate time lag Dt for the next hop t + Dt < tI ? Select a transition t = tIt ← t +Dt noyes Interaction with the lattice
- 32. Physical instabilities of thevoltage drop between contacts vs time suggest the existence of fluctuating MC results:MC results: subthresholdsubthreshold conduction in amorphous GSTconduction in amorphous GST Massimo Rudan University of Bologna 32 existence of fluctuating minimum-energy paths (moving filaments??)
- 33. Our one-dimensional systemconsists of a number of free propagating electrons, all of them in the same energy state, and an electron initially in the ground state of a potential well. The system always operates in a two-particle regime, ManyMany--electronelectron (ME)(ME) detrappingdetrapping. Physical model. Physical model —— II Massimo Rudan University of Bologna 33 in a two-particle regime, that is, a band electron is supposed to interact with the trapped electron only when the previous scattering event is over.
- 34. Within a two-particleregime, the Hamiltonian of the system takes the form: H x1, x2( ) = H0 x1( )+ H0 x2( )+ e2 4πε x1 − x2 , H0 x( ) = − h2 2m∗ ∂2 ∂x2 +VT x( ) ManyMany--electronelectron (ME)(ME) detrappingdetrapping. Physical model. Physical model —— IIII Massimo Rudan University of Bologna 34 H0 x( ) = − 2m∗ ∂x2 +VT x( ) The Schrödinger equation reads: ih ∂ ∂t ψ x1, x2,t( ) = H x1, x2,t( )ψ x1, x2,t( ) Two particle wavefunction Two particle wavefunction At t = 0 ψ x1, x2,0( )= w x1( )χ0 x2( ) w (x1): Gaussian wavepacket χχχχ0(x2): bound ground state
- 35. The Schrödinger equationis solved by means of the Crank- Nicholson finite difference scheme (F. Buscemi et al., PRA 73, 052312, 2006). • A 80 nm-long region is considered. • The space coordinates of the carriers are discretized with an N- point grid corresponding to a spatial resolution ∆x = 0.125 nm. • The time step of the system evolution is taken as ∆t = 0.2 fs. ManyMany--electronelectron (ME)(ME) detrappingdetrapping. Theory. Theory Massimo Rudan University of Bologna 35 • The time step of the system evolution is taken as ∆t = 0.2 fs. ψk+1 = Mbk N*N–element vector describing the two-particle wavefunction at time step k+1 M: diagonal with fringes matrix The known term involving the value of Ψ at time step k The system is solved by means of an iterative algorithm based on the Gauss-Seidel scheme by imposing a closed boundary.
- 36. Single-particle density forthe trapped electron As a consequence of the scattering, the trapped electron can be elevated to non-bound states with high energies, this meaning ResultsResults —— II Massimo Rudan University of Bologna 36 energies, this meaning that its spatial-density probability, initially peaked, broadens up. The broadening of the trapped carrier wavepacket indicates the occurrence of a detrapping process.
- 37. At tf thespatial wavepackets can be used to evaluate the detrapping probability. By ignoring the relaxation of the trapped electrons to the ground level, the two-particle state can be written ψ x1, x2,tf( )= αn n ∑ ϕn x1( )χn x2( ) ϕϕϕϕn(x1): free propagating states χχχχn(x2): bound or non-bound eingestates of the single- particle Hamiltonian H0 Letting K the number of bound states, the ResultsResults —— IIII Massimo Rudan University of Bologna 37 Letting K the number of bound states, the probability PT(1) that a single scattering event leaves the trapped electron in an arbitrary bound state is PT (1) = αn 2 n=0 K−1 ∑ 0 Due to the physical and geometrical parameters of the system under investigation, χ0 is the only bound state.
- 38. newAfter the firstelectron-electron collision has occurred, the new incoming particle, again described by a Gaussian wavepacket, interacts via the Coulomb potential with an electron which is now in a linear superposition of bound and non-bound states. SCATTERINGSCATTERINGINPUT STATE OUTPUT STATE DetrappingDetrapping probability due to eprobability due to e--e interactionse interactions —— II Massimo Rudan University of Bologna 38 α0 ϕ0 χ0 + αn n=1 ∑ ϕnχn wχ0 wχn βn n=1 ∑ ϕnχn (bound + non-bound states) (non-bound states)(non-bound state) (bound state)
- 39. Within the assumptionof independent scattering events, the probability PT(m) that a number m of collisions leaves the trapped electron in the bound ground state is the product of the probabilities of no excitation in each scattering. DetrappingDetrapping probability due to eprobability due to e--e interactionse interactions —— IIII Massimo Rudan University of Bologna 39 PT (m) = α0 2m PD (m) =1− PT (m) =1− α0 2m Detrapping probability after m scattering eventsDetrapping probability after m scattering events
- 40. Detrapping probability asa function of the number of injected electrons The detrapping is more effective when the energy of the injected electrons is larger. DetrappingDetrapping probability due to eprobability due to e--e interactions. Resultse interactions. Results Massimo Rudan University of Bologna 40 At a given injection energy, the detrapping probability increases with m and becomes very close to 1 within a few tens of interactions.
- 41. MeasurementsMeasurements —— II Dueto the negative-slope branch and the need of a current generator, the measuring set up may be unstable. This may be Massimo Rudan University of Bologna 41 may be unstable. This may be exploited by inducing oscillations. Several parameters may be measured from this.
- 42. MeasurementsMeasurements —— IIII Theincrease in temperature due to the oscillatory regime may in turn induce a partial crystallization of the material. This phenomenon Massimo Rudan University of Bologna 42 This phenomenon may also be exploited for determining physical parameters.
- 43. ConclusionsConclusions • PCM area new class of memories that fills a cost & performance gap. • PCM are very promising due to the architectural simplicity and scalability. • Although the devices are well into the R&D phase, Massimo Rudan University of Bologna 43 • Although the devices are well into the R&D phase, several physical mechanisms are still under debate. • Reliability and testing-related issues: threshold behavior, sub-threshold fluctuations, electrical instabilities, partial re-crystallization.