Physical Modeling and Design for Phase Change Memories by M. Rudan at the IEEE VLSI Test Symposium 2012.

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. de Castro” 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 and performance 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
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• Microscopic modeling: Monte Carlo analysis and
detrapping mechanism.
• Measurements.
• Conclusions.

5. CPU vs Hard Drive 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
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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 Hard Drive PerformanceCPU vs Hard Drive Performance —— IIII
CPU with Main
StorageNVM Storage
Massimo Rudan
University of Bologna
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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
Family Example 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
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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 a chalcogenide?What is a chalcogenide?
Chalcos (ore) + Gen (formation) → Chalcogen (ore formation)
Electro-positive Element or + Chalcogen →
Massimo Rudan
University of Bologna
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Chalcogenide: (GeTe)1+x(Sb2Te3)x
x = 1 → Ge2Sb2Te5

9. Why chalcogenides?Why chalcogenides?
H
e
a
t
Crystalline Amorphous
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
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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)Change Memory (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
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exponential
Ohmic
Sub-threshold behavior
(courtesy: D. Ielmini)
THRESHOLD
voltage

11. Metal 1
M
etal 2
Row
Column
Poly
Si-Substrate
Metal 1
M
etal 2
Row
Column
Poly
Si-Substrate
PCM architecturePCM architecture
Massimo Rudan
University of Bologna
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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 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
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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
ME emission
High current
Scheme of the energy transitionsScheme of the energy transitions
Massimo Rudan
University of Bologna
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The electron tunnels between
traps and remains a low-
mobility trap electron
The electron becomes a high-
mobility band electron

14. GC RS
The whole device 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
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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
Large transition intervalLarge transition interval
Massimo Rudan
University of Bologna
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II’ I’’
If IK = I’’ – I’ is large,
the V(I) curve is not N-
shaped.

16. V
Always low
conductance
Always saturated
conductance
Small transition intervalSmall transition interval
Massimo Rudan
University of Bologna
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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
V V
Low conductance
High conductance
Limiting casesLimiting cases
The second case is important for measuring purposes
Massimo Rudan
University of Bologna
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I’=I’’ I I’ I’’ I

18. Given the general expression 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
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I
GC
n << nT ≈ N
µT << µn
nT ≈ 0
µn >> µT
I’ I’’

19. 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
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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 currents there 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
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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 low currents 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
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exponential.
The region where the
conductance saturates is
not affected by the trap
mobility.

22. The comparison with
experiments has 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
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Electrodes are made of a broken carbon nanotube, coated with a 10-
nm GST layer.

23. The geometrical factor A/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
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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 dataFitting experimental data —— IIII
Massimo Rudan
University of Bologna
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25. 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
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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 expression has 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
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Parameter C0 is directly
related to physical
quantities of interest:
0 T
A
C q N
L
µ=

27. One starts from the 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
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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
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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 process in 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
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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
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a-GST layer Metal contactMetal contact
Compensating charge
Filled donor trap
Empty donor trap

31. Generate traps
Move a carrier 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
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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
the voltage 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
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existence of fluctuating
minimum-energy paths
(moving filaments??)

33. Our one-dimensional system consists 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
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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-particle regime, 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
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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 equation is 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
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• 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 for the 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
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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 the spatial 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
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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 first electron-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
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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 assumption of 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
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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 as a 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
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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
Due to 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
The increase in
temperature due to the
oscillatory regime
may in turn induce a
partial crystallization
of the material.
This phenomenon
Massimo Rudan
University of Bologna
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This phenomenon
may also be exploited
for determining
physical parameters.

43. ConclusionsConclusions
• PCM are a 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.