Physical Modeling and Design for Phase Change Memories

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 RudanUniversity of Bologna

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Massimo Rudan

ARCES-University of Bologna, Italy

Phase Change Memories

Hyatt Maui, HI — April 23–26, 2012

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


2

CNR Institute of

Nanosciences (S3)

University of Modena and

Reggio Emilia, Italy

R. BrunettiR. Brunetti C. C. JacoboniJacoboni A. A. CappelliCappelli

… more thanks…… more thanks…

Eric PopEric Pop

University of Illinois at Urbana Champaign


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Part of this work has been carried outunder the contract 347713/2011 of theIntel Corporation, whose support isgratefully acknowledged.

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


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• Microscopic modeling: Monte Carlo analysis and

detrapping mechanism.

• Measurements.

• Conclusions.

CPU vs Hard Drive Performance CPU 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


5

CPU withEmbedded Memory

(SRAM cache)

Main MemoryDIMMS on

MB(DRAM)

StorageHard Disk Drive

(HDD)

Latency = 1x$$/bit = 1x

Latency = 100,000x$$/bit = 0,01x

CPU vs Hard Drive Performance CPU vs Hard Drive Performance —— IIII

CPU with Main StorageNVM Storage


6

CPU withEmbedded Memory

(SRAM cache)

Main MemoryDIMMS on

MB(DRAM)

Hard Disk Drive(HDD)

Latency = 1x$$/bit = 1x

NVM StorageSolid State Disk

(Managed NAND)

Latency = 100,000x$$/bit = 0,01x

Latency = 1000x$$/bit = 0,1x

Possible implementationsPossible implementations

Family Example Mechanism Selector

Phase ChangeChalcogenide 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 RAMSwitching of magnetic resistive

layer by spin-polarized electronsBipolar


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Junction

Electro-

chemicale.g.: CuSiO2

Formation / dissolution of “nano-

bridge” by electrochemistryBipolar

Binary Oxide

(filaments)e.g.: NiO

Reversible filament formation by

oxidation-reductionUnipolar

Interfacial

Resistance

Memristors

e.g.: doped STO, PCMO

Oxygen vacancy drift diffusion

induced barrier modulationBipolar

What is a chalcogenide?What is a chalcogenide?

Chalcos (ore) + Gen (formation) → Chalcogen (ore formation)

Electro-positive Element or + Chalcogen →


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Chalcogenide: (GeTe)1+x(Sb2Te3)x

x = 1 → Ge2Sb2Te5

Why chalcogenides?Why chalcogenides?

H

e

a

t

Crystalline AmorphousHeat reversibly switches

chalcogenides from crystalline

to amorphous.

The ratio ρa/ρc ~102 is the

principle behind phase- change

memories .


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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

PhasePhase--Change Memory (PCM)Change Memory (PCM)

Sub-threshold behavior

SET current

READ voltage

RESET currentRepr. from Wutting and

Yamada, Nature Mater.

6, 824 (2007)


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exponential

Ohmic

Sub-threshold behavior

(courtesy: D. Ielmini)

THRESHOLD voltage

Metal 1Metal 2

Row

Column

Poly

Si-Subst rate

Metal 1Metal 2

Row

Column

Poly

Si-Subst rate

PCM architecturePCM architecture


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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.

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


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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”).

Trap conduction

Low current

ME emission

High current

Scheme of the energy transitionsScheme of the energy transitions


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The electron tunnels between

traps and remains a low-

mobility trap electronThe electron becomes a high-

mobility band electron

GC R

S

The whole device is described as aone-dimensional structure made ofthe series of the amorphous GSTmaterial of area A, length L, withconductance

( ) ( ) 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


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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

SGG

RG

RR+

+=+=11

VAlways low

conductance

Always saturated

conductance

Large transition intervalLarge transition interval


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II’ I’’

If IK

= I’’ – I’ is large,

the V(I) curve is not N-

shaped.

V

Always low

conductance

Always saturated

conductance

Small transition intervalSmall transition interval


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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.

Low conductance

High conductance

V V

Low conductance

High conductance

Limiting casesLimiting cases

The second case is important for measuring purposes


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I’=I’’ I I’ I’’ I

Given the general expression of the conductance,

( ) ( ) nnNnnqLAG TnTTC +=+= , µµ

−+=

N

nNq

L

AG

T

TnTC µ

µµµ 1

GC

IK

Models of the conductance Models of the conductance —— II


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I

GC

n << nT ≈ N

µT << µn

nT ≈ 0

µn >> µT

I’ I’’

The balance of the generation and recombination phenomena yields(with I, IK > 0, neq ≤ n < N)

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2

II

I nNn

nn −+=

( )[ ].1/ exp)( ),( −=+= KCBI IIIrIrnnn

Models of the conductance Models of the conductance —— IIII


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The model is derived by combining the phonon-assisted netrecombination 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

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 relation current relation —— II


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However, it is found from

experiments that the linear

part is followed by an

exponential part.

Thus, improvements are

necessary.

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 relation current relation —— IIII


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exponential.

� The region where the

conductance saturates is

not affected by the trap

mobility.

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


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Electrodes are made of a broken carbon nanotube, coated with a 10-

nm GST layer.

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 data Fitting experimental data —— II


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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).

Fitting experimental data Fitting experimental data —— IIII


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The lower branch at low currents can be considered near toequilibrium. If T becomes larger the concentration n of the bandelectrons 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--dependence dependence of the lower branchof the lower branch —— II


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with kB the Boltzmann constant and C0, C1, Ea parameters. Note thatthis result does not contradict the low-temperature conductanceexpression (Mott’s law of variable-range hopping):

( )1 4

0expCG T T ∝ −

The above expression has been fitted to the experimental data byIelmini 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--dependence dependence of the lower branchof the lower branch —— IIII


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Parameter C0 is directly

related to physical

quantities of interest:

0 T

AC q N

Lµ=

One starts from the V(I) characteristic, where GC incorporates also thedependence of the trap mobility on I :

( )1

S

C

V R IG I

= +

The snap-back point is found by making the derivative dV/dI to vanish.This yields:

dG G

Scaling factorsScaling factors —— II


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d1

d

C CS C

G GR G

I I+ =

The right hand side is invariant with respect to the physical andgeometrical scaling factors of GC and (independently) of I. The resultshows that in this model the snap-back current (obviously notnecessarily the voltage!) is invariant with respect to such scalingfactors.

Scaling factorsScaling factors —— IIII

The scaling of the V(I) characteristic under variation in the geometricalfactors (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-


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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.

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 of Hopping of trap electronstrap electrons


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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.

The numerical approach: the Monte Carlo methodThe numerical approach: the Monte Carlo method


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a-GST layer Metal contactMetal contact

Compensating charge

Filled donor trap

Empty donor trap

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


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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

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


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existence of fluctuating

minimum-energy paths

(moving filaments??)

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


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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.

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( ) = −h

2

2m∗

∂2

∂x2+V

Tx( )

ManyMany--electronelectron (ME) (ME) detrappingdetrapping. Physical model . Physical model —— IIII


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H0 x( ) = −2m∗ ∂x2

+VTx( )

The Schrödinger equation reads:

ih∂∂tψ 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

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


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• The time step of the system evolution is taken as ∆t = 0.2 fs.

ψ k+1 =MbkN*N–element vector describing the

two-particle wavefunction at time

step k+1M: 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.

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

Results Results —— II


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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.

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, t f( ) = αn

n

∑ ϕnx1( ) 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

Results Results —— IIII


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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) = α

n

2

n=0

K−1

∑

0

Due to the physical and geometrical parameters of the system under

investigation, χ0 is the only bound state.

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 interactions e interactions —— II


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α0 ϕ0χ0 + αn

n=1

∑ ϕnχnwχ0

wχn

βn

n=1

∑ ϕnχn

(bound + non-bound states)

(non-bound states)(non-bound state)

(bound state)

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 interactions e interactions —— IIII


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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

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


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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.

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


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may be unstable. This may be

exploited by inducing

oscillations. Several parameters

may be measured from this.

MeasurementsMeasurements —— IIII

The increase in

temperature due to the

oscillatory regime

may in turn induce a

partial crystallization

of the material.

This phenomenon


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This phenomenon

may also be exploited

for determining

physical parameters.

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,


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• 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.

Technology

Physical Modeling and Design for Phase Change Memories