Resistive Random Access Memory (ReRAM, RRAM)

1. Theory

Definition: Resistive random access memory (ReRAM or RRAM) is a two-terminal metal-insulator-metal cell switched between a low-resistance state (LRS) and a high-resistance state (HRS) by moving ions.
Valence change mechanism (VCM): In oxides such as $Hf O_{x}$ , $Ta O_{x}$ and $Ti O_{x}$ , oxygen vacancies drift under the field to form or break a conductive filament; switching is bipolar.
Electrochemical metallization (ECM, CBRAM): An active Ag or Cu electrode oxidizes, metal ions migrate, and a metallic filament grows at the inert electrode.
Thermochemical mechanism (TCM): Joule heating drives unipolar switching.
Filamentary vs interface-type: Filamentary LRS is area-independent and metallic; interface-type resistance scales as 1/A and is thermally activated (Schottky-like).
Forming: A one-time soft breakdown at about $E_{f or m} t_{o x}$ creates the first filament; forming-free stacks use thin or oxygen-deficient layers.
Conductance quantization: An atom-wide filament shows conductance steps near $2 q^{2} / h$ ( $\approx 77.5 μ S$ ).

2. Working Principle

Set: A voltage closes the gap between filament and electrode, abruptly lowering the resistance.
Compliance current: An external limit or a 1T1R select transistor caps the set current; higher compliance gives a thicker filament and lower $R_{L R S}$ .
Reset: Opposite-polarity voltage (bipolar) or Joule heating (unipolar) dissolves part of the filament, opening a tunnelling gap.
HRS conduction: Current tunnels across the gap $g$ , falling exponentially as the gap widens.
Switching kinetics: Ion hopping depends exponentially on field and temperature, making switching abrupt and strongly voltage-dependent.
Multi-level storage: Different compliance currents or reset voltages give intermediate resistance levels.
Memristor model: The linear ion-drift model describes the concept, but real cells follow the exponential hopping kinetics.

3. Applications

Embedded non-volatile memory: Code and data storage in microcontrollers and IoT chips at 40–22 nm, where embedded flash is costly.
Low-power serial memory (CBRAM): Ultra-low-energy standalone non-volatile memory for battery-powered devices.
In-memory and neuromorphic computing (research): Crossbar arrays of analog ReRAM cells for matrix-vector multiplication.

Name	Formula	Related Formulae	Variables/Constants
Forming voltage and breakdown statistics	$V_{f or m} \approx E_{f or m} t_{o x}$	$F (t) = 1 - exp [- (\frac{t}{τ})^{β}]$	$V_{f or m} : v o l t a g e o f t h e f i r s t se t (f or min g) t ha t cr e a t es t h e co n d u c t i v e f i l am e n t in a f r es h o x i d e$ $E_{f or m} : a v er a g e f i e l d a t w hi c h t h e f i r s t f i l am e n t f or m s, a so f t - b r e ak d o w n f i e l d l o w er t han t h e ha r d b r e ak d o w n f i e l d$ $t_{o x} : o x i d e t hi c k n ess$ $F (t) : c u m u l a t i v e p r o babi l i t y t ha t f or min g ha s occ u r r e d a f t er a s t r ess t im e t$ $τ, β : W e ib u l l sc a l e t im e an d s ha p e p a r am e t er o f t h e f or min g t im e d i s t r ib u t i o n$ $(V_{f or m} \propto t_{o x} i s an e m p i r i c a l t r e n d, n o t a l a w; in t h e p er co l a t i o n p i c t u r e β i t se l f g r o w s w i t h t_{o x})$
Resistance of the low-resistance state (cylindrical filament)	$R_{L R S} = \frac{ρ _{f} t _{o x}}{π r _{f}^{2}}$	$R (T) = R_{0} [1 + α_{T} (T - T_{0})]$	$R_{L R S} : r es i s t an ce o f t h e ce l l w i t h a co m pl e t e m e t a l l i c f i l am e n t$ $ρ_{f} : r es i s t i v i t y o f t h e f i l am e n t ma t er ia l$ $t_{o x} : f i l am e n t l e n g t h, e q u a l t o t h e o x i d e t hi c k n ess f or a co m pl e t e f i l am e n t$ $r_{f} : f i l am e n t r a d i u s$ $R (T) : r es i s t an ce o f t h e f i l am e n t a t t e m p er a t u r e T$ $R_{0} : r es i s t an ce a t t h e r e f er e n ce t e m p er a t u r e T_{0}$ $α_{T} : t e m p er a t u r e coe f f i c i e n t o f r es i s t an ce (p os i t i v e f or a m e t a l l i c f i l am e n t, w hi c h i s t h e u s u a l c h ec k t ha t t h e l o w - r es i s t an ce s t a t e i s f i l am e n t a r y)$ $(i d e a l cy l in d er; a r e a l f i l am e n t i s t a p er e d, an d i t s r es i s t i v i t y i s ab o v e t h e b u l k v a l u e b ec a u se o f i t s nan o m e t r e s i z e)$
Conductance quantization of a narrow filament	$G = \frac{2 q ^{2}}{h} \sum_{n} T_{n}$	$G_{0} = \frac{2 q ^{2}}{h} \approx 77.5 μ S$ $R_{0} = \frac{1}{G _{0}} = \frac{h}{2 q ^{2}} \approx 12.9 k Ω$	$G : co n d u c t an ce o f t h e co n s t r i c t i o n (L an d a u er f or m u l a, s p in d e g e n er a cy in c l u d e d)$ $T_{n} : t r an s mi ss i o n p r o babi l i t y o f t h e n - t h co n d u c t i o n c hann e l$ $G_{0} : co n d u c t an ce q u an t u m f or tw o s p in s t a t es$ $R_{0} : r es i s t an ce o f o n e f u l l y t r an s mi tt in g c hann e l$ $q, h : e l e m e n t a r y c ha r g e an d P l an c k co n s t an t$ $(a ppl i es w h e n t h e na r r o w es t p a r t o f t h e f i l am e n t i s ab o u t o n e a t o m w i d e, so t ha t t h e l o w - r es i s t an ce s t a t e f a l l s n e a r in t e g er m u l t i pl es o f G_{0})$
High-resistance state: tunnelling gap	$I = I_{0} exp (- \frac{g}{g _{0}}) sinh (\frac{V}{V _{0}})$	$\frac{R _{H R S}}{R _{L R S}} \approx exp (\frac{g}{g _{0}})$	$I : ce l l c u r r e n t$ $I_{0} : c u r r e n t p r e f a c t or (f i t p a r am e t er)$ $g : l e n g t h o f t h e t u nn e l l in g g a p b e tw ee n t h e t i p o f t h e f i l am e n t an d t h e o pp os i t e e l ec t r o d e$ $g_{0} : c ha r a c t er i s t i c t u nn e l l in g l e n g t h o f t h e g a p (f i t p a r am e t er o f t h e or d er o f a f e w t e n t h s o f a nan o m e t r e)$ $V : v o l t a g e a cr oss t h e ce l l$ $V_{0} : c ha r a c t er i s t i c v o l t a g e o f t h e s inh d e p e n d e n ce (f i t p a r am e t er)$ $R_{H R S}, R_{L R S} : r es i s t an ces a t t h e s am e r e a d v o l t a g e f or g a p g an d f or n o g a p (t h e r a t i o i s f or a s ma l l r e a d v o l t a g e, w h er e s inh i s l in e a r; t h e ser i es r es i s t an ce o f t h e f i l am e n t b o d y i s n e g l ec t e d)$ $(S t an f or d co m p a c t m o d e l o f G u an, Y u an d W o n g)$
Gap growth and dissolution by ion hopping	$\frac{d g}{d t} = - ν_{0} exp (- \frac{E _{a}}{k _{B} T}) sinh (\frac{q a γ V}{t _{o x} k _{B} T})$	$γ = γ_{0} - β (\frac{g}{g _{1}})^{3}$	$g : g a p l e n g t h (i t s h r ink s f or a p os i t i v e se t v o l t a g e an d g r o w s f or a r ese t v o l t a g e o f t h e o pp os i t e s i g n)$ $ν_{0} : a tt e m pt f r e q u e n cy o f an i o n h o p$ $E_{a} : a c t i v a t i o n e n er g y f or t h e h o p o f an o x y g e n v a c an cy or i o n (a d i f f er e n t v a l u e i s u se d f or se t an d r ese t)$ $k_{B}, T : B o l t z mann co n s t an t an d l oc a l t e m p er a t u r e o f t h e f i l am e n t t i p$ $a : h o pp in g d i s t an ce o f t h e i o n$ $γ : l oc a l f i e l d e nhan ce m e n t f a c t or a t t h e t i p, w hi c h d ecr e a ses a s t h e g a p g r o w s$ $γ_{0}, β, g_{1} : f i t p a r am e t er s o f t h e f i e l d e nhan ce m e n t (g_{1} mak es t h e r a t i o d im e n s i o n l ess)$ $(T i s t h e l oc a l f i l am e n t t e m p er a t u r e f r o m t h e J o u l e h e a t in g r o w, so t h e tw o r o w s a r e so l v e d t o g e t h er)$ $V : v o l t a g e a cr oss t h e o x i d e$ $t_{o x} : o x i d e t hi c k n ess, so t ha t V / t_{o x} i s t h e a v er a g e f i e l d$
Joule heating of the filament	$T_{f} = T_{0} + R_{t h} I V$	$P_{r ese t} \approx \frac{T _{r ese t} - T _{0}}{R _{t h}}$	$T_{f} : t e m p er a t u r e o f t h e f i l am e n t$ $T_{0} : ambi e n t t e m p er a t u r e$ $R_{t h} : e f f ec t i v e t h er ma l r es i s t an ce f r o m t h e f i l am e n t t o t h e s u r r o u n d in g s$ $I, V : c u r r e n t an d v o l t a g e o f t h e ce l l (t h e p r o d u c t i s t h e d i ss i p a t e d p o w er)$ $P_{r ese t} : p o w er n ee d e d t o h e a t t h e f i l am e n t t o t h e t e m p er a t u r e T_{r ese t} a t w hi c h i t d i sso l v es (t h e t h er ma l r ese t o f u ni p o l a r ce l l s)$ $T_{r ese t} : f i l am e n t t e m p er a t u r e a t w hi c h t h er ma l d i sso l u t i o n t ak es pl a ce$ $(l u m p e d m o d e l; R_{t h} d e p e n d s o n t h e f i l am e n t r a d i u s an d t h e s u r r o u n d in g ma t er ia l s . I n bi p o l a r ce l l s t h e h e a t in g o n l y s p ee d s u p t h e f i e l d - d r i v e n h o pp in g)$
Low-resistance state against compliance current	$R_{L R S} \approx R_{1} (\frac{I _{1}}{I _{cc}})^{p}$	$I_{r ese t} \approx I_{cc}$	$R_{L R S} : r es i s t an ce o f t h e l o w - r es i s t an ce s t a t e o b t ain e d w i t h t h e co m pl ian ce c u r r e n t I_{cc}$ $R_{1} : r es i s t an ce a t t h e r e f er e n ce co m pl ian ce c u r r e n t I_{1}$ $I_{cc} : co m pl ian ce c u r r e n t a ppl i e d d u r in g se t, w hi c h l imi t s t h e f i l am e n t s i z e$ $p : e x p o n e n t, c l ose t o o n e f or man y o x i d es (an e m p i r i c a l f i t; I e l mini r e p or t s R_{L R S} \approx V_{C} / I_{cc} w i t h V_{C} o f a f e w t e n t h s o f a v o l t)$ $I_{r ese t} : c u r r e n t a t t h e o n se t o f r ese t, w hi c h i s o f t h e or d er o f t h e co m pl ian ce c u r r e n t f or t h e s am e f i l am e n t$
Faraday growth of an electrochemical metallization filament	$\frac{d L}{d t} = \frac{M j _{i o n}}{z F ρ _{m}}$	$j_{i o n} = j_{0} [exp (\frac{α _{a} z F η}{R T}) - exp (- \frac{α _{c} z F η}{R T})]$ $E = E^{0} + \frac{R T}{z F} ln \frac{a _{o x}}{a _{r e d}}$	$L : l e n g t h o f t h e m e t a l l i c f i l am e n t (g r o w s b y r e d u c t i o n o f t h e m e t a l i o n s a t i t s t i p)$ $M : m o l a r ma ss o f t h e f i l am e n t m e t a l$ $j_{i o n} : c u r r e n t d e n s i t y o f t h e i o n r e d u c t i o n a t t h e t i p o f t h e f i l am e n t$ $z : c ha r g e n u mb er o f t h e i o n (o n e f or A g +, tw o f or C u 2 +)$ $F : F a r a d a y co n s t an t$ $ρ_{m} : ma ss d e n s i t y o f t h e f i l am e n t m e t a l$ $j_{0} : e x c han g e c u r r e n t d e n s i t y o f t h e e l ec t r o d e r e a c t i o n$ $α_{a}, α_{c} : an o d i c an d c a t h o d i c c ha r g e t r an s f er coe f f i c i e n t s$ $η : o v er p o t e n t ia l a t t h e t i p (v o l t a g e in e x cess o f t h e e q u i l ib r i u m e l ec t r o d e p o t e n t ia l)$ $R, T : g a s co n s t an t an d t e m p er a t u r e$ $E, E^{0} : e l ec t r o d e p o t e n t ia l an d s t an d a r d e l ec t r o d e p o t e n t ia l (N er n s t e q u a t i o n)$ $a_{o x}, a_{r e d} : a c t i v i t i es o f t h e o x i d i z e d an d r e d u ce d s p ec i es$ $(s t an d a r d e l ec t r o d e p o t e n t ia l s in w a t er : A g^{+} / A g + 0.80 V, C u^{2 +} / C u + 0.34 V; v a l u es in a so l i d e l ec t r o l y t e d i f f er . D e t ai l e d m o d e l s a dd n u c l e a t i o n an d i o n t r an s p or t l imi t s)$
Interface-type switching (area-dependent)	$J = A^{*} T^{2} exp [- \frac{q φ _{B} - q ^{3} E /4 π ε}{k _{B} T}]$	$R_{in t er f a ce} \propto \frac{1}{A}$ $R_{f i l am e n t} in d e p e n d e n t o f A$	$J : c u r r e n t d e n s i t y o v er a S c h o tt k y ba r r i er w i t h ima g e - f or ce l o w er in g (t h er mi o ni c e mi ss i o n)$ $A^{*} : e f f ec t i v e R i c ha r d so n co n s t an t$ $T : t e m p er a t u r e$ $φ_{B} : ba r r i er h e i g h t a t t h e e l ec t r o d e / o x i d e in t er f a ce, w hi c h i s c han g e d b y t h e m o v e m e n t o f o x y g e n v a c an c i es a t t h e in t er f a ce$ $E : f i e l d a t t h e in t er f a ce$ $ε : hi g h - f r e q u e n cy (o pt i c a l) p er mi tt i v i t y o f t h e o x i d e, a s o n t h e D R A M p a g e$ $A : e l ec t r o d e a r e a (t h e r es i s t an ce o f an in t er f a ce - t y p e ce l l sc a l es a s 1/ A, w hi l e t ha t o f a f i l am e n t a r y ce l l d oes n o t d e p e n d o n t h e a r e a, w hi c h i s t h e u s u a l w a y t o t e l l t h e tw o a p a r t)$ $k_{B} : B o l t z mann co n s t an t$
1T1R cell: compliance and switching energy	$I_{cc} = \frac{μ _{n} C _{o x} W}{2 L} (V_{G} - V_{T})^{2}$	$E_{se t} \approx V_{se t} I_{cc} t_{se t}$	$I_{cc} : co m pl ian ce c u r r e n t se t b y t h e ser i es se l ec t t r an s i s t or in s a t u r a t i o n$ $μ_{n}, C_{o x}, W, L : e l ec t r o n m o bi l i t y, g a t e c a p a c i t an ce p er u ni t a r e a, w i d t h an d l e n g t h o f t h e se l ec t t r an s i s t or$ $V_{G}, V_{T} : g a t e v o l t a g e an d t h r es h o l d v o l t a g e o f t h e se l ec t t r an s i s t or$ $E_{se t} : e n er g y o f a se t o p er a t i o n (an a pp r o x ima t i o n, w i t h t h e c u r r e n t t ak e n e q u a l t o t h e co m pl ian ce c u r r e n t d u r in g t h e s w i t c hin g t im e)$ $V_{se t} : v o l t a g e a cr oss t h e ce l l d u r in g se t$ $t_{se t} : t im e t ak e n f or t h e se t t r an s i t i o n$ $(s q u a r e - l a w s a t u r a t i o n; w i t h t h e ce l l o n t h e so u r ce s i d e t h e ce l l v o l t a g e l o w er s V_{GS} . E_{se t} o v er es t ima t es, s in ce t h e c u r r e n t i s b e l o w I_{cc} b e f or e t h e f i l am e n t c l oses)$
Linear ion drift model (memristor)	$v (t) = R_{o n} \frac{w}{D} i (t) + R_{o f f} (1 - \frac{w}{D}) i (t)$	$\frac{d w}{d t} = μ_{v} \frac{R _{o n}}{D} i (t)$	$v (t), i (t) : v o l t a g e an d c u r r e n t o f t h e d e v i ce$ $w : w i d t h o f t h e d o p e d (l o w - r es i s t an ce) r e g i o n, b e tw ee n 0 an d D$ $D : t o t a l t hi c k n ess o f t h e o x i d e f i l m$ $R_{o n}, R_{o f f} : r es i s t an ces o f t h e d e v i ce w h e n w = D an d w h e n w = 0$ $μ_{v} : a v er a g e m o bi l i t y o f t h e d o p an t s (o x y g e n v a c an c i es)$ $(a s im pl e m o d e l t ha t l e a d s t o t h e p in c h e d c u r r e n t - v o l t a g e l oo p; i t l a c k s t h e s t r o n g n o n l in e a r i t y in t h e f i e l d o f t h e i o n h o pp in g r o w)$

Important Questions

Ques: What are the main types of resistive switching in an oxide cell?

The valence change mechanism (VCM) is found in transition-metal oxides such as HfO $_{x}$ , TaO $_{x}$ and TiO $_{x}$ : oxygen vacancies move under the field and form or break a conductive filament of a sub-stoichiometric oxide, and the switching is bipolar. The electrochemical metallization mechanism (ECM, also called conductive-bridge RAM or CBRAM) uses an active electrode such as Ag or Cu: the metal is oxidized at the anode, the ions move through the electrolyte and are reduced at the inert cathode to grow a metallic filament, so the Faraday and electrode-reaction rows apply. In the thermochemical mechanism (TCM), the switching is unipolar and driven by Joule heating, which fits the Joule heating row. A given device can show more than one mechanism, and the polarity of the voltage and the electrode materials are used to tell them apart. Reference: R. Waser, R. Dittmann, G. Staikov and K. Szot, “Redox-based resistive switching memories - nanoionic mechanisms, prospects, and challenges,” Adv. Mater. 21, 2632 (2009); R. Waser and M. Aono, “Nanoionics-based resistive switching memories,” Nat. Mater. 6, 833 (2007).

Ques: Why is a compliance current needed, and why does the low-resistance state depend on it?

During set the resistance falls abruptly when the filament closes the gap, and without a limit the current would rise until the filament over-grows or the oxide breaks down permanently. A compliance current (set by an external limit or by the select transistor of a 1T1R cell) limits the power dissipated after the filament forms. A larger compliance current gives a thicker filament, so $R_{L R S}$ falls roughly as $1/ I_{cc}$ in the empirical row, and the reset current rises in proportion. Typical HfO $_{x}$ cells set and reset at about $\pm 0.5$ to $1.5$ V with compliance currents of tens to a few hundred microamperes, switch in nanoseconds, and give resistance ratios of about 10 to 100. The high-resistance state is set by the gap $g$ and does not follow the compliance in the same way. This dependence also allows several resistance levels in a cell by changing $I_{cc}$ . Reference: D. Ielmini, “Resistive switching memories based on metal oxides: mechanisms, reliability and scaling,” Semicond. Sci. Technol. 31, 063002 (2016); H.-S.P. Wong et al., “Metal-oxide RRAM,” Proc. IEEE 100, 1951 (2012); X. Guan, S. Yu and H.-S.P. Wong, “On the switching parameter variation of metal-oxide RRAM - Part I: physical modeling and simulation methodology,” IEEE Trans. Electron Devices 59, 1172 (2012) (gap model).

Ques: What is forming, and why do some cells not need it?

A fresh oxide contains few oxygen vacancies and is insulating. The forming step applies a voltage of about $E_{f or m} t_{o x}$ to create the first filament by soft dielectric breakdown, which is why it is higher than the later set voltage. A thicker oxide has a higher forming voltage, and the forming time is statistical (Weibull row). Forming-free cells have a thin oxide, a stack with an oxygen-deficient layer (for example a Ta $_{2}$ O $_{5 - x}$ /TaO $_{2 - x}$ bilayer, in which the oxygen-deficient TaO $_{2 - x}$ acts as a vacancy reservoir), or a higher defect density introduced during growth, so that the first set voltage is close to the later ones. Reference: H.-S.P. Wong et al., Proc. IEEE 100, 1951 (2012); R. Waser et al., Adv. Mater. 21, 2632 (2009); M.-J. Lee et al., “A fast, high-endurance and scalable non-volatile memory device made from asymmetric Ta $_{2}$ O $_{5 - x}$ /TaO $_{2 - x}$ bilayer structures,” Nat. Mater. 10, 625 (2011).

Ques: Why does a very narrow filament show conductance quantization?

When the narrowest part of the filament is about one atom wide, the electron wavelength is comparable to the constriction and the conductance is given by the Landauer formula $G = (2 q^{2} / h) \sum T_{n}$ , with steps near integer multiples of $G_{0} \approx 77.5 μ$ S, that is a resistance of 12.9 k $Ω$ per channel. The effect was seen in atomic switches and in some oxide and CBRAM cells at a low compliance current. In practice the observed levels depend on the transmission $T_{n}$ , so the steps are not always integers, and the effect is a fingerprint of a single-atom contact, not of the switching mechanism. Reference: K. Terabe, T. Hasegawa, T. Nakayama and M. Aono, “Quantized conductance atomic switch,” Nature 433, 47 (2005); S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, 1995.

Ques: How can one tell a filamentary cell from an interface-type cell?

In a filamentary cell the low-resistance state is carried by a narrow filament, so its resistance does not depend on the electrode area, shows a positive temperature coefficient (metallic) and depends on the compliance current. In an interface-type cell the resistance is set by a barrier at the electrode, so it scales as $1/ A$ , shows a thermally activated current (the Schottky row) and changes gradually, without an abrupt set. Measuring cells of several areas is the usual test, and the electrode and oxide combination decides which is found. Reference: R. Waser et al., Adv. Mater. 21, 2632 (2009); D. Ielmini, Semicond. Sci. Technol. 31, 063002 (2016).

Ques: How is the 1T1R cell used, and what does the transistor do?

A select transistor in series with the resistive element limits the compliance current during set, as in the 1T1R row, and blocks the current in unselected cells. The gate voltage chooses the compliance, so $V_{G}$ is a way to set the low-resistance level. The cost is the area of the transistor, and the transistor must supply the reset current. In the usual layout the cell sits on the drain side; for a bipolar reset the current reverses, so the transistor terminal next to the cell becomes its source, which is lifted by the cell voltage, and the gate overdrive is smaller than during set. Two-terminal selectors for crosspoint arrays are covered on the Selector pages. Reference: H.-S.P. Wong et al., Proc. IEEE 100, 1951 (2012); S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

Ques: What is the memristor model, and how does it relate to real devices?

The linear ion drift model gives the resistance as a mixture of $R_{o n}$ and $R_{o f f}$ set by the state variable $w$ , which moves in proportion to the current. Chua proposed the memristor as the missing circuit element in 1971, and the identification with a titanium dioxide cell was made in 2008. Real cells switch much more abruptly than the linear drift model, because the ion mobility depends exponentially on the field and temperature (the hopping row), so the linear model is used for the concept and the hopping model for device design. Reference: D. Ielmini, Semicond. Sci. Technol. 31, 063002 (2016); L.O. Chua, “Memristor - the missing circuit element,” IEEE Trans. Circuit Theory 18, 507 (1971); D.B. Strukov, G.S. Snider, D.R. Stewart and R.S. Williams, “The missing memristor found,” Nature 453, 80 (2008).

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