Ferroelectric Random Access Memory (FeRAM)

1. Theory

Definition: Ferroelectric random access memory (FeRAM) stores a bit as the direction of remanent polarization in a ferroelectric capacitor connected to a bit line through an access transistor.
Cell forms: 1T1C (one transistor, one capacitor, compared against a reference) or 2T2C (two complementary cells compared directly).
Ferroelectric bistability: The Landau free energy has two minima, $\pm P$ , so the polarization stays at $\pm P_{r}$ with no field and no refresh.
Coercive field: The polarization reverses only above $E_{c}$ ; the coercive voltage is $V_{c} = E_{c} t_{F E}$ .
Materials: Perovskites (PZT, SBT) have large $P_{r}$ and low $E_{c}$ but need thick films and special electrodes; doped $Hf O_{2}$ (e.g. HZO) is CMOS-compatible at 5 to 20 nm but has $E_{c} \approx 1$ MV/cm.
Depolarization field: Interfacial dead layers create a field opposing the polarization, which grows as the film is thinned.

2. Working Principle

Write: A plate-line or bit-line pulse above the coercive voltage sets the polarization to $+ P_{r}$ or $- P_{r}$ .
Destructive read: A pulse drives the cell toward $+ P_{s}$ ; a cell storing $- P_{r}$ switches and releases about $2 P_{r} A$ more charge than one storing $+ P_{r}$ .
Read-restore: A switched cell must be rewritten after every read, which also consumes endurance.
Bit-line signal: The charge divides onto the bit line, giving $Δ V \approx 2 P_{r} A / (C_{B L} + C_{F E})$ ; a large $P_{r} A$ is needed against a large bit-line capacitance.
Switching kinetics: Domains nucleate and grow (Merz law, KAI model), so switching time is spread and the write pulse must cover the slowest cells.
Low write energy: Writing needs only about $2 P_{r} A V_{w r i t e}$ , with no charge pump or high voltage.

3. Applications

Fast, high-endurance non-volatile logging: Data loggers, smart meters, industrial controllers and automotive event recorders.
Contactless smart cards and RFID: Low-energy writes suit cards powered by the reader’s field, such as transit cards.
Medical and battery-powered devices: Instant, low-power non-volatile storage for implants, hearing aids and wearables.

Name	Formula	Related Formulae	Variables/Constants
Landau-Devonshire free energy and the polarization-field relation	$F = \frac{α}{2} P^{2} + \frac{β}{4} P^{4} + \frac{γ}{6} P^{6} - E P$	$E = α P + β P^{3} + γ P^{5}$ $α = α_{0} (T - T_{0})$	$F : f r ee e n er g y d e n s i t y o f t h e f er r oe l ec t r i c a s a f u n c t i o n o f t h e p o l a r i z a t i o n$ $P : p o l a r i z a t i o n a l o n g t h e p o l a r a x i s$ $E : a ppl i e d e l ec t r i c f i e l d, f o u n d f r o m t h e co n d i t i o n d F / d P = 0$ $α, β, γ : L an d a u coe f f i c i e n t s (α < 0 b e l o w t h e C u r i e t e m p er a t u r e; β > 0 f or a seco n d - or d er t r an s i t i o n an d β < 0, γ > 0 f or a f i r s t - or d er t r an s i t i o n)$ $α_{0} : p os i t i v e co n s t an t r e l a t e d t o t h e C u r i e co n s t an t (α_{0} = 1/ ε_{0} C)$ $T_{0} : C u r i e - W e i ss t e m p er a t u r e$
Spontaneous polarization and coercive field (second-order case)	$P_{s} = - \frac{α}{β}$	$E_{c} = \frac{2 ∣ α ∣ ^{3/2}}{3 3 β}$ $P_{s} \propto (T_{C} - T)^{1/2}$	$P_{s} : s p o n t an eo u s p o l a r i z a t i o n o f t h e h o m o g e n eo u s f er r oe l ec t r i c a t z er o f i e l d, o b t ain e d f or γ = 0 an d α < 0$ $E_{c} : in t r in s i c coer c i v e f i e l d, a t w hi c h t h e s t or e d s t a t e b eco m es u n s t ab l e (d E / d P = 0)$ $T_{C} : C u r i e t e m p er a t u r e, ab o v e w hi c h t h e ma t er ia l i s p a r a e l ec t r i c$ $(m e a s u r e d coer c i v e f i e l d s in t hin f i l m s a r e f a r l o w er t han t hi s in t r in s i c v a l u e b ec a u se s w i t c hin g ha pp e n s b y n u c l e a t i o n an d d o main w a l l m o t i o n)$
Dielectric susceptibility and small-signal capacitance	$χ_{F E} = \frac{d P}{d E} = \frac{1}{α + 3 β P ^{2} + 5 γ P ^{4}}$	$C_{F E} = \frac{A}{t _{F E}} (ε_{0} ε_{b} + χ_{F E})$ $ε_{r} (T) = \frac{C}{T - T _{0}} (T > T_{C})$	$χ_{F E} : d i f f er e n t ia l s u sce pt ibi l i t y (c han g e o f t h e p o l a r i z a t i o n p er u ni t f i e l d) a t t h e p o l a r i z a t i o n P, in F / m (t h e d im e n s i o n l ess s u sce pt ibi l i t y i s χ_{F E} / ε_{0})$ $C_{F E} : s ma l l - s i g na l c a p a c i t an ce o f t h e f er r oe l ec t r i c c a p a c i t or a t a g i v e n p o in t o f t h e h y s t er es i s l oo p$ $A, t_{F E} : e l ec t r o d e a r e a an d t hi c k n ess o f t h e f er r oe l ec t r i c l a y er$ $ε_{b} : ba c k g r o u n d r e l a t i v e p er mi tt i v i t y o f t h e l a tt i ce (n o t in c l u d in g t h e p o l a r i z a t i o n r es p o n se)$ $C : C u r i e co n s t an t o f t h e C u r i e - W e i ss l a w$ $(t h e c a p a c i t an ce i s l a r g es t n e a r t h e coer c i v e f i e l d, an d t h e r e g i o n w i t h α + 3 β P^{2} < 0 i s n o t r e a c h e d in a s t ab l e c a p a c i t or)$
Coercive voltage and operating voltage	$V_{c} = E_{c} t_{F E}$	$V_{w r i t e} ≳ V_{c} (with margin for the distribution of the switching voltage)$	$V_{c} : coer c i v e v o l t a g e a cr oss t h e f er r oe l ec t r i c c a p a c i t or (t h e m e a s u r e d E_{c} t im es t h e t hi c k n ess)$ $E_{c} : m e a s u r e d coer c i v e f i e l d o f t h e t hin f i l m$ $t_{F E} : t hi c k n ess o f t h e f er r oe l ec t r i c l a y er$ $V_{w r i t e} : v o l t a g e a ppl i e d a cr oss t h e c a p a c i t or in a w r i t e o p er a t i o n, w hi c h s h o u l d e x cee d V_{c} so t ha t t h e s w i t c hin g f ini s h es in t h e a v ai l ab l e t im e$ $(t hinn er f i l m s g i v e a l o w er coer c i v e v o l t a g e, b u t n o t in s t r i c t p r o p or t i o n, b ec a u se t h e m e a s u r e d E_{c} r i ses a s t h e f i l m t hin s : t h e K a y - D u nn sc a l in g f or p er o v s k i t es g i v es E_{c} \propto t_{F E}^{- 2/3})$
Switching charge and read signal	$Q_{r e v} = 2 P_{r} A$	$Q_{s w} = (P_{s} + P_{r}) A + C_{l in} V$ $Q_{n s w} = (P_{s} - P_{r}) A + C_{l in} V$ $Δ Q = Q_{s w} - Q_{n s w} = 2 P_{r} A$	$Q_{r e v} : c ha r g e r e l e a se d w h e n t h e p o l a r i z a t i o n r e v er ses f r o m - P_{r} t o + P_{r}$ $P_{r} : r e man e n t p o l a r i z a t i o n (t h e p o l a r i z a t i o n l e f t a t z er o f i e l d)$ $P_{s} : s a t u r a t i o n p o l a r i z a t i o n r e a c h e d a t t h e r e a d v o l t a g e$ $A : a r e a o f t h e f er r oe l ec t r i c c a p a c i t or$ $Q_{s w}, Q_{n s w} : c ha r g e d e l i v er e d b y a p os i t i v e r e a d p u l se t o a ce l l t ha t s w i t c h es (s t or e d - P_{r}) an d t o o n e t ha t d oes n o t (s t or e d + P_{r})$ $C_{l in} V : l in e a r (n o n - s w i t c hin g) c ha r g e a t t h e r e a d v o l t a g e V, w hi c h i s co mm o n t o b o t h s t a t es$ $Δ Q : d i f f er e n ce b e tw ee n t h e tw o s t a t es, w hi c h i s t h e s i g na l a v ai l ab l e t o t h e se n se c i r c u i t i f b o t h s t a t es s a t u r a t e$ $(r ec t an g u l a r, s a t u r a t e d l oo p a ss u m e d; a f ini t e l oo p s l o p e or p a r t ia l s w i t c hin g g i v es a s ma l l er s i g na l)$
Bit line load line	$C_{B L} V_{B L} = A [D (V_{P L} - V_{B L}) - D_{0}]$	$V_{F E} = V_{P L} - V_{B L}$ $V_{B L, 1} - V_{B L, 0} \approx \frac{2 P _{r} A}{C _{B L} + C _{F E}}$	$V_{B L} : bi t l in e v o l t a g e a f t er t h e pl a t e l in e p u l se, f o u n d a s t h e in t er sec t i o n o f t h e f er r oe l ec t r i c p o l a r i z a t i o n c u r v e w i t h t h e l o a d l in e o f t h e bi t l in e c a p a c i t an ce$ $C_{B L} : c a p a c i t an ce o f t h e bi t l in e$ $D (V_{F E}) : c ha r g e p er u ni t a r e a (d i s pl a ce m e n t, a l m os t e q u a l t o P) o f t h e f er r oe l ec t r i c a t t h e v o l t a g e V_{F E} a cr oss i t (t h e b r an c h d e p e n d s o n t h e s t or e d s t a t e)$ $D_{0} : ini t ia l v a l u e b e f or e t h e p u l se, - P_{r} or + P_{r} f or t h e tw o s t or e d s t a t es, so o n l y t h e c ha r g e r e l e a se d i s co u n t e d$ $V_{P L} : am pl i t u d e o f t h e pl a t e l in e p u l se$ $V_{F E} : v o l t a g e t ha t r e main s a cr oss t h e c a p a c i t or a f t er c ha r g e s ha r in g$ $V_{B L, 1} - V_{B L, 0} : v o l t a g e d i f f er e n ce b e tw ee n a s w i t c hin g an d a n o n - s w i t c hin g ce l l, a pp r o x ima t e d w i t h t h e s ma l l - s i g na l c a p a c i t an ce C_{F E} in t h e s a t u r a t e d s t a t e$
Required ferroelectric area	$A ≳ \frac{Δ V _{min} ( C _{B L} + C _{F E} )}{2 P _{r}}$	$A = \frac{Δ V _{min} C _{B L}}{2 P _{r}} (C_{F E} ≪ C_{B L})$	$A : c a p a c i t or a r e a n ee d e d f or a g i v e n bi t l in e v o l t a g e d i f f er e n ce$ $Δ V_{min} : s ma l l es t v o l t a g e d i f f er e n ce t ha t t h e se n se c i r c u i t c an r eso l v e w i t h ma r g in f or o f f se t s$ $C_{B L} : bi t l in e c a p a c i t an ce$ $C_{F E} : s ma l l - s i g na l c a p a c i t an ce o f t h e f er r oe l ec t r i c c a p a c i t or$ $P_{r} : r e man e n t p o l a r i z a t i o n, so a l a r g er P_{r} a l l o w s a s ma l l er ce l l$
Switching kinetics	$τ_{s w} = τ_{\infty} exp (\frac{E _{a}}{E})$	$Δ P (t) = 2 P_{s} [1 - exp (- (\frac{t}{τ})^{n})]$	$τ_{s w} : s w i t c hin g t im e o f t h e p o l a r i z a t i o n u n d er a f i e l d E (M er z l a w)$ $τ_{\infty} : s w i t c hin g t im e a t v er y hi g h f i e l d$ $E_{a} : a c t i v a t i o n f i e l d o f t h e n u c l e a t i o n p r ocess$ $E : a ppl i e d f i e l d, w hi c h m u s t b e w e l l ab o v e E_{c} f or a f a s t s w i t c hin g$ $Δ P (t) : s w i t c h e d p o l a r i z a t i o n a t t im e t in t h e K o l m o g or o v - A v r ami - I s hiba s hi (K A I) m o d e l$ $τ : c ha r a c t er i s t i c s w i t c hin g t im e o f t h e K A I m o d e l$ $n : A v r ami e x p o n e n t o f t h e K A I m o d e l, w hi c h d e p e n d s o n t h e d im e n s i o na l i t y o f t h e d o main g r o w t h an d o n t h e n u c l e a t i o n r a t e : n = d f or p r e - e x i s t in g n u c l e i an d n = d + 1 f or a co n s t an t n u c l e a t i o n r a t e, w i t h d t h e d im e n s i o na l i t y o f g r o w t h$ $(K A I s u i t s e p i t a x ia l f i l m s; p o l y cr y s t a l l in e f i l m s a r e b e tt er d escr ib e d b y t h e n u c l e a t i o n - l imi t e d s w i t c hin g m o d e l o f T a g an t se v e t a l ., w i t h a d i s t r ib u t i o n o f s w i t c hin g t im es)$
Energy of a switching operation	$E_{s w} = \int V d Q \approx 2 P_{r} A V_{w r i t e}$	$E_{l oo p} = A t_{F E} \oint E d P \approx 4 P_{r} E_{c} A t_{F E}$	$E_{s w} : e n er g y s u ppl i e d b y t h e d r i v er t o r e v er se t h e p o l a r i z a t i o n, f or a s w i t c hin g c ha r g e 2 P_{r} A a t t h e v o l t a g e V_{w r i t e}$ $V_{w r i t e} : v o l t a g e o f t h e w r i t e p u l se$ $E_{l oo p} : e n er g y d i ss i p a t e d p er cy c l e b y a r ec t an g u l a r h y s t er es i s l oo p, e q u a l t o t h e l oo p a r e a t im es t h e v o l u m e$ $E_{c} : coer c i v e f i e l d o f t h e l oo p$ $(t h e e n er g y o f c ha r g in g t h e l in e a r c a p a c i t an ce i s n o t in c l u d e d; f or V_{w r i t e} = V_{c} o n e cy c l e ha s tw o s w i t c hin g s, 2 \times 2 P_{r} A V_{c} = 4 P_{r} E_{c} A t_{F E}, so t h e tw o r o w s a g r ee)$
Depolarization field with an interfacial dead layer	$E_{F E} = - \frac{P t _{d}}{ε _{0} ( ε _{b} t _{d} + ε _{d} t _{F E} )}$	$E_{d} = - E_{F E} \frac{t _{F E}}{t _{d}}$	$E_{F E} : f i e l d in t h e f er r oe l ec t r i c w h e n t h e e l ec t r o d es a r e s h or t e d, o pp os i t e t o t h e p o l a r i z a t i o n$ $P : p o l a r i z a t i o n o f t h e f er r oe l ec t r i c$ $t_{d}, ε_{d} : t hi c k n ess an d r e l a t i v e p er mi tt i v i t y o f t h e n o n - f er r oe l ec t r i c in t er f a c ia l (d e a d) l a y er$ $t_{F E}, ε_{b} : t hi c k n ess an d ba c k g r o u n d r e l a t i v e p er mi tt i v i t y o f t h e f er r oe l ec t r i c$ $E_{d} : f i e l d in t h e d e a d l a y er$ $(t h e f i e l d g r o w s w i t h t h e d e a d - l a y er t hi c k n ess an d w i t h P, an d i t p u s h es t h e p o l a r i z a t i o n t o w a r d z er o; im p er f ec t scr ee nin g in t h e e l ec t r o d es ha s t h e s am e e f f ec t)$

Important Questions

Ques: How does a ferroelectric capacitor store a bit without power?

A ferroelectric has a spontaneous electric polarization with two stable orientations, +P and -P, separated by an energy barrier in the free energy $F (P)$ . When the field is removed the polarization settles at the remanent values $\pm P_{r}$ , so the capacitor holds a bit as the sign of the polarization, and no refresh is needed. The state can be reversed only by applying a field above the coercive field, which moves the polarization along the hysteresis loop from one stable branch to the other. The stored state is read by a polarity-dependent charge flow when the polarization is switched. Reference: J.F. Scott, Ferroelectric Memories, Springer, 2000.

Ques: Why is reading an FeRAM destructive, and what signal is read?

To read the cell a voltage pulse is applied to the plate line that would drive the polarization to $+ P_{s}$ . If the cell stores $- P_{r}$ , the polarization reverses and releases $Q_{s w} = (P_{s} + P_{r}) A$ plus the linear charge. If it stores $+ P_{r}$ , only $(P_{s} - P_{r}) A$ plus the linear charge flows. The difference $Δ Q \approx 2 P_{r} A$ is the signal. A cell that held $- P_{r}$ is left in the $+ P_{r}$ state, so the data must be rewritten after every read, which makes the access a read-restore cycle, and which also consumes endurance. Reference: H. Ishiwara, M. Okuyama and Y. Arimoto (eds.), Ferroelectric Random Access Memories, Springer, 2004.

Ques: How does the bit line capacitance affect the read voltage?

The charge from the capacitor is collected by the bit line capacitance $C_{B L}$ , which raises the bit line voltage until the voltage across the ferroelectric falls. The final $V_{B L}$ is the intersection of the load line $C_{B L} V_{B L} = A [D (V_{P L} - V_{B L}) - D_{0}]$ with the polarization curve of the stored state, so the signal is $V_{B L, 1} - V_{B L, 0} \approx 2 P_{r} A / (C_{B L} + C_{F E})$ . Since $C_{B L}$ is usually much larger than the capacitance of one cell, a large $P_{r}$ and area $A$ are needed to reach the signal required by the sense circuit. Reference: J.F. Scott, Ferroelectric Memories, Springer, 2000.

Ques: What is the difference between the 1T1C and 2T2C cells?

The 1T1C cell has one access transistor and one ferroelectric capacitor and needs a reference level between the voltages of the two states, which is usually generated by another capacitor or by a reference cell. The 2T2C cell uses two transistors and two capacitors that store opposite polarizations, so the sense circuit compares them directly, which doubles the signal and needs no reference. This makes the read more tolerant to variation of $P_{r}$ , but the cell is about twice as large. Reference: H. Ishiwara, M. Okuyama and Y. Arimoto (eds.), Ferroelectric Random Access Memories, Springer, 2004.

Ques: How do the perovskite and the hafnium-oxide ferroelectrics compare for memory capacitors?

Perovskite films such as Pb(Zr,Ti)O $_{3}$ (PZT) and SrBi $_{2}$ Ta $_{2}$ O $_{9}$ (SBT) have a large remanent polarization (of the order of 10 to 40 $μ$ C/cm $^{2}$ ) and a low coercive field (tens of kV/cm), but they must be tens to a hundred nanometers thick and need a noble-metal or oxide electrode, which makes integration with CMOS difficult. Doped HfO $_{2}$ (for example Hf $_{0.5}$ Zr $_{0.5}$ O $_{2}$ ) is ferroelectric in films of 5 to 20 nm and is compatible with CMOS materials, but its coercive field is about 1 MV/cm, so the coercive voltage $V_{c} = E_{c} t_{F E}$ of a 10 nm film is about 1 V. The remanent polarization is of the order of 10 to 25 $μ$ C/cm $^{2}$ . Reference: T.S. Böscke, J. Müller, D. Bräuhaus, U. Schröder and U. Böttger, “Ferroelectricity in hafnium oxide thin films,” Appl. Phys. Lett. 99, 102903 (2011).

Ques: What limits the scaling of a ferroelectric capacitor?

The required area is $A ≳ Δ V_{min} (C_{B L} + C_{F E}) /2 P_{r}$ , so a smaller cell needs either a larger $P_{r}$ , a lower bit line capacitance or a three-dimensional capacitor with a larger area for the same footprint. Thinning the film lowers $V_{c}$ but increases the depolarization field of the interface layers, $E_{F E} = - P t_{d} / ε_{0} (ε_{b} t_{d} + ε_{d} t_{F E})$ , which can destabilize the stored state. The switching also becomes a nucleation process in a small volume, so the switching voltage and time spread widens as the grains and the capacitor shrink. Reference: J.F. Scott, Ferroelectric Memories, Springer, 2000; R. Mehta, B.D. Silverman and J.T. Jacobs, “Depolarization fields in thin ferroelectric films,” J. Appl. Phys. 44, 3379 (1973).

Ques: Why does the switching of a ferroelectric capacitor take a distribution of times?

Polarization reversal happens by the nucleation of reversed domains and the growth of their walls, and the nucleation rate rises exponentially with the field. The Merz law $τ_{s w} = τ_{\infty} e^{E_{a} / E}$ describes the typical time in a given field, and the KAI model $Δ P = 2 P_{s} [1 - e^{- (t / τ)^{n}}]$ gives the growth of the switched fraction in a uniform film. In a real polycrystalline film the nucleation sites, grain size and local fields vary, so the switching is spread over a distribution of times. The write pulse must be long enough for the slowest cells to switch. The KAI picture fits epitaxial films; for polycrystalline films the nucleation-limited switching model, which assigns each region its own switching time, is more accurate. Reference: W.J. Merz, Phys. Rev. 95, 690 (1954); Y. Ishibashi and Y. Takagi, J. Phys. Soc. Jpn. 31, 506 (1971); A.K. Tagantsev, I. Stolichnov, N. Setter, J.S. Cross and M. Tsukada, “Non-Kolmogorov-Avrami switching kinetics in ferroelectric thin films,” Phys. Rev. B 66, 214109 (2002).

Quartz 5

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