Carbon Nanotube Random Access Memory (NRAM)

1. Theory

Definition: NRAM is a non-volatile memory that stores a bit in the contact or separation of carbon nanotubes and reads it as a resistance.
Original form: A suspended nanotube over a crossing nanotube or electrode, either straight (gap) or deflected into contact.
Commercial form: A fabric of randomly oriented nanotubes between two electrodes, whose network connectivity sets the state.
Placement: It sits in the mechanical folder because the state is stored in the contact geometry of a moving part held by adhesion; the resistance is only the read-out.
Bistability: The contact state is stable when van der Waals adhesion energy exceeds the elastic energy of the bent tube.
Nanotube resistance: Each tube has a quantum resistance of about $h /4 q^{2} \approx 6.45 kΩ$ plus a length-dependent scattering term.

2. Working Principle

Write (set): An electrostatic force pulls the tube (or tubes in the fabric) into contact, where adhesion holds them.
Write (reset): An opposite or larger pulse separates the tubes, restoring the high-resistance state.
Read: The resistance is low in contact and exponentially higher across a tunnelling gap, changing by a factor of about 2.8 per 0.1 nm of gap.
Fabric conduction: The network conducts above a percolation threshold, with tube-to-tube junctions dominating the resistance.
Electrostatic force: Falls quickly with tube height, so gaps of a few to tens of nanometres are needed for practical voltages.
Mechanism uncertainty: Whether fabric switching is purely mechanical is not fully settled.

3. Applications

Current status: In development and early licensing; not yet a volume memory product.
Embedded and standalone NVM (target): Fast, high-endurance, radiation- and temperature-tolerant non-volatile memory compatible with CMOS back-end processing.

Name	Formula	Related Formulae	Variables/Constants
Resistance of a single carbon nanotube	$R_{C N T} = \frac{h}{4 q ^{2}} (1 + \frac{L}{λ})$	$R_{0} = \frac{h}{4 q ^{2}} \approx 6.45 k Ω$	$R_{C N T} : tw o - t er mina l r es i s t an ce o f a m e t a l l i c s in g l e - w a l l nan o t u b e o f l e n g t h L w i t h i d e a l co n t a c t s$ $h, q : P l an c k co n s t an t an d e l e m e n t a r y c ha r g e$ $R_{0} : q u an t u m r es i s t an ce o f t h e f o u r co n d u c t in g c hann e l s (tw o ban d s an d tw o s p in s) o f a m e t a l l i c nan o t u b e in t h e ba l l i s t i c l imi t$ $L : l e n g t h o f t h e nan o t u b e b e tw ee n t h e co n t a c t s$ $λ : e l ec t r o n m e an f r ee p a t h (t h e r es i s t an ce r i ses l in e a r l y w i t h L b ey o n d i t)$
Tunnelling conductance of a nanotube-nanotube junction	$G_{j} = G_{c} exp (- β d)$	$β = 2 κ = \frac{2 2 m ^{*} φ}{ℏ}$	$G_{j} : co n d u c t an ce o f a j u n c t i o n b e tw ee n tw o nan o t u b es se p a r a t e d b y a g a p d$ $G_{c} : co n d u c t an ce a t co n t a c t (z er o g a p, l imi t e d b y t h e q u an t u m r es i s t an ce an d t h e o v er l a p)$ $d : g a p b e tw ee n t h e nan o t u b es$ $β : d ec a y co n s t an t o f t h e t u nn e l l in g co n d u c t an ce w i t h d i s t an ce$ $κ : d ec a y co n s t an t o f t h e w a v e f u n c t i o n in t h e g a p$ $m^{*}, φ : t u nn e l l in g e f f ec t i v e ma ss an d ba r r i er h e i g h t o f t h e g a p$ $ℏ : r e d u ce d P l an c k co n s t an t$
Percolation in a nanotube fabric	$σ_{s} = σ_{0} (n - n_{c})^{s}$	$n_{c} \approx \frac{5.64}{L _{t}^{2}}$ $R_{s} = \frac{1}{σ _{s}}$	$σ_{s} : s h ee t co n d u c t an ce o f a n e tw or k o f nan o t u b es ab o v e t h e p er co l a t i o n t h r es h o l d$ $σ_{0} : p r e f a c t or, se t b y t h e r es i s t an ces o f t h e t u b es an d o f t h e j u n c t i o n s$ $n : a r e a l d e n s i t y o f nan o t u b es in t h e f ab r i c$ $n_{c} : cr i t i c a l d e n s i t y a t w hi c h t h e n e tw or k f i r s t co nn ec t s t h e e l ec t r o d es$ $s : cr i t i c a l e x p o n e n t o f t h e co n d u c t an ce, ab o u t 1.3 f or tw o - d im e n s i o na l p er co l a t i o n (H u e t a l . m e a s u r e d ab o u t 1.33)$ $(n_{c} L_{t}^{2} \approx 5.64 f or z er o - w i d t h s t i c k s : J . L i an d S . - L . Z han g, P h y s . R e v . E 80, 040104 (2009); t h e e a r l i er M o n t e C a r l o v a l u e o f G . E . P ik e an d C . H . S e a g er, P h y s . R e v . B 10, 1421 (1974), i s c l ose t o i t)$ $L_{t} : l e n g t h o f t h e nan o t u b es (a l o n g er t u b e co nn ec t s a t a l o w er d e n s i t y)$ $R_{s} : s h ee t r es i s t an ce o f t h e f ab r i c$
Capacitance and electrostatic force of a tube over a plane	$C^{'} = \frac{2 π ε _{0}}{c o s h ^{- 1} ( z / r )}$	$F^{'} = \frac{π ε _{0} V ^{2}}{[ c o s h ^{- 1} ( z / r ) ] ^{2} z ^{2} - r ^{2}}$	$C^{'} : c a p a c i t an ce p er u ni t l e n g t h o f a cy l in d er o f r a d i u s r w h ose a x i s i s a t a h e i g h t z ab o v e a co n d u c t in g pl an e$ $ε_{0} : v a c uu m p er mi tt i v i t y$ $z : h e i g h t o f t h e a x i s o f t h e nan o t u b e ab o v e t h e e l ec t r o d e (z i s u se d so a s n o t t o c l a s h w i t h t h e P l an c k co n s t an t h)$ $r : r a d i u s o f t h e nan o t u b e$ $F^{'} : a tt r a c t i v e e l ec t r os t a t i c f or ce p er u ni t l e n g t h t o w a r d t h e pl an e a t t h e v o l t a g e V, f r o m F^{'} = (V^{2} /2) (- d C^{'} / d z), u s in g d cosh^{- 1} (z / r) / d z = 1/ z^{2} - r^{2}$ $V : v o l t a g e b e tw ee n t h e nan o t u b e an d t h e e l ec t r o d e$
Stiffness of a suspended nanotube	$k = \frac{192 E I}{L ^{3}}$	$I = \frac{π r ^{4}}{4}$ $I = \frac{π ( r _{o}^{4} - r _{i}^{4} )}{4}$	$k : s p r in g co n s t an t f or a f or ce a t t h e ce n t r e o f a b e am c l am p e d a t b o t h e n d s$ $E : Y o u n g m o d u l u s o f t h e nan o t u b e$ $I : seco n d m o m e n t o f a r e a o f t h e cr oss - sec t i o n$ $L : l e n g t h o f t h e s u s p e n d e d p a r t$ $r : r a d i u s (so l i d b e am m o d e l)$ $r_{o}, r_{i} : o u t er an d inn er r a d i u s o f a h o l l o w t u b e (t h e w a l l t hi c k n ess o f a s in g l e - w a l l nan o t u b e i s ambi g u o u s, so t h e v a l u e o f I i s m o d e l - d e p e n d e n t)$
Bistability condition (adhesion against elasticity)	$γ_{s} ℓ_{c} > \frac{1}{2} k δ^{2}$	$δ = z - d_{c}$	$γ_{s} : v an d er W aa l s a d h es i o n e n er g y p er u ni t l e n g t h o f t h e co n t a c t b e tw ee n t h e nan o t u b e an d t h e e l ec t r o d e or an o t h er nan o t u b e$ $ℓ_{c} : l e n g t h o f t h e nan o t u b e in co n t a c t$ $k : s p r in g co n s t an t o f t h e s u s p e n d e d nan o t u b e (p r e v i o u s r o w)$ $δ : d e f l ec t i o n o f t h e nan o t u b e a t t h e ce n t r e w h e n i t t o u c h es t h e e l ec t r o d e$ $z : h e i g h t o f t h e nan o t u b e ab o v e t h e e l ec t r o d e in t h e o p e n s t a t e$ $d_{c} : e q u i l ib r i u m se p a r a t i o n in co n t a c t$ $(a l u m p e d e n er g y co m p a r i so n, in t h e s p i r i t o f t h e t o t a l - e n er g y c a l c u l a t i o n o f R u ec k es e t a l . (2000) : w h e n i t h o l d s t h e c l ose d s t a t e ha s t h e l o w er e n er g y . N o n - v o l a t i l i t y n ee d s t h e c l ose d s t a t e t o b e a l oc a l minim u m, i . e . t h e a d h es i o n f or ce m u s t e x cee d t h e r es t or in g f or ce a t co n t a c t, a s in t h e N E M S r e l e a se r o w; t h e p o in t - l o a d k i s o n l y an es t ima t e f or a d i s t r ib u t e d co n t a c t)$

Important Questions

Ques: How does a carbon nanotube random access memory (NRAM) store a bit?

The original concept is a suspended carbon nanotube over a crossing nanotube or electrode. The two states are the nanotube straight (a gap) and the nanotube deflected into contact, and they are stored by the balance of the van der Waals adhesion and the elastic energy (the bistability row). The electrostatic force pulls the tube in for a write, and in the contact state the junction has a low resistance; in the open state the resistance is that of a tunnelling gap, which is exponentially larger. In the commercial form the suspended tubes are replaced by a fabric of randomly oriented nanotubes between two electrodes, where the state is the connection of the network, set by the contact or separation of the tubes in the junctions. Reference: T. Rueckes et al., “Carbon nanotube-based nonvolatile random access memory for molecular computing,” Science 289, 94 (2000); P.L. McEuen and J.-Y. Park, “Electron transport in single-walled carbon nanotubes,” MRS Bull. 29, 272 (2004).

Ques: Why does the resistance of the fabric depend on the junctions?

A random network conducts through many nanotubes in series and parallel, and the resistance of each nanotube is of the order of $h /4 q^{2} \approx 6.45$ k $Ω$ plus the length term. The junctions between the nanotubes are tunnelling or contact resistances with a conductance $G_{j} = G_{c} exp (- β d)$ , so a change in the gap of a fraction of a nanometre changes the junction resistance by a large factor (for example $β \approx 10$ nm $^{- 1}$ for a 1 eV barrier and the free electron mass changes it by a factor of about 2.8 per 0.1 nm). The network resistance follows percolation above the critical density $n_{c}$ , and the junctions dominate the resistance for a dense fabric of semiconducting and metallic tubes. Reference: L. Hu, D.S. Hecht and G. Gruner, “Percolation in transparent and conducting carbon nanotube networks,” Nano Lett. 4, 2513 (2004); A. Javey et al., “Ballistic carbon nanotube field-effect transistors,” Nature 424, 654 (2003).

Ques: How does the NRAM differ from the NEMS cantilever memory and from ReRAM?

In both NEMS and NRAM the stored information is the position or contact of a moving part, held by adhesion, so both are placed in the mechanical folder. The NEMS beam is a macroscopic (micron-sized) cantilever with a contact resistance described by Holm theory, while the nanotube is a molecular-scale beam with a quantum resistance and tunnelling gaps. A ReRAM stores the state in the atomic arrangement of a filament (ions or vacancies move) and not in the position of a body whose motion can be described by beam mechanics, although the dividing line is thin at the nanometre scale, and the mechanism of the fabric switching is not settled in the literature. Reference: T. Rueckes et al., Science 289, 94 (2000); O. Loh and H.D. Espinosa, “Nanoelectromechanical contact switches,” Nat. Nanotechnol. 7, 283 (2012).

Ques: What is the electrostatic force on a nanotube, and why is the geometry important?

A cylinder of radius $r$ with its axis at a height $z$ above a conducting plane has the capacitance per unit length $C^{'} = 2 π ε_{0} / cosh^{- 1} (z / r)$ , and the force per unit length follows from $F^{'} = \frac{1}{2} V^{2} (- d C^{'} / d z)$ , which gives the expression in the force row. The force falls quickly as the height above the electrode increases, so a large voltage is needed unless the gap is small, which is why the proposed cells have gaps of a few nanometres to tens of nanometres. The one-dimensional model neglects the fringing field at the ends and the screening of the field by the nanotube’s finite density of states. Reference: T. Rueckes et al., Science 289, 94 (2000); S.D. Senturia, Microsystem Design, Kluwer, 2001.

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