Mixed Ionic Electronic Conduction Selector (MIEC)

1. Theory

Definition: A mixed ionic-electronic conduction (MIEC) selector is a two-terminal access device made of a thin Cu-containing layer in which both mobile ions and electrons carry current.
Smooth nonlinearity: Its current-voltage curve is symmetric and strongly nonlinear, with no threshold, snap-back or negative resistance.
Partial conductivities: Total conductivity is $σ_{e} + σ_{i}$ , and the ionic transference number $t_{i} = σ_{i} / σ$ measures the ionic share.
Coupled transport: Ions and compensating electrons move together, so redistribution follows the chemical diffusion coefficient $\tilde{D} \approx t_{e} θ_{i} D_{i} + t_{i} θ_{e} D_{e}$ .
Empirical model: The page uses a generic mixed-conductor description and a $sinh$ fit, since the exact physical origin of the nonlinearity is not settled.

2. Working Principle

Low bias: Very little current flows, which isolates half-selected cells.
High bias: Ion redistribution and field-enhanced hopping make the current rise roughly as $sinh (V / V_{0})$ .
Selectivity: $S = I (V) / I (V /2) = 2 cosh (V /2 V_{0})$ ; a smaller $V_{0}$ gives a higher selectivity at a given voltage, or the same selectivity at a lower voltage.
Ion transport: Drift (field-enhanced hopping) and diffusion (Nernst-Planck flux) redistribute Cu ions on a time scale $τ_{i}$ .
No memory allowed: The ion profile must relax fully and quickly when bias is removed, so the selector does not retain a state.
High current density: Supports the large currents needed for phase-change programming in a back-end-compatible layer.

3. Applications

Current status: Research and development; demonstrated in large test arrays but not in commercial products.
Crosspoint memory access (target): Symmetric, back-end-compatible selector for bipolar ReRAM and high-current phase-change crosspoint arrays.

Name	Formula	Related Formulae	Variables/Constants
Partial conductivities of a mixed conductor	$σ = σ_{e} + σ_{i}$ $t_{i} = \frac{σ _{i}}{σ _{e} + σ _{i}}$	$σ_{i} = \frac{n _{i} z ^{2} q ^{2} D _{i}}{k _{B} T}$ $t_{e} = 1 - t_{i}$	$σ, σ_{e}, σ_{i} : t o t a l, e l ec t r o ni c an d i o ni c co n d u c t i v i t y o f t h e mi x e d i o ni c - e l ec t r o ni c co n d u c t or$ $t_{i}, t_{e} : i o ni c an d e l ec t r o ni c t r an s f er e n ce n u mb er s (t h e f r a c t i o n o f t h e c u r r e n t c a r r i e d b y i o n s an d b y e l ec t r o n s)$ $n_{i} : co n ce n t r a t i o n o f t h e m o bi l e i o n s (f or e x am pl e C u +)$ $z : c ha r g e n u mb er o f t h e m o bi l e i o n$ $q : e l e m e n t a r y c ha r g e$ $D_{i} : d i f f u s i o n coe f f i c i e n t o f t h e m o bi l e i o n$ $k_{B}, T : B o l t z mann co n s t an t an d t e m p er a t u r e (t h e e x p r ess i o n f or σ_{i} i s t h e N er n s t - E in s t e in r e l a t i o n, w i t h D_{i} t h e c ha r g e (co n d u c t i v i t y) d i f f u s i o n coe f f i c i e n t, w hi c h e q u a l s t h e t r a cer v a l u e d i v i d e d b y t h e H a v e n r a t i o)$
Ion drift velocity (hopping)	$v_{d} = 2 a ν exp (- \frac{E _{a}}{k _{B} T}) sinh (\frac{z q a E}{2 k _{B} T})$	$μ_{i} = \frac{z q D _{i}}{k _{B} T} (l o w f i e l d)$	$v_{d} : d r i f t v e l oc i t y o f t h e m o bi l e i o n s in t h e f i e l d E$ $a : h o pp in g d i s t an ce b e tw ee n tw o i o n s i t es$ $ν : a tt e m pt f r e q u e n cy o f t h e h o p$ $E_{a} : a c t i v a t i o n e n er g y o f t h e i o n h o p a t z er o f i e l d$ $z, q : c ha r g e n u mb er o f t h e i o n an d e l e m e n t a r y c ha r g e$ $E : e l ec t r i c f i e l d in t h e mi x e d co n d u c t or$ $k_{B}, T : B o l t z mann co n s t an t an d t e m p er a t u r e$ $μ_{i} : i o ni c m o bi l i t y in t h e l o w - f i e l d l imi t, f r o m t h e N er n s t - E in s t e in r e l a t i o n$ $D_{i} : i o n d i f f u s i o n coe f f i c i e n t, D_{i} = a^{2} ν e^{- E_{a} / k_{B} T} in o n e d im e n s i o n, w hi c h mak es t h e l o w - f i e l d l imi t o f v_{d} e q u a l t o μ_{i} E$
Ion flux (Nernst-Planck)	$J_{i} = - D_{i} \frac{\partial c}{\partial x} + \frac{z q D _{i}}{k _{B} T} c E$	$\frac{\partial c}{\partial t} = - \frac{\partial J _{i}}{\partial x}$	$J_{i} : f l ux o f t h e m o bi l e i o n s (n u mb er p er u ni t a r e a an d t im e)$ $D_{i} : i o n d i f f u s i o n coe f f i c i e n t$ $c (x, t) : co n ce n t r a t i o n o f t h e m o bi l e i o n s a t p os i t i o n x an d t im e t$ $z, q : c ha r g e n u mb er o f t h e i o n an d e l e m e n t a r y c ha r g e$ $E : e l ec t r i c f i e l d in t h e l a y er$ $k_{B}, T : B o l t z mann co n s t an t an d t e m p er a t u r e$ $(t h e f i r s t t er m i s d i f f u s i o n, t h e seco n d i s d r i f t; t h e seco n d e q u a t i o n i s t h e co n t in u i t y e q u a t i o n; i d e a l d i l u t e i o n s, z > 0 d r i f t in g a l o n g E . T h e f i e l d i t se l f f o l l o w s f r o m P o i sso n^{'} s e q u a t i o n w i t h t h e i o n an d e l ec t r o n c ha r g e, w hi c h m u s t b e so l v e d t o g e t h er w i t h t h ese tw o)$
Chemical diffusion and ion-redistribution time	$\tilde{D} \approx t_{e} θ_{i} D_{i} + t_{i} θ_{e} D_{e}$	$τ_{i} \approx \frac{t ^{2}}{D ~}$ $θ_{i} = \frac{\partial l n a _{i}}{\partial l n c}$	$\tilde{D} : c h e mi c a l d i f f u s i o n coe f f i c i e n t, w hi c h d escr ib es t h e r e l a x a t i o n o f t h e co n ce n t r a t i o n o f t h e m o bi l e s p ec i es in a mi x e d co n d u c t or$ $t_{e}, t_{i} : e l ec t r o ni c an d i o ni c t r an s f er e n ce n u mb er s$ $θ_{i}, θ_{e} : t h er m o d y nami c f a c t or s o f t h e i o n s an d o f t h e e l ec t r o ni c c a r r i er s (1 f or i d e a l d i l u t e s p ec i es)$ $D_{i}, D_{e} : i o n an d e l ec t r o ni c d i f f u s i o n coe f f i c i e n t s$ $(f or i d e a l d i l u t e s p ec i es t hi s i s t h e ambi p o l a r coe f f i c i e n t D_{i} D_{e} (c_{i} + c_{e}) / (c_{i} D_{i} + c_{e} D_{e}))$ $τ_{i} : t im e f or t h e i o n s t o r e d i s t r ib u t e a cr oss a l a y er o f t hi c k n ess t$ $a_{i} : a c t i v i t y o f t h e m o bi l e i o n$ $c : co n ce n t r a t i o n o f t h e m o bi l e i o n$ $(f or a p u l se m u c h s h or t er t han τ_{i} t h e i o n p r o f i l e s t a y s a t i t s ini t ia l s t a t e, an d f or a l o n g er p u l se t h e p r o f i l e an d t h e c u r r e n t c han g e d u r in g t h e p u l se)$
Current-voltage characteristic	$J = J_{0} sinh (\frac{V}{V _{0}})$	$J \approx \frac{J _{0}}{2} exp (\frac{V}{V _{0}}) (V ≫ V_{0}, V > 0)$ $J \propto exp (- \frac{E _{a}^{'}}{k _{B} T})$	$J : c u r r e n t d e n s i t y t h r o ug h t h e se l ec t or, o dd in t h e v o l t a g e (bi d i r ec t i o na l)$ $J_{0} : c u r r e n t d e n s i t y p r e f a c t or (a f i t p a r am e t er)$ $V : v o l t a g e a cr oss t h e se l ec t or$ $V_{0} : c ha r a c t er i s t i c v o l t a g e o f t h e e x p o n e n t ia l r i se (a f i t p a r am e t er, w hi c h se t s t h e n o n l in e a r i t y)$ $E_{a}^{'} : a pp a r e n t a c t i v a t i o n e n er g y o f t h e c u r r e n t a t a f i x e d v o l t a g e$ $k_{B}, T : B o l t z mann co n s t an t an d t e m p er a t u r e$ $(an e m p i r i c a l d escr i pt i o n : t h e c ha r a c t er i s t i c r i ses s m oo t h l y an d w i t h o u t s na p - ba c k or h y s t er es i s; t h e a c t i v a t e d f or m a t f i x e d v o l t a g e i s a l so o n l y a f i t)$
Selectivity of the sinh characteristic	$S = \frac{I ( V )}{I ( V /2 )} = 2 cosh (\frac{V}{2 V _{0}})$	$S \approx exp (\frac{V}{2 V _{0}}) (V ≫ V_{0})$ $V \approx 2 V_{0} ln (S) (S ≫ 1)$	$S : se l ec t i v i t y in t h e V /2 bia s sc h e m e$ $I (V), I (V /2) : c u r r e n t s a t t h e r e a d v o l t a g e an d a t ha l f o f i t$ $V : r e a d v o l t a g e a cr oss t h e se l ec t or$ $V_{0} : c ha r a c t er i s t i c v o l t a g e o f t h e s inh c ha r a c t er i s t i c$ $(t h e l a s t r e l a t i o n g i v es t h e v o l t a g e n ee d e d f or a c h ose n se l ec t i v i t y; t h e a r r a y - s i z e l imi t N_{ma x} i s a s o n t h e O v o ni c T h r es h o l d S w i t c h (O T S) p a g e)$

Important Questions

Ques: What is a mixed ionic-electronic conduction (MIEC) selector?

A MIEC selector is a two-terminal device with a thin layer of a material in which both mobile ions (usually Cu) and electrons or holes carry current. It is placed in series with a memory element, and it has a strongly nonlinear, symmetric characteristic: the current is very low at small voltage and rises exponentially at a higher one. The device was proposed as an access device for phase-change memory crosspoint arrays, and it works at a high current density, which a diode made in a back-end process does not provide. The description on this page is a device-level one (the partial conductivities, the ion hopping and the sinh characteristic), because the exact physical origin of the nonlinearity is not settled. Reference: K. Gopalakrishnan et al., “Highly-scalable novel access device based on mixed ionic electronic conduction (MIEC) materials for high density phase change memory (PCM) arrays,” VLSI Tech. Dig. (2010), p. 205; G.W. Burr et al., “Access devices for 3D crosspoint memory,” J. Vac. Sci. Technol. B 32, 040802 (2014).

Ques: How does a MIEC selector differ from an OTS?

An OTS has a threshold, a snap-back to a holding voltage and a negative differential resistance, with a delay time. A MIEC selector has a smooth, monotonic characteristic with no snap-back and no negative resistance, so the snap-back spike of the OTS (the spike charge row of that page) does not occur, and its selectivity is set by the slope parameter $V_{0}$ and not by a threshold. In exchange, its off-state current at the half-select voltage is set by the exponential tail rather than cut off below a threshold. A smaller $V_{0}$ gives a higher selectivity at a given voltage, and reaches a chosen $S$ at a lower voltage ( $V \approx 2 V_{0} ln S$ ), but the half-select current never drops as sharply as below an OTS threshold. The OTS is a chalcogenide, and the MIEC is a Cu-containing mixed conductor. Reference: G.W. Burr et al., J. Vac. Sci. Technol. B 32, 040802 (2014); G.W. Burr et al., “Large-scale (512kbit) integration of multilayer-ready access-devices based on mixed-ionic-electronic-conduction (MIEC) at 100% yield,” VLSI Tech. Dig. (2012), p. 41.

Ques: Why is the ion transport important in a selector that has no memory?

The mobile ions set the nonlinearity and the speed: under a bias they redistribute in the layer by drift and diffusion (the Nernst-Planck row) with the time constant $τ_{i}$ , and the current depends on how far this has gone. For a pulse much shorter than $τ_{i}$ the profile is at its initial state and the characteristic is that of the start condition, and for a pulse longer than $τ_{i}$ the current changes during the pulse. The selector should therefore not retain a state (no memory), and the requirement is that the redistribution reverses completely and quickly when the bias is removed, which is a condition on $τ_{i}$ and on the symmetry of the device that belongs to the reliability pages (stability and endurance of Cu-based selectors). Reference: J. Maier, Physical Chemistry of Ionic Materials: Ions and Electrons in Solids, Wiley, 2004; N.F. Mott and R.W. Gurney, Electronic Processes in Ionic Crystals, Oxford University Press, 1940.

Ques: What is the chemical diffusion coefficient and why does it differ from the ion diffusion coefficient?

In a mixed conductor the mobile ions cannot move alone, because the electric field of an ion cloud is compensated by electrons that move with them. The relaxation of the concentration is then controlled by the coupled (ambipolar) transport and described by the chemical diffusion coefficient, $\tilde{D} \approx t_{e} θ_{i} D_{i} + t_{i} θ_{e} D_{e}$ in the chemical diffusion row. When electrons dominate the conduction ( $t_{e} \to 1$ ), $\tilde{D} \approx θ_{i} D_{i}$ and the ions respond at their own rate. When ions dominate ( $t_{i} \to 1$ ), $\tilde{D} \approx θ_{e} D_{e}$ : the redistribution is limited by the minority electronic carriers that must move with the ions. A thermodynamic factor above one, as in a non-ideal solution, speeds up the relaxation. Reference: J. Maier, Physical Chemistry of Ionic Materials: Ions and Electrons in Solids, Wiley, 2004; C. Wagner, Z. Phys. Chem. B 21, 25 (1933).

Ques: How does a MIEC selector compare with a diode and a transistor?

A MIEC selector is symmetric in voltage and so works with bipolar memory elements, it can be built in the back end at a low temperature, and it supports a high on-current density, which a polysilicon diode cannot (the current is limited by the series resistance). Its disadvantages relative to an OTS are the lower selectivity for a given voltage (the exponential tail) and the need for Cu in the layer, which brings contamination and stability issues. Against a transistor, it takes much less area, but it gives no control of the compliance current of a ReRAM cell. Reference: G.W. Burr et al., J. Vac. Sci. Technol. B 32, 040802 (2014); S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

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