Diode-Type Selector (PN and Schottky)

1. Theory

Definition: A diode-type selector is a rectifying two-terminal device (a PN junction or Schottky diode, in single-crystal or polysilicon) placed in series with a memory element to block sneak currents in a crosspoint array.
Rectification: Forward current is large and reverse current small, so unselected cells on the selected lines can be reverse biased.
Unipolar elements only: A single diode suits memory elements switched with one polarity (phase-change, unipolar ReRAM); bipolar elements need an antiparallel diode pair or a symmetric selector.
Thermal-limit selectivity: Current rises at most one decade per $2.3 n k_{B} T / q$ (about 60 mV), so selectivity $S$ needs $V \approx (2 n k_{B} T / q) ln S$ (about 0.71 V for $S = 1 0^{6}$ ).
Polysilicon diodes: Stacked diodes must be deposited, and grain-boundary barriers lower mobility and raise leakage compared with single-crystal diodes.
Schottky option: Lower turn-on voltage and no stored charge, but higher reverse leakage and barrier variation.

2. Working Principle

Selected cell: The diode is forward biased and passes the programming or read current to the memory element.
Half-selected cells: Their diodes are reverse biased or at low forward bias, passing only small saturation or generation current.
Current-density requirement: In a $4 F^{2}$ cell the diode must deliver about $1 0^{6}$ to $1 0^{7} A/c m^{2}$ ; series resistance and high-level injection then limit the on-current.
Leakage: Reverse saturation, generation current and polysilicon trap-assisted leakage set the off-state floor.
Bidirectional pair: Antiparallel diodes give a symmetric $2 I_{s} sinh (q V / n k_{B} T)$ characteristic for bipolar memory elements.

3. Applications

Current status: Largely displaced by ovonic threshold switches in modern crosspoint memories.
Historical role: Diode-switched phase-change memory arrays and stacked polysilicon-diode one-time-programmable 3D memory.

Name	Formula	Related Formulae	Variables/Constants
Diode with series resistance	$V = \frac{n k _{B} T}{q} ln (\frac{I}{I _{s}} + 1) + I R_{s}$	$I = I_{s} [exp (\frac{q V _{D}}{n k _{B} T}) - 1]$ $V_{D} = V - I R_{s}$	$V : v o l t a g e a ppl i e d a cr oss t h e d i o d e an d i t s ser i es r es i s t an ce$ $I : d i o d e c u r r e n t$ $I_{s} : s a t u r a t i o n c u r r e n t$ $n : i d e a l i t y f a c t or$ $k_{B}, T : B o l t z mann co n s t an t an d t e m p er a t u r e$ $q : e l e m e n t a r y c ha r g e$ $R_{s} : ser i es r es i s t an ce o f t h e d i o d e (t h e n e u t r a l r e g i o n s an d t h e co n t a c t s)$ $V_{D} : v o l t a g e a cr oss t h e j u n c t i o n i t se l f$ $(s in g l e i d e a l i t y f a c t or; n r i ses t o w a r d 2 a t hi g h inj ec t i o n, see t h e o n - c u r r e n t r o w)$
Saturation and generation currents	$I_{s} = q A (\frac{D _{p} p _{n 0}}{L _{p}} + \frac{D _{n} n _{p 0}}{L _{n}})$ $I_{g e n} = \frac{q A n _{i} W}{2 τ _{g}}$	$I_{s} \propto T^{3} exp (- \frac{E _{g}}{k _{B} T})$ $I_{g e n} \propto T^{3/2} exp (- \frac{E _{g}}{2 k _{B} T})$ $W = \frac{2 ε _{s} ( V _{bi} + V _{R} )}{q} (\frac{1}{N _{A}} + \frac{1}{N _{D}})$	$I_{s} : s a t u r a t i o n c u r r e n t o f t h e i d e a l d i o d e, w hi c h se t s t h e f or w a r d c u r r e n t an d t h e r e v er se d i f f u s i o n l e ak a g e$ $I_{g e n} : r e v er se g e n er a t i o n c u r r e n t in t h e d e pl e t i o n r e g i o n$ $A : j u n c t i o n a r e a$ $D_{p}, D_{n} : d i f f u s i o n coe f f i c i e n t s o f h o l es an d e l ec t r o n s$ $p_{n 0}, n_{p 0} : min or i t y c a r r i er co n ce n t r a t i o n s a t e q u i l ib r i u m in t h e n - s i d e an d t h e p - s i d e$ $L_{p}, L_{n} : min or i t y c a r r i er d i f f u s i o n l e n g t h s$ $n_{i} : in t r in s i c c a r r i er co n ce n t r a t i o n$ $W : w i d t h o f t h e d e pl e t i o n r e g i o n a t t h e r e v er se v o l t a g e V_{R}$ $τ_{g} : g e n er a t i o n l i f e t im e$ $E_{g} : ban d g a p$ $ε_{s}, V_{bi}, N_{A}, N_{D} : p er mi tt i v i t y, b u i l t - in v o l t a g e an d a cce pt or an d d o n or d e n s i t i es (t h e s t an d a r d p - n j u n c t i o n r e l a t i o n s, a s o n t h e P N J u n c t i o n p a g e)$ $(l o n g - d i o d e f or m; in a t hin p o l y s i l i co n or s h or t - ba se d i o d e L_{p} an d L_{n} a r e r e pl a ce d b y t h e n e u t r a l ba se w i d t h s . T h e T^{3} an d T^{3/2} e x p o n e n t s i g n or e t h e t e m p er a t u r e d e p e n d e n ce o f D, L an d τ)$ $k_{B} : B o l t z mann co n s t an t$
Schottky diode	$I = A A^{*} T^{2} exp (- \frac{q φ _{B}}{k _{B} T}) [exp (\frac{q V _{D}}{n k _{B} T}) - 1]$	$I_{s, S B D} = A A^{*} T^{2} exp (- \frac{q φ _{B}}{k _{B} T})$	$I : c u r r e n t o f a m e t a l - se mi co n d u c t or d i o d e w i t h t h er mi o ni c e mi ss i o n o v er t h e ba r r i er$ $A : co n t a c t a r e a$ $A^{*} : e f f ec t i v e R i c ha r d so n co n s t an t$ $φ_{B} : S c h o tt k y ba r r i er h e i g h t$ $V_{D}, n : v o l t a g e a cr oss t h e ba r r i er an d i d e a l i t y f a c t or$ $I_{s, S B D} : s a t u r a t i o n c u r r e n t o f t h e S c h o tt k y d i o d e, w hi c h i s or d er s o f ma g ni t u d e l a r g er t han t ha t o f a p - n d i o d e o f t h e s am e a r e a f or a ba r r i er b e l o w t h e ban d g a p (n o min or i t y c a r r i er s t or a g e; f or e x am pl e ab o u t 1 0^{- 5} A / c m^{2} f or a 0.7 V ba r r i er o n S i a g ain s t ab o u t 1 0^{- 11} A / c m^{2} f or a s i l i co n p - n j u n c t i o n)$ $k_{B} : B o l t z mann co n s t an t$
Rectification ratio and selectivity	$R_{r ec t} = \frac{I _{f} ( V )}{I _{r} ( V )}$	$S = \frac{I ( V )}{I ( V /2 )} \approx exp (\frac{q V}{2 n k _{B} T})$ $V \approx \frac{2 n k _{B} T}{q} ln S$	$R_{r ec t} : r a t i o o f t h e f or w a r d c u r r e n t t o t h e r e v er se c u r r e n t a t t h e s am e v o l t a g e ma g ni t u d e (f or a u ni p o l a r m e m or y e l e m e n t t h e u n se l ec t e d ce l l s a r e r e v er se bia se d)$ $I_{f} (V), I_{r} (V) : f or w a r d c u r r e n t a t + V an d ma g ni t u d e o f t h e r e v er se c u r r e n t a t - V$ $S : se l ec t i v i t y in t h e V /2 bia s sc h e m e (v a l i d w h e n t h e f or w a r d c u r r e n t i s e x p o n e n t ia l in v o l t a g e a t b o t h V an d V /2)$ $V : r e a d v o l t a g e a cr oss t h e j u n c t i o n$ $n, k_{B}, T, q : i d e a l i t y f a c t or, B o l t z mann co n s t an t, t e m p er a t u r e an d e l e m e n t a r y c ha r g e$
On-current limit from series resistance and high injection	$I_{o n} \approx \frac{V _{a pp} - V _{o n}}{R _{s}}$	$R_{s} = \frac{ρ _{b} t _{b}}{A} + \frac{ρ _{c}}{A}$ $n \to 2 (p_{n} ≳ N_{D})$	$I_{o n} : f or w a r d c u r r e n t a t a hi g h v o l t a g e, w h er e t h e ser i es r es i s t an ce l imi t s t h e c u r r e n t$ $V_{a pp} : a ppl i e d v o l t a g e a cr oss t h e d i o d e$ $V_{o n} : t u r n - o n v o l t a g e o f t h e j u n c t i o n (c l ose t o t h e b u i l t - in v o l t a g e a t a hi g h c u r r e n t)$ $R_{s} : ser i es r es i s t an ce$ $ρ_{b}, t_{b} : r es i s t i v i t y an d t hi c k n ess o f t h e l i g h tl y d o p e d ba se r e g i o n$ $ρ_{c} : s p ec i f i c co n t a c t r es i s t i v i t y o f t h e co n t a c t s (t h e co n t a c t t er m i s d i v i d e d b y A t o g i v e a r es i s t an ce)$ $A : j u n c t i o n a r e a$ $p_{n}, N_{D} : inj ec t e d h o l e co n ce n t r a t i o n an d d o n or d e n s i t y o f t h e ba se (w h e n t h e inj ec t e d d e n s i t y r e a c h es t h e d o p in g, t h e i d e a l i t y f a c t or r i ses t o w a r d tw o)$ $(s p r e a d in g r es i s t an ce a t t h e co n t a c t s i s n e g l ec t e d)$
Polysilicon grain boundaries	$μ_{e f f} = \frac{q L _{g}}{2 π m ^{*} k _{B} T} exp (- \frac{E _{b}}{k _{B} T})$	$E_{b} = \frac{q ^{2} N _{t}^{2}}{8 ε _{s} N _{D}} (N_{t} < L_{g} N_{D})$ $E_{b} = \frac{q ^{2} L _{g}^{2} N _{D}}{8 ε _{s}} (N_{t} > L_{g} N_{D})$ $ρ_{b} = \frac{1}{q N _{D} μ _{e f f}}$	$μ_{e f f} : e f f ec t i v e m o bi l i t y in p o l y s i l i co n l imi t e d b y t h er mi o ni c e mi ss i o n o v er t h e g r ain b o u n d a r y ba r r i er s$ $L_{g} : g r ain s i z e$ $m^{*} : co n d u c t i v i t y e f f ec t i v e ma ss o f t h e c a r r i er s$ $E_{b} : g r ain b o u n d a r y ba r r i er h e i g h t (in e n er g y; S e t o^{'} s p o t e n t ia l ba r r i er t im es q), p a r tl y d e pl e t e d g r ain s in t h e f i r s t f or m an d f u l l y d e pl e t e d g r ain s in t h e seco n d, w i t h t h e ma x im u m a t N_{D} = N_{t} / L_{g}$ $N_{t} : a r e a l d e n s i t y o f c a r r i er t r a p s a t t h e g r ain b o u n d a r y$ $N_{D} : d o p in g d e n s i t y$ $ε_{s} : p er mi tt i v i t y o f s i l i co n$ $ρ_{b} : r es i s t i v i t y o f t h e p o l y s i l i co n ba se (t hi s r ai ses R_{s} in t h e p r e v i o u s r o w f or a p o l y s i l i co n d i o d e)$ $k_{B} : B o l t z mann co n s t an t$
Required current density in a crosspoint cell	$J_{r e q} = \frac{I _{p r o g}}{A _{ce l l}}$	$A_{ce l l} = 4 F^{2}$	$J_{r e q} : c u r r e n t d e n s i t y t ha t t h e se l ec t or m u s t c a r r y t o s u ppl y t h e p r o g r ammin g c u r r e n t o f t h e m e m or y e l e m e n t$ $I_{p r o g} : p r o g r ammin g c u r r e n t o f t h e m e m or y e l e m e n t$ $A_{ce l l} : f oo tp r in t o f t h e ce l l, e q u a l t o 4 F^{2} in a s in g l e - l a y er cr oss p o in t a r r a y$ $F : minim u m f e a t u r e s i z e (ha l f o f t h e l in e p i t c h)$ $(t h e se l ec t or a r e a c ann o t b e l a r g er t han t h e ce l l f oo tp r in t, so J_{r e q} r i ses a s t h e ce l l i s sc a l e d a t a co n s t an t p r o g r ammin g c u r r e n t; I_{p r o g} o f ab o u t 100 μ A i s a t y p i c a l p ha se - c han g e r ese t v a l u e)$
Antiparallel diode pair (bidirectional selector)	$I = I_{s} [exp (\frac{q V}{n k _{B} T}) - exp (- \frac{q V}{n k _{B} T})] = 2 I_{s} sinh (\frac{q V}{n k _{B} T})$	$V_{0} = \frac{n k _{B} T}{q}$ $S = 2 cosh (\frac{q V}{2 n k _{B} T})$	$I : c u r r e n t o f tw o i d e n t i c a l d i o d es co nn ec t e d in an t i p a r a l l e l, o dd in v o l t a g e$ $I_{s} : s a t u r a t i o n c u r r e n t o f o n e d i o d e$ $V : v o l t a g e a cr oss t h e p ai r (ser i es r es i s t an ce an d t h e r e v er se l e ak a g e o f t h e b l oc k in g d i o d e n e g l ec t e d)$ $n, k_{B}, T, q : i d e a l i t y f a c t or, B o l t z mann co n s t an t, t e m p er a t u r e an d e l e m e n t a r y c ha r g e$ $V_{0} : c ha r a c t er i s t i c v o l t a g e in t h e s inh f or m o f t h e M i x e d I o ni c E l ec t r o ni c C o n d u c t i o n S e l ec t or (M I E C) p a g e, e q u a l t o n k_{B} T / q f or an i d e a l p ai r; ser i es r es i s t an ce mak es t h e e f f ec t i v e v a l u e l a r g er . T h e p ai r n ee d s tw o se p a r a t e j u n c t i o n s in t h e ce l l)$ $S : se l ec t i v i t y in t h e V /2 sc h e m e$

Important Questions

Ques: Why can a diode be used as a selector, and what kind of memory element does it need?

A diode passes a large current in the forward direction and a small one in the reverse direction, so in a crosspoint array the unselected cells on the selected lines can be biased in reverse. A single diode in series with the memory element therefore works when the memory element is switched with one polarity (unipolar), as in phase-change memory and in unipolar resistive cells. A bipolar element needs a bidirectional selector, which can be an antiparallel pair of diodes (the last row), an OTS or a MIEC selector. Of these, the diode is the most mature device, with well-established equations, and it was used in diode-switch phase-change memory. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007; J.H. Oh et al., “Full integration of highly manufacturable 512Mb PRAM based on 90nm technology,” IEDM Tech. Dig. (2006).

Ques: How large a selectivity does an exponential diode give?

A diode current rises by one decade for each $2.3 n k_{B} T / q$ of voltage, about 60 mV at room temperature for $n = 1$ . The selectivity of the selectivity row is therefore $S \approx exp (q V /2 n k_{B} T)$ , and a chosen $S$ needs $V \approx (2 n k_{B} T / q) ln S$ . For example, $S = 1 0^{6}$ at room temperature with $n = 1$ (taking $k_{B} T / q = 25.85$ mV) needs $V \approx 0.71$ V. The exponential cannot be made steeper than the thermal limit, so a threshold device such as an OTS, whose selectivity is set by a threshold between $V /2$ and $V$ , can reach a higher selectivity at a lower voltage. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007; G.W. Burr et al., “Access devices for 3D crosspoint memory,” J. Vac. Sci. Technol. B 32, 040802 (2014).

Ques: Why is the current density the main requirement for a diode selector?

The diode must supply the programming current of the memory element within the footprint of the cell. In a crosspoint array $A_{ce l l} = 4 F^{2}$ , so for $F = 20$ nm the footprint is $1.6 \times 1 0^{- 11}$ cm $^{2}$ , and a programming current of 100 $μ$ A needs $J \approx 6 \times 1 0^{6}$ A/cm $^{2}$ . At this current density the series resistance limits the current (the on-current row) and the junction is in high-level injection, so the diode has to be made of a heavily doped, short, low-resistivity structure. A reset current of a phase-change cell reduces with scaling, but the footprint reduces faster, so the requirement rises. Reference: G.W. Burr et al., J. Vac. Sci. Technol. B 32, 040802 (2014); K. Gopalakrishnan et al., VLSI Tech. Dig. (2010), p. 205.

Ques: Why is polysilicon worse than single-crystal silicon for the diode?

A diode stacked above the first layer of the array cannot be grown on the substrate, so it is made of deposited polysilicon (or recrystallized silicon). Polysilicon has grain boundaries with trap states that capture carriers and form potential barriers, which lowers the effective mobility as in the grain boundary row, $μ_{e f f} \propto L_{g} exp (- E_{b} / k_{B} T)$ , so the base resistivity and the series resistance are higher, and the trap-assisted generation raises the reverse leakage. Larger grains reduce the number of barriers in the path, and a higher doping reduces the barrier once the grains are only partly depleted ( $L_{g} N_{D} > N_{t}$ ), and a single-crystal diode in the substrate layer avoids it, but cannot be stacked. Reference: T.I. Kamins, Polycrystalline Silicon for Integrated Circuit Applications, Kluwer, 1988; J.Y.W. Seto, “The electrical properties of polycrystalline silicon films,” J. Appl. Phys. 46, 5247 (1975).

Ques: How does a Schottky diode compare with a p-n diode as a selector?

A Schottky diode is a majority-carrier device without minority carrier storage, so it switches fast and has a lower turn-on voltage at the same current, which is useful for a high current at a low voltage. Its saturation current for a barrier below the bandgap is larger than that of a p-n junction, so its reverse leakage is higher for the same area, and the leakage of the unselected cells must be checked against the selectivity requirement. It can be made at low temperature from a metal and a deposited semiconductor, which suits a back-end process, but its barrier height depends on the interface quality and varies from device to device. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

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