CODING ASSISTED ADAPTIVE THRESHOLDING FOR SNEAK-PATH MITIGATION IN RESISTIVE MEMORIES Zehui Chen UCLA Clayton Schoeny UCLA Lara Dolecek UCLA
Resistive memory and the sneak-path problem • Crossbar structure for emerging non- volatile memory (ReRAM, PCRAM). • Simple • High Density [Zidan et al. 13] 2
Resistive memory and the sneak-path problem • Crossbar structure for emerging non- High Resistance State (HRS) – binary 0 Low Resistance State (LRS) – binary 1 volatile memory (ReRAM, PCRAM). • Simple • High Density • Sneak path(s) due to lack of isolation. • Causing read errors • Severe for HRS cells (binary 0) 3
Resistive memory and the sneak-path problem • Crossbar structure for emerging non- High Resistance State (HRS) – binary 0 Low Resistance State (LRS) – binary 1 volatile memory (ReRAM, PCRAM). • Simple • High Density • Sneak path(s) due to lack of isolation. • Causing read errors • Severe for HRS cells (binary 0) Desired Path (0 is read) 4
Resistive memory and the sneak-path problem • Crossbar structure for emerging non- High Resistance State (HRS) – binary 0 Low Resistance State (LRS) – binary 1 volatile memory (ReRAM, PCRAM). • Simple • High Density • Sneak path(s) due to lack of isolation. • Causing read errors • Severe for HRS cells (binary 0) Sneak Path (1 is read) 5
Sneak-path modeling • 1D1R (1 Diode 1 Resistor) structure with High Resistance State (HRS) – binary 0 Low Resistance State (LRS) – binary 1 unreliable diode. Sneak Path 6
Sneak-path modeling • 1D1R (1 Diode 1 Resistor) structure with High Resistance State (HRS) – binary 0 Low Resistance State (LRS) – binary 1 unreliable diode. • Reliable diodes eliminate sneak-path problem. Sneak Path 7
Sneak-path modeling • 1D1R (1 Diode 1 Resistor) structure with High Resistance State (HRS) – binary 0 Low Resistance State (LRS) – binary 1 unreliable diode. • Reliable diodes eliminate sneak-path problem. 8
Sneak-path modeling • 1D1R (1 Diode 1 Resistor) structure with High Resistance State (HRS) – binary 0 Low Resistance State (LRS) – binary 1 unreliable diode. • Reliable diodes eliminate sneak-path problem. • With unreliable diode, sneak-path problem reappear. • We assume diode fails to open position with probability 𝑞 𝑔 . Diode fails to open position 9
Sneak-path modeling • 1D1R (1 Diode 1 Resistor) structure with High Resistance State (HRS) – binary 0 Low Resistance State (LRS) – binary 1 unreliable diode. • Reliable diodes eliminate sneak-path problem. • With unreliable diode, sneak-path problem reappear. • We assume diode fails to open position with probability 𝑞 𝑔 . Diode fails to open position Sneak Path 10
Sneak-path modeling • Model sneak-path event as Boolean RV. High Resistance State (HRS) – binary 0 Low Resistance State (LRS) – binary 1 By our definition, a sneak-path event occurs at cell (𝑗, 𝑘) if the following three conditions are met: 1) The bit value stored is 0 . 2) There exists at least one combination of 𝑑, 𝑠 ∈ 1, … , 𝑜 , 𝑑 ≠ 𝑘, 𝑠 ≠ 𝑗 that induces a sneak-path defined by 𝑠 𝐵 𝑗𝑑 = 𝐵 𝑠𝑑 = 𝐵 𝑠𝑘 = 1. Diode 3) The diode at cell location (𝑠, 𝑑) fails to open fails to open position position. We use Boolean RV 𝑓 𝑗𝑘 to denote the occurrence of sneak-path event. 𝑑 Sneak Path 11
Parallel resistance interference • Effect of sneak-path event is modeled as parallel resistance interference [Ben- Hur and Cassuto, 17]. Original cell Sneak-path Original cell resistance resistance resistance [Zidan et al. 13] 12
Parallel resistance interference • Effect of sneak-path event is modeled as parallel resistance interference [Ben- Hur and Cassuto, 17]. • With additive Gaussian measurement noise, the resistance of cell (𝑗, 𝑘) is: −1 + 𝑓 𝑗𝑘 1 + 𝜃, 𝑥ℎ𝑓𝑜 0 𝑗𝑡 𝑡𝑢𝑝𝑠𝑓𝑒, 𝑠 𝑗𝑘 = 𝑆 0 𝑆 𝑡 𝑆 1 +𝜃, 𝑥ℎ𝑓𝑜 1 𝑗𝑡 𝑡𝑢𝑝𝑠𝑓𝑒. 𝑆 0 : HRS resistance 𝑆 1 : LRS resistance 𝑆 𝑡 : Sneak-path resistance 𝜃: Gaussian measurement noise with variance 𝜏 2 13
Parallel resistance interference • Effect of sneak-path event is modeled as parallel resistance interference [Ben- Hur and Cassuto, 17]. • With additive Gaussian measurement noise, the resistance of cell (𝑗, 𝑘) is: −1 + 𝑓 𝑗𝑘 1 + 𝜃, 𝑥ℎ𝑓𝑜 0 𝑗𝑡 𝑡𝑢𝑝𝑠𝑓𝑒, 𝑠 𝑗𝑘 = 𝑆 0 𝑆 𝑡 𝑆 1 +𝜃, 𝑥ℎ𝑓𝑜 1 𝑗𝑡 𝑡𝑢𝑝𝑠𝑓𝑒. 𝑆 1 𝑆 0 𝑆 0 : HRS resistance 𝑆 1 : LRS resistance 𝑆 𝑡 : Sneak-path resistance 𝜃: Gaussian measurement noise with variance 𝜏 2 14
Parallel resistance interference • Effect of sneak-path event is modeled as parallel resistance interference [Ben- Hur and Cassuto, 17]. • With additive Gaussian measurement noise, the resistance of cell (𝑗, 𝑘) is: Sneak-path Event −1 + 𝑓 𝑗𝑘 1 + 𝜃, 𝑥ℎ𝑓𝑜 0 𝑗𝑡 𝑡𝑢𝑝𝑠𝑓𝑒, 𝑠 𝑗𝑘 = 𝑆 0 𝑆 𝑡 𝑆 1 +𝜃, 𝑥ℎ𝑓𝑜 1 𝑗𝑡 𝑡𝑢𝑝𝑠𝑓𝑒. 𝑆 1 𝑆 0 −1 1 + 1 𝑆 0 : HRS resistance 𝑆 0 𝑆 𝑡 𝑆 1 : LRS resistance 𝑆 𝑡 : Sneak-path resistance 𝜃: Gaussian measurement noise with variance 𝜏 2 15
A detection problem • Need to estimate the bit value stored. Posterior functions: No side information / 𝑑 = : Λ 0 𝑠 𝑗𝑘 = 1 − 𝑟 ൣ𝑔 𝑠 𝑗𝑘 − 𝑆 0 𝑄 𝑓 𝑗𝑘 = 0 𝐵 𝑗𝑘 = 0, 𝑑 −1 1 1 𝑄 𝑓 𝑗𝑘 = 1 𝐵 𝑗𝑘 = 0, 𝑑 ሿ, +𝑔 𝑠 𝑗𝑘 − 𝑆 0 + 𝑆 𝑡 and Λ 1 𝑠 𝑗𝑘 = 𝑟𝑔 𝑠 𝑗𝑘 − 𝑆 1 . 𝑔(∙): Gaussian density function with variance 𝜏 2 𝑟 : prior probability of 1 being stored 𝑑 : side information Decide 1 Decide 0 • A threshold detector is used. 16
Coding assisted adaptive thresholding • Sneak-path probabilities on same row/column are highly dependent [Cassuto, Kvatinsky and Yaakobi 16]. 17
Coding assisted adaptive thresholding • Sneak-path probabilities on same row/column are highly dependent. • Proposed Construction: • Main diagonal is all 0s (pilots). • Provides side information. 𝑜−1 • Low redundancy overhead (rate: 𝑜 ). 18
Characterizing inter-cell dependency • We characterize the dependency between two cells by computing 𝑄(𝑓 𝑗𝑘 |𝐵 𝑗𝑘 = 0, 𝑓 𝑗𝑗 ) . Targeted cell Reference cell 19
Characterizing inter-cell dependency • We characterize the dependency between two cells by computing 𝑄(𝑓 𝑗𝑘 |𝐵 𝑗𝑘 = 0, 𝑓 𝑗𝑗 ) . • We characterize the dependency between three cells by computing 𝑄(𝑓 𝑗𝑘 |𝐵 𝑗𝑘 = 0, 𝑓 𝑗𝑗 , 𝑓 𝑘𝑘 ) . Targeted cell Reference cell 20
Characterizing inter-cell dependency • We characterize the dependency between two cells by computing Posterior functions : 𝑄(𝑓 𝑗𝑘 |𝐵 𝑗𝑘 = 0, 𝑓 𝑗𝑗 ) . Λ 0 𝑠 𝑗𝑘 = 1 − 𝑟 ൣ𝑔 𝑠 𝑗𝑘 − 𝑆 0 𝑄 𝑓 𝑗𝑘 = 0 𝐵 𝑗𝑘 = 0, 𝑑 −1 𝑆 0 + 1 1 𝑄 𝑓 𝑗𝑘 = 1 𝐵 𝑗𝑘 = 0, 𝑑 ሿ, +𝑔 𝑠 𝑗𝑘 − • We characterize the dependency 𝑆 𝑡 and between three cells by computing Λ 1 𝑠 𝑗𝑘 = 𝑟𝑔 𝑠 𝑗𝑘 − 𝑆 1 . 𝑄(𝑓 𝑗𝑘 |𝐵 𝑗𝑘 = 0, 𝑓 𝑗𝑗 , 𝑓 𝑘𝑘 ) . • Different realization of 𝑑 induces 𝑑 = No side information: 𝑑 = {𝑓 𝑗𝑗 /𝑓 𝑘𝑘 } different distributions, which result Single reference cell: 𝑑 = 𝑓 𝑗𝑗 , 𝑓 Two reference cells: in different thresholds (independent 𝑘𝑘 of data, can be precalculated). 21
Adaptive thresholding procedures 1) Measure resistances of cells on the 𝑓 𝑗𝑗 , 𝑗 ∈ diagonal and determine Ƹ 1, … , 𝑜 using a threshold detector. 𝑓 𝑗𝑗 = 1 𝑓 𝑗𝑗 = 0 Decide Ƹ Decide Ƹ 22
Adaptive thresholding procedures 1) Measure resistances of cells on the 𝑓 𝑗𝑗 , 𝑗 ∈ diagonal and determine Ƹ 1, … , 𝑜 using a threshold detector. 2) To read cell (𝑗, 𝑘) , choose the appropriate threshold: 1) Double threshold scheme: same threshold is used for each row/column 𝑓 𝑗𝑗 / Ƹ 𝑓 based on Ƹ 𝑘𝑘 . Targeted cell Reference cell 23
Adaptive thresholding procedures 1) Measure resistances of cells on the 𝑓 𝑗𝑗 , 𝑗 ∈ diagonal and determine Ƹ 1, … , 𝑜 using a threshold detector. 2) To read cell (𝑗, 𝑘) , choose the appropriate threshold: 1) Double threshold scheme: same threshold is used for each row/column 𝑓 𝑗𝑗 / Ƹ 𝑓 based on Ƹ 𝑘𝑘 . 2) Triple threshold scheme: select 𝑓 𝑗𝑗 and Ƹ 𝑓 threshold based on Ƹ 𝑘𝑘 . Targeted cell Reference cells 24
Example: Double threshold scheme • Example: 𝑆 1 = 100Ω, 𝑆 0 = 1000Ω, 𝑆 𝑡 = 250Ω, 𝑞 𝑔 = 10 −3 , 𝜏 = 30, 𝑜 = 8. 𝐶𝐹𝑆 = 2.304 × 10 −4 25
Example: Double threshold scheme • Example: 𝑆 1 = 100Ω, 𝑆 0 = 1000Ω, 𝑆 𝑡 = 250Ω, 𝑞 𝑔 = 10 −3 , 𝜏 = 30, 𝑜 = 8. 𝐶𝐹𝑆 = 2.304 × 10 −4 𝐶𝐹𝑆 = 1.602 × 10 −4 26
BER improvement with adaptive thresholding • BER vs. Noise: • Large improvement for moderate noise. • Our schemes prevent saturation of BER in high noise region. 27
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