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Journal of Semiconductors
Vol. 43, Issue 2, 023101 (2022)

A review of compact modeling for phase change memory

Feilong Ding¹, Baokang Peng¹, Xi Li², Lining Zhang¹, Runsheng Wang³, Zhitang Song², and Ru Huang³

Author Affiliations

¹School of Electronic and Computer Engineering, Peking University, Shenzhen 518055, China

²Shanghai Institute of Micro-System and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China

³Institute of Microelectronics, Peking University, Beijing 100871, China

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DOI: 10.1088/1674-4926/43/2/023101 Cite this Article

Feilong Ding, Baokang Peng, Xi Li, Lining Zhang, Runsheng Wang, Zhitang Song, Ru Huang. A review of compact modeling for phase change memory[J]. Journal of Semiconductors, 2022, 43(2): 023101 Copy Citation Text

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Abstract

Phase change memory (PCM) attracts wide attention for the memory-centric computing and neuromorphic computing. For circuit and system designs, PCM compact models are mandatory and their status are reviewed in this work. Macro models and physics-based models have been proposed in different stages of the PCM technology developments. Compact modeling of PCM is indeed more complex than the transistor modeling due to their multi-physics nature including electrical, thermal and phase transition dynamics as well as their interactions. Realizations of the PCM operations including threshold switching, set and reset programming in these models are diverse, which also differs from the perspective of circuit simulations. For the purpose of efficient and reliable designs of the PCM technology, open issues and challenges of the compact modeling are also discussed.

$ \Delta G = \frac{{16}}{3}\frac{{\pi {\gamma ^3}}}{{d{g^2}}} . $

(1)

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$ {P_{\rm n}} = \alpha \Delta t \, \exp \left[ { - \beta \left( {{E_{\rm{a}1}} + \Delta G} \right)} \right] , $

(2)

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$ {c_N} = r_0^N \; \exp \left[ { - \beta \left( {{E_{{\text{a}}1}} + \Delta G} \right)} \right] . $

(3)

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$ {v_{\rm g}} = f{a_0}\alpha \left[ {1 - \exp \left( { - \beta \left| {dg} \right|} \right)} \right]\exp \left( { - \beta {E_{\rm{a}2}}} \right), $

(4)

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$ {P_{\rm g}} = pa\Delta t\left[ {1 - \exp \left( { - \beta \left| {d{{g}}} \right|} \right)} \right]\exp \left( { - \beta {E_{\rm{a}2}}} \right), $

(5)

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$ {c_G} = r_0^G\left[ {1 - \exp ( - \beta \left| {dg} \right|)} \right]\exp \left( { - \beta {E_{\rm{a}2}}} \right), $

(6)

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$ X\left( t \right) = 1 - \exp \left[ { - {{(At)}^n}} \right] , $

(7)

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$ A\left( T \right) = w\,\exp \left( { - \frac{{{E_{\rm a}}}}{{kT}}} \right) , $

(8)

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$\begin{array}{l} {I_\alpha }\left( {{V_1},{V_{\rm{status}}}} \right) = \dfrac{1}{{{n_{\rm{status}}} {R_0}\left( {{V_{\rm{status}}}} \right) \cdot \exp \left[ {\dfrac{{{E_{\rm{a,status}}}}}{k}\left( {\dfrac{1}{T} - \dfrac{1}{{{T_{\rm{ref}}}}}} \right)} \right]}} \\ \qquad\qquad\qquad\times\left[ {\exp \left( {{n_{\rm{status}}} {V_1}} \right) - 1} \right] . \end{array}$

(9)

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$ {I_1} = {I_\alpha }\left( {{V_1},{C_x}} \right) + F\left( {{I_1};{I_{\rm{th}}}} \right) \cdot \left[ { - {I_\alpha }\left( {{V_1},{C_x}} \right) + F\left( {{V_1};{V_{\rm{hold}}}} \right) \cdot {I_{\rm{on}}}\left( {{V_1}} \right)} \right] . $

(10)

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$ I = - \frac{V}{{{R_{\rm{load}}}}} + \frac{{{V_{\rm{th}}}}}{{{R_{\rm{load}}}}} , $

(11)

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$ {C_x} = 1 - \exp \left( { - \frac{t}{\tau }} \right) . $

(12)

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$ {R_x} = {R_{0\rm{c}}} + \left( {{R_{0\rm{a}}} - {R_{0\rm{c}}}} \right) \left( {1 - {C_x}} \right), $

(13)

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$ \frac{{\rm{d}F}}{{\rm{d}t}} = - \frac{{F - \theta \left( {{I_{\rm{gst}}} - {I_{\rm{th}}}} \right)}}{{{\tau _f}}} , $

(14)

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$ {\left( {\frac{{\rm{d}{V_{\rm{a}}}}}{{\rm{d}t}}} \right)_{\rm{c}}} = - \left({P_{\rm{n}}}{V_{\rm{n}}}\frac{{{V_{\rm{a}}}}}{{{V_{\rm{m}}}}} + {S_{\rm{a}}}{v_{\rm{g}}} \right) . $

(15)

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$ {\left( {\frac{{\rm{d}{V_{\rm{a}}}}}{{\rm{d}t}}} \right)_{\rm{c}}} = - \frac{{{V_{\rm{a}}}}}{\tau } . $

(16)

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$ {\left( {\frac{{\rm{d}{V_{\rm{a}}}}}{{\rm{d}t}}} \right)_{\rm{a}}} = \frac{{{T_{\rm{a}}} - {T_{\rm{m}}}}}{{{R_{\rm{t}}}\Delta {h_{\rm{1}}}}} , $

(17)

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$ \frac{{\rm{d}{V_{\rm{a}}}}}{{\rm{d}t}} = {\left( {\frac{{\rm{d}{V_{\rm{a}}}}}{{\rm{d}t}}} \right)_{\rm{c}}}\theta \left( {{T_{\rm{m}}} - {T_{\rm{a}}}} \right) + {\left( {\frac{{\rm{d}{V_{\rm{a}}}}}{{\rm{d}t}}} \right)_{\rm{a}}}\theta \left( {{T_{\rm{a}}} - {T_{\rm{m}}}} \right) . $

(18)

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$ {\tau _{\rm{m}}}\frac{{\partial {F_{\rm{m}}}}}{{\partial t}} + {F_{\rm{m}}} = {\left[ {1 + \exp \left( {\frac{{{T_{\rm{m}}} - T}}{{{\sigma _{\rm{m}}}}}} \right)} \right]^{ - 1}} , $

(19)

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$ {\tau _{\rm c}}\frac{{\partial {F_{\rm c}}}}{{\partial t}} + {F_{\rm c}} = 1 - {F_{\rm m}} , $

(20)

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$ {F_{\rm a}} = 1 - {F_{\rm m}} - {F_{\rm c}} . $

(21)

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$ {\tau _{\rm{set}}}\frac{{\partial {F_{\rm c}}}}{{\partial t}} = {F_{\rm a}} b \, \exp \left( {1 - b {F_{\rm a}}} \right) , $

(22)

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$ {\tau _{\rm{set}}} = {\tau _{\rm{HT}}} + {\tau _{\rm{LT}}} = {\tau _{0\rm{HT}}}\exp \left( {\frac{{{E_{\rm{AHT}}}}}{{kT}}} \right) + {\tau _{0\rm{LT}}}\exp \left( {\frac{{{E_{\rm{ALT}}}}}{{kT}}} \right), $

(23)

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$ {I_{\rm{PF}}} = {A_{\rm{kPF}}} F \left( { - \frac{{{\Phi _{\rm{PF}}} - {\beta _{\rm{PF}}}\sqrt F }}{{kT}}} \right) , $

(24)

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$ {\Phi _{\rm{PF}}} = {E_{\rm{a}0}} - \frac{{{a_{\rm{va}}}{T^2}}}{{{b_{\rm{va}}} + T}} , $

(25)

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$ \frac{{\rm{d}\Omega_i}}{{\rm{d}t}} = \sum\limits_{i \ne j} {\sum\limits_{e,c \in P} {\left( {{c_{j \to i}}{\Omega _j} - {e_{i \to j}}{\Omega _i}} \right)} } , $

(26)

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$\begin{array}{l} I = 2eA \dfrac{{\Delta z}}{{{\tau _0}}}\left[ {n_{\rm{T}1}}\exp \left( { - \dfrac{{{E_{\rm c}} - {E_{\rm{T}1}}}}{{kT}}} \right) + \left( {{n_{\rm{T}2}} + {n_{\rm{T}2}}'} \right)\exp \left( { - \dfrac{{{E_{\rm c}} - {E_{\rm{T}2}}}}{{kT}}} \right) \right] \\ \qquad\times\, {\rm{sinh}} \left( {\dfrac{{dU}}{{kT}}} \right) , \end{array}$

(27)

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$ {H_{\rm{a,OFF}}} = \frac{{{E_{\rm{T}2}} - {E_{\rm{T}1}}}}{{kT}} , $

(28)

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$ {H_{\rm{a,ON}}} = {H_{\rm{gst,a}}} - {H_{\rm{a,OFF}}} , $

(29)

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$ V = {F_{\rm{OFF}}} {H_{\rm{a,OFF}}} + {F_{\rm{ON}}} {H_{\rm{a,ON}}} , $

(30)

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$ {\mu _{\Delta {G_N}}} = {m_1}{G_{N - 1}}\left( {{t_0}} \right) + \left( {{c_1} + {A_1}{P_{\rm{mem}}}} \right) , $

(31)

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$ {\sigma _{\Delta {G_N}}} = {m_2}{G_{N - 1}}\left( {{t_0}} \right) + \left( {{c_2} + {A_2}{P_{\rm{mem}}}} \right) , $

(32)

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$ \Delta {G_N} = {\mu _{\Delta {G_N}}} + {\sigma _{\Delta {G_N}}}\chi , $

(33)

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$ {G_N}\left( {{t_0}} \right) = {G_{N - 1}}\left( {{t_0}} \right) + \Delta {G_N} . $

(34)

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$ R\left( t \right) = {R_1}{\left( {\frac{t}{{{t_1}}}} \right)^v} , $

(35)

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Feilong Ding, Baokang Peng, Xi Li, Lining Zhang, Runsheng Wang, Zhitang Song, Ru Huang. A review of compact modeling for phase change memory[J]. Journal of Semiconductors, 2022, 43(2): 023101

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