Conductance Quantization in Resistive Random Access Memory

Abstract

The intrinsic scaling-down ability, simple metal-insulator-metal (MIM) sandwich structure, excellent performances, and complementary metal-oxide-semiconductor (CMOS) technology-compatible fabrication processes make resistive random access memory (RRAM) one of the most promising candidates for the next-generation memory. The RRAM device also exhibits rich electrical, thermal, magnetic, and optical effects, in close correlation with the abundant resistive switching (RS) materials, metal-oxide interface, and multiple RS mechanisms including the formation/rupture of nanoscale to atomic-sized conductive filament (CF) incorporated in RS layer. Conductance quantization effect has been observed in the atomic-sized CF in RRAM, which provides a good opportunity to deeply investigate the RS mechanism in mesoscopic dimension. In this review paper, the operating principles of RRAM are introduced first, followed by the summarization of the basic conductance quantization phenomenon in RRAM and the related RS mechanisms, device structures, and material system. Then, we discuss the theory and modeling of quantum transport in RRAM. Finally, we present the opportunities and challenges in quantized RRAM devices and our views on the future prospects.

Keywords: Conductance quantization; Conductive filament (CF); Resistive random access memory (RRAM); Resistive switching (RS).

Fig. 1

Schematic I–V curves of resistive switching process in a CF-type bipolar RRAM device. Insets A–C show the different resistance states of the device during the switching process. In most cases, the fresh RRAM device shows a very high initial resistance state (IRS) with few defects (inset A). In a positive bias sweep, when the voltage increases to a comparative high voltage (V _Forming), the device switches to the low resistance state (LRS) with a conducting filament formed in the RS layer (inset B). Then, in a negative voltage sweep, when the voltage reaches a critical value (V _RESET), the device switches from LRS to the high resistance state (HRS), corresponding to a RESET transition in which the CF is ruptured (inset C). At last, in another positive sweep, the device will switch to LRS again, with the filament reconnected (inset B). This process is called as SET, with the SET voltage (V _SET) much lower than (V _Forming). If the device has a good endurance, the above SET and RESET switching can be reproducibly and successively carried out for a large number of cycles

Fig. 2

Conductance quantization phenomenon measured in the RESET process of a Pt/HfO₂/Pt unipolar RRAM device. a I − V curves in four RESET cycles. b G − V curves corresponding to (a). G is defined as I/V. The RESET transients of the device show discrete states with half-integer multiples of the conductance quantum. The red lines in (a) and (b) are “guides to the eye” and correspond to I = nG ₀ V or G = nG ₀ with n = 1, 1.5, 2, 2.5, 3, 4, 7, and 10. Discrete conductance levels at about 1 G ₀, 1.5 G ₀, 2 G ₀, 2.5 G ₀, 3 G ₀, 4 G ₀, etc. are evident. c Evolution of CF conductance in the last stage of 100 successive RESET switching cycles of the Pt/HfO₂/Pt device. Discrete conductance levels at 1 G ₀, 2 G ₀, 3 G ₀, 4 G ₀, etc. are revealed. d Histogram of normalized conductance collected at the step-like gradual RESET phase in 100 successive RESET cycles. Conductance peaks at integer and semi-integer multiples of G ₀ are clearly present

Fig. 3

Three characteristic length scales related to quantum conductance phenomenon. The three characteristic length scales are the following: (1) the de Broglie wavelength, which is related to the kinetic energy of the electrons; (2) the mean free path, which is the distance that an electron travels before its initial momentum is destroyed; and (3) the phase-relaxation length, which is the distance that an electron travels widely from one material to another and is also strongly affected by temperature, magnetic field, etc. A conductor will show conductance quantization behavior if any of its three dimensions is smaller than the three characteristic length scales mentioned above. Reproduced with permission [143]

Fig. 4

Switching characteristics and conductance quantization observed in nanoscale junctions with a structure of tungsten tip/ionic conductor layer/silver film [191]. a Conductance change during SET and RESET operation which shows atomic-scale conductance switching. Green lines act as guides to the eye representing a series of conductance levels with equal interval of 1 G ₀. b Conductance change from high resistance state to low resistance state following voltage sweep for three independent conductance states. c Histogram of the conductance difference between high resistance states and the low resistance states during the voltage sweep. The histogram consists of 130 independent I − V curves with initial ON-state conductance smaller than 10 G ₀. ΔG is the difference between the conductance and the zero-bias conductance. The inset histogram shows 5000 repeated and closing cycles at a constant bias voltage of 100 mV. Reproduced with permission

Fig. 5

Histogram showing typical conductance quantization phenomenon [171]. The data are extracted from pulse stimuli results of Ti/Ta₂O₅/Pt-structured memory cells. The data are grouped in every 0.2 G ₀. Histogram consists of 662 conductance values in 22 memory cells. The dashed curve which represents Gaussian fitting curve of the histogram acts as a guides to the eye. Reproduced with permission

Fig. 6

Quantized conductance phenomenon observed by pulse stimuli operation method in Ag/poly(3-hexylthiophene): [6, 6]-phenyl-C61-butyric acid methyl ester/indium–tin oxide sandwich structured devices [183]. Conductance quantization is observed under a successive positive pulses and b successive negative pulses. Positive voltage pulses are 1 μs wide and negative voltage pulses are 5 ms wide. Two adjacent positive or negative pulses are with an interval of 2 s and an increment of 0.05 V. The conductance is read under a basal voltage of 0.1 V. Reproduced with permission

Fig. 7

Quantized conductance steps observed in poly-Si/SiO_x/p-type Si-structured RRAM devices [186]. a Current–voltage curves showing nonlinear behavior with inset showing the relation between conductance and voltage. Several conductance quantization levels can be seen at both integer and half-integer multiples of G ₀. b Histogram consisting of about 1000 conductance steps in which half-integer multiples of G ₀ are clearly revealed. A series of Gaussian distributions act as guides to the eye (dotted lines). Reproduced with permission

Fig. 8

Typical conductance-voltage and current–voltage curves corresponding to RESET process of Pt/HfO₂/Pt devices [70]. Black curves show the usually observed abrupt RESET switching. Green curves display several successive jumps and red curves show progressive RESET process. Insets A–D show the different stages of the CF during the RESET process. The quantized conductance states in the step-like or progressive RESET processes are the intermediate states between low and high resistance states. A CF with conductance of the order of G ₀ = 2e ²/h is the natural boundary between the LRS and HRS states. The step-like or progressive RESET transition finalizes with an abrupt conductance drop of several orders of magnitude. This final drop corresponds to the opening of a spatial gap (potential barrier) in the CF. Discrete changes of conductance of the order of G ₀ recorded during the step-like or progressive RESET transitions are interpreted as the signature of atomic-sized variations of the conducting filament (CF) nanostructure. Reproduced with permission

Fig. 9

Typical RRAM device structures showing conductance quantization effect. a A commonly used sandwich RRAM structure [171]. b A crossbar structure [180]. c An ultra-small-sized RRAM device using CAFM tip as top electrode [170]. Reproduced with permission

Fig. 10

Typical forming, SET, and RESET characteristics of fresh RRAM devices with different initial resistance states. a Switching process illustration of Nb/ZnO/Pt device with an initial high resistance state [170]. Higher voltage in forming is needed compared to that in the SET process. b Switching process illustration of a free-forming p⁺Si/NiSi₂/SiO₂/CeO_x/W device [179]. The Forming process and the SET process show no obvious difference. c Schematic illustration of Ag₂S-based QCAS device and switching behavior between OFF- and ON-state [169]. The initial state of the device is ON-state and a RESET process is needed to start the switching cycles. Reproduced with permission

Fig. 11

Typical conductance quantization phenomenon observed in different structured devices under voltage sweeping mode. a Current jump observed in Ti/HfO₂/TiN-structured memristor during RESET process. The inset diagram indicates the discrete resistance change due to quantum atomic reaction during RESET process [188]. b Conductance quantization observed in Nb/ZnO/Pt during SET process. [170] The inset shows the current–voltage curve in a larger voltage range from 0 to 4 V. c Progressive RESET process in Pt/HfO₂/Pt devices [69]. The dashed line corresponds to the current–voltage curve of 1 G ₀. d Detail of the current–voltage evolution of (c). Reproduced with permission

Fig. 12

Quantized conductance observed in Ag/Ta₂O₅/Pt-structured ECM devices under voltage pulse operation mode [181]. a The value of conductance increases at steps of integer multiples of conductance quantum G ₀ in the SET process under positive pulses with a width of 20 ms at an interval of 2 s. In order to prevent hard breakdown of RRAM device, a current-limiting resistor of 3 kΩ was connected in series with the device. b Quantized conductance decrease phenomenon observed in the RESET process under reversed voltage polarity. No current-limiting resistor is needed in the negative pulse stimuli mode. Reproduced with permission

Fig. 13

Quantized conductance observed in Ti/Ta₂O₅/Pt-structured VCM devices under voltage pulse operation mode [171]. a The value of conductance increases at steps of integer multiples of G ₀ in the SET process under positive pulses with a width of 100 ns at interval of 2 s. b The value of conductance decreases at steps of integer multiples of G ₀ in the RESET process under negative pulses with a width of 1 μs at an interval of 2 s. Reproduced with permission

Fig. 14

Quantized conductance change behavior under different time intervals [181]. a The conductance change under ten successive voltage pulses of 0.4 V with a pulse width of 20 ms at intervals of 2 s. The conductance state could increase to about 2 G ₀ under the input pulses but immediately decays to zero after each input pulse is completed. b The conductance evolution under ten successive voltage pulses of 0.4 V with a width of 20 ms at intervals of 0.2 s. In this case, the conductance gradually increases and maintains at about 1 G ₀ for more than 60 s after the tenth input pulse. Reproduced with permission

Fig. 15

Quantized conductance phenomenon observed in Ag/AgI/Pt devices under current sweep mode [180, 195]. It can been seen that more than five resistance levels which are integer multiples of conductance quantum G ₀ have been observed. Reproduced with permission

Fig. 16

Conductance jump at the integer multiples of G ₀ under voltage bias in Ag/Ta₂O₅/Pt device [181]. a Conductance increase under positive voltage bias of +0.07 V. b Conductance decrease under negative voltage bias of −0.07 V. Reproduced with permission

Fig. 17

Current evolution of Pt/HfO₂/Pt device under constant voltage bias for RESET process indicating quantum level change of the conductance [69]. Reproduced with permission

Fig. 18

First observation of conductance quantization by Van Wees et al. [146]. The resistance of the point contact is a function of gate voltage at 0.6 K. The electron gas under the gate is depleted at −0.6 V when electrons only transport through the point contact and the contact is fully pinched off at −2.2 V. The inset shows the layout of the point contact. Reproduced with permission

Fig. 19

One-dimensional potential model for the forming process of a fresh RRAM cell [175]. The calculation was carried out under the assumption that the electric field E is zero within the filament and constant between the anode surface and the tip of the filament. The filament was treated as a one-dimensional atomic chain. Reproduced with permission

Fig. 20

The schematic of the RRAM device with a narrow CF and the corresponding energy band diagram of the quantum point contact model. a Schematic structure of the RRAM device with a narrow CF. b The dispersion curves of the first four electronic subbands under the confinement of CF in certain z. c The dependence of the energy level of the bottom of the subbands on z. The transmission probability T(E) of the bottom of ground quantized subband of a parabolic potential barrier is used for the calculation of electrical transport. The shaded regions are the states occupied by electrons. The number of the subbands is N _ch, with each one contributing to a conducting mode. In this figure, four subbands are shown. V is the applied voltage. V ₀ is the voltage dropped on TE and BE, represented by the two blue oblique lines. Since V ₀ is much lower than V, usually it can be neglected in the calculation. β is the fraction of voltage that drops at the BE interface, E _F is the Fermi level, E _F,TE and E _F,BE are the TE and BE quasi-Fermi levels, t _B is the width of the potential barrier at the equilibrium Fermi energy (E = 0), and Φ _B is the height of the potential barrier, i.e., the bottom of the first subband. The barrier height is different between high resistance state and low resistance state, which leads to different current expressions. In the deep OFF-state, the barrier thickness t _B is equal to the gap length t _gap

Fig. 21

E–k relationship for a narrow and b wide constriction, respectively [179]. The solid and dashed lines correspond to the bottom of the longitudinal and transversal subbands, respectively. The shaded regions are the states occupied by electrons. The indices indicate the subband number. A tighter constriction leads to higher energy levels. Reproduced with permission

Fig. 22

Quantized conductance effect based on the quantum point contact model [186]. a Schematic illustration of a conducting filament with a lateral constriction of one or several atoms at the narrowest part of the filament. b-I Dispersion curves of the first four electronic subbands at the edge of the constriction. b-II Dispersion curves of the first three subbands at the center of the constriction where the confinement is stronger which leads to a spacing out of the subbands. c-I When the difference in chemical potential between the left and right reservoirs is small, both the left-going and right-going electron modes fall within the same subband. c-II When the difference in chemical potential between the two reservoirs is large, the left-going and right-going electron modes fall into different subbands. Reproduced with permission

Fig. 23

Calculated band structures for crystalline m-HfO₂ with O vacancies and the corresponding conductance [69]. a–d Crystalline m-HfO₂ band structure with different oxygen vacancies separated by 4 a ₀, 2 a ₀, a ₀, and a ₀/2. a ₀ is the length of the c-axis vector for the m-HfO₂ primitive cell (0.5296 nm). e Conductance as a function of energy corresponding to a HfO₂ matrix where one, two, or three O atom rows are removed. The rows subsequently removed are shown in the instate (marked as “1,” “2,” and “3”), where red and white spheres correspond to O and Hf atoms, respectively. Reproduced with permission

Fig. 24

Equivalent circuit model for ECM device [195]. a Equivalent circuit model for an ECM cell including a nanobattery V _emf with an external circuit. R _i is the total resistance of the ionic current path. R _ext is the external resistance, e.g., from the neighboring cells in an array or a sense amplifier. b SPICE simulation results showing a staircase-like change of the cell conductance resulted from the discharging of V _emf. c Evolution of the conductance of a Ag/SiO₂/Pt cell in crossbar structure under a negative cell current I _cell. Reproduced with permission