^{1,2,a)}, Yao Shuai

^{3}, Wenbo Luo

^{3}, Christian Mayr

^{4}, René Schüffny

^{4}, Oliver G. Schmidt

^{1,2}and Heidemarie Schmidt

^{1}

### Abstract

Chua [IEEE Trans. Circuit Theory18, 507–519 (Year: 1971)10.1109/TCT.1971.1083337] predicted rather simple charge-flux curves for active and passive memristors (short for memory resistors) and presented active memristor circuit realizations already in the 1970 s. The first passive memristor has been presented in 2008 [D. B. Strukov, G. S. Snider, and D. R. Williams, Nature (London)453, 80–83 (Year: 2008)10.1038/nature06932]. Typically, memristors are traced in complicated hysteretic current-voltage curves. Therefore, the true essence of many new memristive devices has not been discovered so far. Here, we give a practical guide on how to use normalized charge-flux curves for the prediction of hysteretic current-voltage characteristics of memristors. In the case of memristive BiFeO_{3} thin film capacitor structures, the normalized charge-flux curves superimpose for different numbers of measurement points *N* _{ s } and a different measurement time per measurement point *T* _{ s }. Such normalized charge-flux curves can be used for the prediction of current-voltage characteristics for input signals with arbitrarily chosen *N* _{ s } and *T* _{ s }.

H.S. greatly acknowledges financial support from the Deutsche Forschungsgemeinschaft (DFG) (DFGSCHM1663/4-1). N.D. and W.L. acknowledge funding by the Initiative and Networking Fund of the Helmholtz Association (VH-VI-422).

I. INTRODUCTION

II. DATA RECORDING AND ANALYSIS

III. EXPERIMENTAL RESULTS

A. Input signal

B. Normalized memristance

IV. MODELING OF HYSTERETIC (I-V) DATA

V. CONCLUSIONS AND OUTLOOK

### Key Topics

- Time measurement
- 21.0
- Electric measurements
- 17.0
- Resistors
- 13.0
- Thin film structure
- 10.0
- Charged currents
- 9.0

## Figures

Combined block diagram of the memristance measurement setup and flow chart of validated memristance measurements. The experimental setup uses a Keithley source meter to define the input voltage *v* _{ in }(*t*) or the input current *i* _{ in }(*t*) with respect to its amplitude *V* _{ dc } and *I* _{ dc } and to its shape, e.g., linear, exponential, or sinusoidal, to its number of measurement points *N* _{ s } and to the measurement time *T* _{ s } per measurement point for voltage correct (left) and for current correct (right) measurements. The measurement is controlled by a LabVIEW program. This program also collects the output current *i*(*t*) (left) and output voltage *v*(*t*) (right) data, performs the integration according to Eqs. (4) and (5) and the normalization according to Eqs. (6) and (7) , stores the normalized memristance data in a lookup table and predicts (*i*-*v*) curves from normalized memristance curves for input voltage *v* _{ in } (left) and input current *i* _{ in } (right) with corresponding shape and amplitude and arbitrarily chosen *N* _{ s } and *T* _{ s }.

Combined block diagram of the memristance measurement setup and flow chart of validated memristance measurements. The experimental setup uses a Keithley source meter to define the input voltage *v* _{ in }(*t*) or the input current *i* _{ in }(*t*) with respect to its amplitude *V* _{ dc } and *I* _{ dc } and to its shape, e.g., linear, exponential, or sinusoidal, to its number of measurement points *N* _{ s } and to the measurement time *T* _{ s } per measurement point for voltage correct (left) and for current correct (right) measurements. The measurement is controlled by a LabVIEW program. This program also collects the output current *i*(*t*) (left) and output voltage *v*(*t*) (right) data, performs the integration according to Eqs. (4) and (5) and the normalization according to Eqs. (6) and (7) , stores the normalized memristance data in a lookup table and predicts (*i*-*v*) curves from normalized memristance curves for input voltage *v* _{ in } (left) and input current *i* _{ in } (right) with corresponding shape and amplitude and arbitrarily chosen *N* _{ s } and *T* _{ s }.

Charge-flux (*q*-φ) curve from a 100 Ω and 200 Ω resistor. The insets show the linear (lin), sinusoidal (sin), and exponential (exp) (a) input voltage *v* _{ in } (32 s period *T*, 7 V amplitude *V* _{ dc }) and (b) input current *i* _{ in } (32 s period *T*, 70 mA amplitude *I* _{ dc } for 100 Ω and 35 mA amplitude *I* _{ dc } for 200 Ω) for CW and CCW looping. The number of measurement points in each period amounts to *N* _{ s } = 64 and the measurement time *T* _{ s } per measurement point amounts to *T* _{ s } = 0.5 s. The (*q*-φ) curve from a resistor lies in the 1st quadrant (I) and in the 3rd quadrant (III), if the input signals are looped CW and CCW, respectively.

Charge-flux (*q*-φ) curve from a 100 Ω and 200 Ω resistor. The insets show the linear (lin), sinusoidal (sin), and exponential (exp) (a) input voltage *v* _{ in } (32 s period *T*, 7 V amplitude *V* _{ dc }) and (b) input current *i* _{ in } (32 s period *T*, 70 mA amplitude *I* _{ dc } for 100 Ω and 35 mA amplitude *I* _{ dc } for 200 Ω) for CW and CCW looping. The number of measurement points in each period amounts to *N* _{ s } = 64 and the measurement time *T* _{ s } per measurement point amounts to *T* _{ s } = 0.5 s. The (*q*-φ) curve from a resistor lies in the 1st quadrant (I) and in the 3rd quadrant (III), if the input signals are looped CW and CCW, respectively.

(a) and (b) Unnormalized charge-flux curves from a BFO memristor with a nominal top contact area of 8.92 × 10^{−2} mm^{2}. The insets show the sinusoidal (sin), linear (lin), and exponential (exp) (a) input voltage *v* _{ in } (32 s period *T*, 7 V amplitude *V* _{ dc }) and (b) input current *i* _{ in } (32 s period *T*, +3 × 10^{−5} A maximum positive input current and −2 × 10^{−6} A minimum negative input current). The number of measurement points in each CW and CCW cycle amounts to *N* _{ s } = 64 and the measurement time *T* _{ s } per measurement point amounts to *T* _{ s } = 0.5 s.

(a) and (b) Unnormalized charge-flux curves from a BFO memristor with a nominal top contact area of 8.92 × 10^{−2} mm^{2}. The insets show the sinusoidal (sin), linear (lin), and exponential (exp) (a) input voltage *v* _{ in } (32 s period *T*, 7 V amplitude *V* _{ dc }) and (b) input current *i* _{ in } (32 s period *T*, +3 × 10^{−5} A maximum positive input current and −2 × 10^{−6} A minimum negative input current). The number of measurement points in each CW and CCW cycle amounts to *N* _{ s } = 64 and the measurement time *T* _{ s } per measurement point amounts to *T* _{ s } = 0.5 s.

Normalized charge-flux curves from the BFO memristor in Fig. 3 for (a) input voltage and (b) input current. The number of measurement points in each CW and CCW cycle amounts to *N* _{ s } = 64 and the step length amounts to *T* _{ s } = 0.5 s. The turning points of the normalized CW and CCW memristance curve lie at and , respectively.

Normalized charge-flux curves from the BFO memristor in Fig. 3 for (a) input voltage and (b) input current. The number of measurement points in each CW and CCW cycle amounts to *N* _{ s } = 64 and the step length amounts to *T* _{ s } = 0.5 s. The turning points of the normalized CW and CCW memristance curve lie at and , respectively.

(a) Normalized memristance curve in the first (I) quadrant from a BFO memristor with a nominal top contact area of 8.92 × 10^{−2} mm^{2} recorded with CW 7 V input voltage *v* _{ in } of different shape. The turning point lies at . (b) Sinusoidal (sin), linear (lin), and exponential (exp) input voltage *v* _{ in } (64 measurement steps *N* _{ s }, 32 ms period *T*, 7 V amplitude *V* _{ dc }, measurement time *T* _{ s } = 2 s). (c) Normalized charge curves and (d) normalized flux curves for the input voltage *v* _{ in } represented in (b).

(a) Normalized memristance curve in the first (I) quadrant from a BFO memristor with a nominal top contact area of 8.92 × 10^{−2} mm^{2} recorded with CW 7 V input voltage *v* _{ in } of different shape. The turning point lies at . (b) Sinusoidal (sin), linear (lin), and exponential (exp) input voltage *v* _{ in } (64 measurement steps *N* _{ s }, 32 ms period *T*, 7 V amplitude *V* _{ dc }, measurement time *T* _{ s } = 2 s). (c) Normalized charge curves and (d) normalized flux curves for the input voltage *v* _{ in } represented in (b).

(a) Normalized current curves obtained by differentiating the normalized charge curve (Fig. 5(c) ) recorded on a BFO memristor with a CW input voltage *v* _{ in } of different shape and with the same amplitude. (b) Normalized input voltage curves obtained by differentiating the normalized flux curve (Fig. 5(d) ). Predicted (solid lines) and experimental (symbols) unnormalized output current (c) versus time on a linear scale and (d) versus input voltage on a logarithmic scale.

(a) Normalized current curves obtained by differentiating the normalized charge curve (Fig. 5(c) ) recorded on a BFO memristor with a CW input voltage *v* _{ in } of different shape and with the same amplitude. (b) Normalized input voltage curves obtained by differentiating the normalized flux curve (Fig. 5(d) ). Predicted (solid lines) and experimental (symbols) unnormalized output current (c) versus time on a linear scale and (d) versus input voltage on a logarithmic scale.

(a) Normalized memristance curve from a BFO memristor with a nominal top contact area of 8.92 × 10^{−2} mm^{2} recorded with a CW linear (lin) input voltage *v* _{ in } of the same shape and amplitude and different measurement time per measurement point *T* _{ s } = 0.5, 1, 2 s. The turning point lies at . (b) Linear (lin) input voltage *v* _{ in } (64 measurement steps *N* _{ s }, 7 V amplitude *V* _{ dc }, 0.5, 1, and 2 s measurement time *T* _{ s }). Predicted (solid lines) and experimental (symbols) unnormalized output current (c) versus time on a linear scale and (d) versus input voltage on a logarithmic scale.

(a) Normalized memristance curve from a BFO memristor with a nominal top contact area of 8.92 × 10^{−2} mm^{2} recorded with a CW linear (lin) input voltage *v* _{ in } of the same shape and amplitude and different measurement time per measurement point *T* _{ s } = 0.5, 1, 2 s. The turning point lies at . (b) Linear (lin) input voltage *v* _{ in } (64 measurement steps *N* _{ s }, 7 V amplitude *V* _{ dc }, 0.5, 1, and 2 s measurement time *T* _{ s }). Predicted (solid lines) and experimental (symbols) unnormalized output current (c) versus time on a linear scale and (d) versus input voltage on a logarithmic scale.

(a) Normalized memristance curve from a BFO memristor with a nominal top contact area of 8.92 × 10^{−2} mm^{2} recorded with a CW input voltage *v* _{ in } of the same shape and of different amplitude. The turning point of the normalized CW curves of the memristor lies at = (1,1.85). (b) Linear (lin) input voltage *v* _{ in } with different amplitude (64 measurement steps *N* _{ s }, 32 s period *T*, measurement time *T* _{ s } = 2 s). Predicted (solid lines) and experimental (symbols) unnormalized output current (c) versus time on a linear scale and (d) versus input voltage *v* _{ in } on a logarithmic scale.

(a) Normalized memristance curve from a BFO memristor with a nominal top contact area of 8.92 × 10^{−2} mm^{2} recorded with a CW input voltage *v* _{ in } of the same shape and of different amplitude. The turning point of the normalized CW curves of the memristor lies at = (1,1.85). (b) Linear (lin) input voltage *v* _{ in } with different amplitude (64 measurement steps *N* _{ s }, 32 s period *T*, measurement time *T* _{ s } = 2 s). Predicted (solid lines) and experimental (symbols) unnormalized output current (c) versus time on a linear scale and (d) versus input voltage *v* _{ in } on a logarithmic scale.

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