SPICE Model of Memristor and Its Application Kaida Xu1,2, Yonghong Zhang1
Lin Wang2, William T. Joines2, Qing Huo Liu2
1
2
EHF Key Laboratory of Fundamental Science University of Electronic Science and Technology of China Chengdu, Sichuan, China
[email protected]
Department of Electrical and Computer Engineering Duke University Durham, NC, USA
memristors [3]-[5]. Most of the models using SPICE language were simulated in some derivatives of SPICE, e.g., SPICE3, PSPICE, LTSPICE, NGSPICE. Meanwhile, many researchers had started to exploit the memristor for analog and digital information progressing [6-7] applications due to the unusual electronic properties of the memristor. Moreover, the memristors can be also employed in spintronic and neuromorphic applications [8-9]. To the authors’ knowledge, however, very little exploration has been done on how to utilize the memristor in microwave device and circuit fields [10-11].
Abstract—In this paper, a circuit model of the memristor using SPICE is presented, which expands the hitherto methods to solve the memristor’s modeling equations presented by HP lab. This kind of the memristor model can not only be encoded in SPICE and satisfy the properties of the general memristive systems, but also use few components and simulate fast. In order to further explore the nonlinear and switching characteristics of the memristor model, a directly modulated patch antenna with one memristor is designed by using finite-difference timedomain (FDTD) simulator integrated with the nonlinear SPICE circuit solver.
I.
This paper proposed a new kind of memristor SPICE model, and presents both basic characteristics and applications of the proposed model.
INTRODUCTION
Most people learn from textbooks that there are three fundamental two-terminal circuit elements: resistors (R), capacitors (C), and inductors (L), which are defined by Equation (1a), (1b), and (1c), respectively. Based on symmetry consideration, Leon Chua first postulated the existence of the fourth passive electrical circuit element— memristor (M), acronym for memory resistor in 1971 [1]. Therefore, the link between the electrical charge q and magnetic flux ϕ could be established and the sixth equation (Equation (1f)) could be complemented according to the definition of the memristor besides Equations (1d) and (1e). In 2008, HP researchers Strukov et. al successfully fabricated a nano-scale electronic device based on a thin film of titanium dioxide [2], whose physical properties of the practical system can perfectly match with the memristor mathematical theory. Resistor: dv = Rdi Capacitor: dq = Cdv
(1a) (1b)
Inductor: dϕ = Ldi
(1c)
Definition of current: dq = idt
(1d)
Faraday’s law: dϕ = vdt
(1e)
Memristor: dϕ = Mdq
II.
We followed the published mathematical equations [2], whose physical model can be seen in Fig. 1. Let w and D denote the thickness of the doped region and the sandwiched region in the TiO2 memristor, respectively. And let RON and ROFF denote the low resistance and high resistance at high and low dopant concentration regions, respectively. The equations of the physically realized memristor can be shown in Equation (2a) and (2b).
v (t ) = ( RON dw(t )
=
w( t )
+ ROFF (1 −
D w(t )( D − w(t ))
⋅ μv
w (t ) D
RON
))i (t )
(2a)
i (t ) (2b) dt D D where (2a) and (2b) are port equation and state equation, 2 respectively, with a window function w(t )( D − w(t )) / D . To simplify the equations, we can define the normalized w(t ) doped state variable x (t ) = , whose range is located at D (0,1). So the equations can be changed as follows if we let R k = μv ON2 , D
(1f)
After the HP invention, widespread research related to memristor and its applications has been developing rapidly. First of all, due to the cost and technical difficulties in fabricating the nano-scale memristor, the scholars started research on the unique electronic properties of the memristor and created diverse behavior models for the resistive This research is supported in part by the China Scholarship Council (CSC).
978-1-4799-0066-4/13/$31.00 ©2013 IEEE
MEMRISTOR SPICE MODEL
53
2
v (t ) = ROFF i (t ) + ( RON − ROFF ) ⋅ x ⋅ i (t )
(3a)
dx (t )
= kx (t )(1 − x (t )) ⋅ i (t ) (3b) dt The total resistance of the memristor Rmem can be given by Rmem =
v (t ) i (t )
= ROFF + ( RON − ROFF ) ⋅ x
(4)
Figure 3. The proposed circuit model of the memristor.
The behaviors of the proposed memristor model in SPICE excited by the sinusoidal voltage source can be seen in Figs. 4-7. Here, we give the state variable x0 = 0.05 as the initial value. When input voltage is vs (t ) = sin(2π fo t ) , the simulated transient i-v characteristics of the memristor is shown in Fig. 4, where the current flowing across the memristor is obtained for the inputs with different frequencies. The pinched hysteretic loops clearly appear at low frequencies fo which are collapsed to that of a linear resistor for a tenfold increase. Fig. 5 illustrates the dynamical properties of the normalized doped state variable x (t ) as well
Figure 1. Memristor model with time-varying voltage stimulus.
The aim is to model the above equations in SPICE to obtain the physical properties of the memristor. The first equation (Equation (3a)) can be modeled as a simple series of the resistor ROFF and the other special resistor with state variable. To solve the differential state Equation (3b), we can design an integrator model if it is written in an integral form: x (t ) = ∫ kx (t )(1 − x (t )) ⋅ i (t ) dt + x0 , where x0 is the initial
7
as the input voltage vs (t ) = sin(2π ⋅ 10 t ) . According to the Equation (4), the curve of the corresponding Rmem is shown in Fig. 6. Furthermore, Fig. 7 illustrates a nonlinear currentvoltage characteristic of the memristor, where a periodic pinched hysteresis loop with hard switching emerges. This memristor model, satisfying the properties of the general memristive systems [1], has zero-crossing property in a form of i-v Lissajous figure.
state of the x (t ) . The integrator model consists of one capacitor, one resistor, one Current Dependent Current Source (CDCS), and one Voltage Dependent Voltage Source (VDVS) as shown in Fig. 2. The relations between input and output voltage are 1 vout (t ) = (5) ∫ vin (t ) dt + V0 RC where V0 is the initial voltage. Thus, it can replace the integral form of Equation (3b) if RC = 1 s. This kind of the integrator can not only be encoded in SPICE and produce the basic behavior of the integrator, but also use few components and simulate fast. Combining the Equations (3a) and (3b), therefore, the circuit SPICE model of the memristor is created as shown in Fig. 3. The parameters of the memristor model are as follows, RC = 1 s, ROFF =10 kΩ , RON =100 Ω ,
Current (uA)
140 120 100 2f0 80 10f0 f 60 0 40 20 0 -20 -40 -60 -80 -100 -120 -140 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Voltage (V)
12
k = 1 × 10 .
Figure 4. Simulated transient i-v characteristics of the memristor driven by sinusoidal voltage with different frequencies, where fo =12 MHz.
Figure 2. Simple model of integrator.
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III.
1.0
1.0
One technique that demonstrates the benefits of integrating high-speed semiconductor switching devices with an antenna structure is known as direct antenna modulation (DAM) [12-13]. This modulation technique involves driving an antenna at resonance with a high-frequency carrier wave signal and then modulating the antenna directly with a semiconductor switching device that is controlled by a lowfrequency baseband information signal. In order to explore the nonlinear and switching characteristics of the memoristor, first we design a square half-wavelength microstrip patch antenna with a resonant frequency of 2.46 GHz, and then connect a memristor between the ground plane and patch of the antenna. Similar to DAM using diode switching techniques [13], we incorporate a memristor instead of two diodes into the patch antenna as shown in Fig. 8.
0.8
0.6 w/D
Voltage (V)
0.5
0.0 0.4 -0.5
0.2
-1.0
0
50
100 Time (ns)
0.0 200
150
APPLICATION OF THE MEMRISTOR IN DIRECTLY MODULATED ANTENNA
Figure 5. Time dependence of the state variable x (t ) driven by sinusoidal 7
voltage v s (t ) = sin(2π ⋅ 10 t ) . 10
Rmem (kOhm)
8
6
4
2
0 0
50
100 Time (ns)
150
200
Figure 8. Geometry of the memristor incorporated into the directly modulated patch antenna
Figure 6. Time dependence of the total resistance of the memristor Rmem
In order to demonstrate DAM, a well-designed antenna is necessary to serve as the transmitter antenna in the basic communication system. The square half-wavelength microstrip patch antenna with a resonant frequency of 2.46 GHz was designed on the substrate G-10 epoxy glass with dielectric constant 4.24 by using WavenologyTM EM [14], a commercial full-wave transient field simulator integrated with SPICE circuit solver. As shown in Fig. 8 without memristor, the length and width of the patch are both set 28 mm, slightly smaller than half wavelength for a 2.46 GHz square patch antenna due to the fringing field effect. The probe feed point is located at (14.3, -14.3) mm along the diagonal of the patch plane (approximately a quarterwavelength in the dielectrics), where the feedpoint impedance is approximately 50 Ω . Fig. 9 shows this microstrip patch antenna resonates at 2.46 GHz.
7
driven by sinusoidal voltage v s (t ) = sin(2π ⋅ 10 t ) . 4 3
Current (mA)
2 1 0 -1 -2 -3 -4 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Voltage (V)
By utilizing WavenologyTM EM, finally, a memristor modeled in SPICE is incorporated into the patch antenna. The memristor is placed at (14, -14) mm. To demonstrate the DAM effect of the baseband signal-controlled memristor switching on the radiated 2.46-GHz carrier wave, the timedomain near-field waveform at the observer point of (0, 0, 20) mm is depicted in Fig. 10. Observe that when Rmem is high
Figure 7. Periodic pinched hysteresis loop of i-v characteristics with hard switching. It is a good approximation of the measurement of the real memristor [2], produced by HP Lab.
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problems, we create a novel and simple integrator, and then establish a memristor model to realize the characteristics of the memristor. By applying the hybrid circuit-field simulator WavenologyTM EM, finally, we incorporate the memristor into the patch antenna to achieve direct antenna modulation successfully.
(the resistance of the memristor is close to ROFF ), there is almost no effect on the antenna’s radiation, and the received signal amplitude of the E-field averages around 8 V/m (peak to peak). Then, when Rmem becomes very low (the resistance of the memristor is close to RON ), the charges accumulated on the top radiation edge of the antenna will quickly flow through this path to the ground plane, so that the carrier-wave radiation is minimized and the received signal amplitude of the E-field averages about 2 V/m (peak to peak). Compared with the high-to-low signal ratio of 4:1 in [13], this is also approximately 4:1. Additionally, a clean, easily detectable time-domain amplitude-modulated signal is obtained because the enveloped wave of Fig. 10 agrees well with the timedependent curve of Rmem in Fig. 6.
ACKNOWLEDGMENT The authors sincerely thank Wave Computation Technologies Inc. for providing 3D full-wave simulation software, and Dr. Mengqing Yuan, Dr. Tian Xiao of Wave Computation Technologies Inc. for providing useful suggestion during the design process. REFERENCES [1]
0
[2]
Magnitude (dB)
-5
[3] -10
[4]
-15
[5] -20
[6] -25 2.0
2.2
2.4 2.6 Frequency (GHz)
2.8
3.0
[7]
Figure 9. Reflection coefficient of the square patch antenna without memristor 10
[8]
8 6
[9]
Ex (V/m)
4 2
[10]
0 -2
[11]
-4 -6
[12]
-8 -10 500
550
600
650 700 Time (ns)
750
800
[13]
Figure 10. Received time-domain near-field Ex waveform at (0, 0, 20) mm. [14]
IV. CONCLUSION In this paper, using SPICE without compatibility
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