An Introduction to Memimpedance and Memadmittance Systems Analysis

The magic peanut butter and jelly model of memristive electronic systems.

Last year (May 2008) researchers at HP Labs revealed a physical realization of a new type of circuit element called a memristor which was originally predicted in 1971 by Leon Chua and which possesses both memory storage and signal processing capabilities. In early 2009 Massimiliano Di Ventra, Yuriy V. Pershin, and Leon Chua published another article noting memcapacitors and meminductors as additional potential circuit elements. This article extends the model of memristive switching found by HP Labs to cover combined memristive/memcapacitive systems.

Thanks to Prof. Wei Wang, Prof. Massimiliano Di Ventra, and Prof. Leon Chua for providing some useful feedback.
 

Background

  
In a 1971 paper "Memristor-the Missing Circuit Element" a professor at UC Berkeley named Leon Chua proposed that the existing passive circuit elements (resistors, capacitors, and inductors) were insufficient to characterize all electrical interactions. A new circuit element was proposed named the "memristor" linking flux (the time integral of voltage) and charge. At the time Prof. Chua considered this a fourth fundamental circuit element and it was shown by further research that the memristor was part of a broader range of dynamic systems important to the characterization of a wide variety of phenomena including the Hodgkin-Huxley model used to model neurons. However, there was a lack of a physical realization of memristors so research was mostly limited to a theoretical discussion. 
 
This situation changed on May 1, 2008 when researchers at HP Labs published a paper in Nature entitled "The missing memristor found" which was the first to make the connection between memristors and an actual physical electronic device. The type of memristor originally discovered by HP was based on titanium oxide films formed with nanometer dimension thicknesses and including ions formed from oxygen depletion. It was shown in the HP paper that the ionic drift was responsible for the memristive effect.
 
While HP Labs was the first to make the connection between the memristor theory and a physical realization it has since been noted that many earlier discovered material systems also exhibit memristive effects (see memristor wikipedia page). The development of these new materials has also been connected to a new application in non-volatile memory (RRAM), neuromorphic computer architectures, and analog electronic systems as discussed in my other knol.
 
In early 2009 Massimiliano Di Ventra, Yuriy V. Pershin, and Leon Chua (who originally conceptualized the memristor) made another prediction in the paper "Circuit elements with memory: memristors, memcapacitors and meminductors" introducing two new circuit elements based on generalizations of capacitors and inductors include memory effects similar to that found in memristors. This article is indicative of the idea that the characteriztion of memristors as a fourth fundamental circuit element is incorrect. A better characterization is that memristors, memcapacitors, and meminductors are all each generalizations of the three basic passive circuit elements. However, this does not indicate a weakness in the memristor theory and actually enhances the importance of these materials since they can perform not just resistance functions but capacitive and inductive functions as well.
 
The possibility thus exists for a new type of electrical framework based on reconfigurable and adaptive electronic designs that may become increasing important  as Moore's Law becomes more difficult to fulfill. 
 
Since the scientists at HP labs were unaware of memcapacitive or meminductive effects when writing their original paper they did not take this into account in their model. This knol extends the concepts of memristors, memcapacitors, and meminductors to a generalized system of memimpedance and memadmittance.     

 

Non-Mathematical Explanation of Memimpedance and Memadmittance

The purpose of this section is to provide a layperson overview of what is meant by a memimpedance or memadmittance system. For more mathematical rigor see the next section.
 
A good introduction to memristors, as understood by HP Labs, can be found in a video on Youtube entitled "6-minute Memristor Guide". Below is a further explanation of a system incorporating memcapacitive effects.
 
Imagine you have a peanut butter and jelly sandwich.
 
The peanut butter and jelly sandwich has four components - an upper piece of bread, peanut butter, jelly, and a lower piece of bread. The two pieces of bread are analogous to electrodes which are metal conducting elements. The peanut butter is analogous to an electrical property called resistance which controls how much current can flow between the two pieces of bread (electrodes). The jelly is analogous to an electrical property called capacitance which determines how much charge is stored between the two pieces of bread (electrodes). 
 
However, this is not an ordinary peanut butter and jelly sandwich. This is a magic peanut butter and jelly sandwich in which peanut butter can magically transform into jelly and jelly can magically be reverted back into peanut butter by controlling the voltages applied to the two pieces of bread. The total amount of peanut butter plus jelly remains constant but the ratio of peanut butter to jelly can be changed so that you can be eating a sandwich that is mostly peanut butter or mostly jelly depending on your choice. By placing this type of sandwich into an electrical control system capacitance and resistance can both be controllably varied to create adaptive control systems providing novel functions such as self-healing, pattern recognition, and learning for future electronic hardware.
 
HP Labs knew about the peanut butter but they didn't know about the jelly.    
 

Mathematical Explanation of Memimpedance and Memadmittance

For a combined memristive and memcapacitive metal oxide junction it would be expected that the electron transport in the junction would be a function of the flux and the voltage
Taking the total derivative of q with respect to time produces:
Rewriting in terms of memconductance and memcapacitance:
This type of system may be classified as a parallel connected memristor and memcapacitor.
Similar analysis can be performed in terms of flux
Taking the total derivative of j with respect to time produces:
Rewriting in terms of memristance and memcapacitance:
This type of system may be classified as a memimpedance system equivalent to a series connected memristor and memcapacitor

The relationship of the above theoretical systems to actual physical system models could be based on either ionic or filamentary metal oxide resistance switching. One good example is discussed in the article by Tsui et al. “Field-Induced Resistive Switching in Metal-Oxide Interfaces” in which both resistance and capacitance hysteretic effects are noted in a perovskite system.   

For ionic metal oxide resistance switching mechanisms an extension of the analysis by HP Labs group would be based on the thickness variation parameter w expressed as a function of flux and voltage.

Taking the total derivative of w with respect to time produces:

 
In the case of a uniform electric field in the junction the ionic mobility mv is a proportionality constant between the velocity of ion drift and the electric field and can be expressed as the ratio of the voltage (v) and the total thickness (D).  Expressed mathematically:
This can be rewritten in terms of flux:

To account for memcapacitive effects it is noted that increasing the thickness of the non-ionic region w(t) increases the effective dielectric thickness w(t). Assuming the parallel plate capacitor model:

 
 

where Q is the total available ionic charge in the metal oxide junction, A is the cross-sectional area of the upper and lower electrodes that sandwich the metal oxide regions, and e is the permittivity of the metal oxide material. It may be more useful to rewrite this in terms of the total available ionic charge density s = Q/(AD) so that:

 

Multiplying both sides by the denominator and differentiating produces:

 
Combining Eq. 8, 10, and 13 produces:
A very thin metal oxide junction may be characterized as a parallel memresistor and memcapacitor since it can be capable of both charge transfer and charge storage. By noting COFF and CON as the maxima high/low capacitances associated with the different ionic states and noting that the equivalent capacitance of two capacitors in series is the parallel combination, current can be expressed as a function of w as: 
Rewriting (Eq. 14):
Eq. 16 and 17 define a memadmittance system. Note this indicates that memcapacitance should be enhanced by a square in the reduction in film thickness for ionic thin films in a similar manner to memristance. Also note that the results of the Nature paper from HP Labs is only a special case of this equation. It is possible that at higher frequencies the memcapacitive effects will appear. The memcapacitive effects also may be able to be amplified at low frequencies by decreasing the ion mobility (such as by using nanoparticles as the ion carrier), by decreasing the ion density or by using higher permittivity materials. 
  

Alternative Memadmittance Equations

Eq. 16 and 17 represent one simple example of a memadmittance system but may not be fully applicable to actual thin film memadmittance systems since the assumption of linearity underlies Eq. 16 and it is also assummed that the maximum value of the ionic drift w is equal to the total film thickness D which may not be true in practice. In general a set of memadmittance equations take the form given by Eq. 18 and Eq. 19. Eq.18 represents the electron drift, Eq. 19 repesents the ionic or vacancy drift and Eq. 20 represents the bounded limit of the ionic or vacancy drift.

 i(t) = electron current
q = electron charge
v(t) = voltage
R(w) = memristance function
C(w) = memcapacitance function
f(w,v,dv/dt) = ionic/vacancy drift function
wmax = maximum drift of ions or vacancies 
 
For different physical systems the memristance, memcapacitance, and drift functions may be different. Eq. 16 and 17 assume a linear behavior for these functions, however this may not be realistic for actual systems. Eq. 21 and 22 are possible alternative memristance/memcapacitance functions in terms of measurable material parameters based on an exponential dependance which may be applicable to systems based on electron tunneling. Eq. 23 and 24 represent an example of other functions which may be applicable to filamentary resistance switching systems in which the switching is based on a drift threshold dependent fuse mechanism. 
 
r = material resistivity
A = junction area
 RON = measured ON resistance
 ROFF = measured OFF resistance
 CON = measured ON capacitance
 COFF = measured OFF capacitance

 

 

Set vs. Reset Time

 
In equation 22 above no distinction was made between set and reset times. However, in reality there is such a distinction based on the internal fields generated by the ions which supplement the external field produced by the applied voltage. To compensate for the internal fields a correction factor needs to be added to the drift time. The figure below illustrates the basic reset/set states for resistance memory based on positively charged oxygen vacancies.
 
(A) Reset (high resistance) state. Oxygen vacancies are distributed throughout a region having a thickness D.
 
(B) Set (low resistance) state. Oxygen vacancies are pushed downward by external electric field (Eext). As the thickness of the vacancy region decreases the density of vacancies increases and the probability of electron tunneling between the oxygen vacancies also increases. An internal electric field (Eint) is also produced due to electrostatic repulsive forces generated by the positively charged ion vacancies. Disturbance of this state either by applying an additional voltage pulse in the same direction (in the case of unipolar switching) or applying an additional voltage pulse in the opposite direction (in the case of bipolar switching) reverts back to original state.  In the case of metal filamentary type RRAM (i.e. programmable metallization cells) the total electric field (external+internal) controls the breaking and reformation of the filaments.      
 
While the above model is framed in terms of oxygen vacancies it may equally be valid for the drift of atoms, molecules, nanoparticles, or other ion carriers capable of supporting electron transport in a thin film. The ionic drift equation is defined by a linear relationship between the time derivative of the average drift displacement w and the total electric field in the thickness D with the proportionality constant determined by the ionic mobility (Eq. 30).
 
The external field Eext may be calculated based on an externally applied voltage V reduced by a threshold voltage VT resultant from any difference between the work functions of the material forming the anode and the material forming the oxide (Eq. 31).  

 

The internal field Eint may be derived based on Gauss’s law in one dimension in terms of the ionic charge density sv of the vacancies and the permittivity e of the material being used. The ionic charge density may also be expressed in terms of the total ionic charge Qv and the volume of the ionic region (A=cross sectional area) (Eq. 32, 33).

 
 
 
 
Combining Eq. 30, 31, and 33 produces an equation for the ionic drift (Eq. 34).
 
 
From the above equation as the drift distance increases the rate of change of the drift decreases (since the natural log of a number less than 1 is negative). It would be expected that if the total field falls below a threshold field Eth than atomic or molecular electrostatic forces would become dominant and drift would not continue (i.e. dw/dt = 0). This condition for drift stability may be expressed as Eq.35. Upon removal of the external field the internal field must be less than Eth otherwise drift would resume but in the opposite direction. The drift stability condition under zero bias can then be expressed as Eq. 36. Solving Eq.36 for w provides an indication of the maximum allowable drift wmax. In the case of bipolar switching, an opposite polarity voltage can revert the drift direction back to the initial state. In the case of unipolar switching, an additional external voltage with the same polarity as the initial voltage is applied. If this voltage has sufficiently high magnitude and short duration it can move the ion drift to a point beyond wmax and provide a sufficient increase in the internal electric field so the drift reverts direction and returns to the initial state.
 
 

 

a) Filamentary ReRAM Setting

For a metal filamentary system the formation of the filament can be expected to be dependent on the electric field in the junction. If the field is above a critical level the filament will form. The ionic drift distance associated with the critical setting field (Eset)may be calculated using Eq. 33 resulting in Eq.38. The exact time necessary to reach this drift level can be found by solving the nonlinear differential equation of (Eq34). However, an approximate solution can be found assuming that the logarithmic term is slowly varying over a limited range in which case it can be approximated as a constant average value equal to half the critical field Eset.  In this case the magnitude of (Eq.34) may be expressed in terms of Eq. 39 in which voltage drop VT1 is due to any work function difference between the electrode on which the voltage is applied and the resistance switching material. Solving for the switching time by integrating (Eq.39) in the range 0<w<wset produces Eq. 40.

Combining Eq. 40 with Eq. 38 results in Eq. 41. For a sufficiently high ionic density EintAe/Qv <<1 and Eq. 11 may be approximated as Eq. 42 where sv = Qv/(DA) is the ionic charge density. It is noted that Eq. 41 and 42 assume temperature independence for the setting time. However, near the threshold level (V=VT1+EsetD/2) thermal energy associated with the oxygen vacancies introduce a probability of sub-threshold switching. To account for this effect Eq.42 can be modified in accordance with a Maxwell-Boltzmann distribution (Eq. 43) where k is the Boltzmann constant, T is the temperature, and qv is the effective charge associated with the vacancies.

 

 
Similar analysis for filamentary resetting produces similar results (Eq. 44-46) in terms of a reset electric field threshold (Ereset) required for breaking the filament.
 
  
 

Experimental Verification

 
The memadmittance model described above may be applicable to several material systems exhibiting memresistive or memcapacitive effects or combinations of these effects. One paper providing some data showing similarities to a parallel memresistive/memcapacitive is "Electric-pulse-induced capacitance change effect in perovskite oxide thin films" published by University of Houston researchers in the Journal of Applied Physics in 2006. 
 
 

Memadmittance Systems

 
While physical examples of memristors and memcapacitors have already been published in the literature there is yet no physical example of a meminductor. However, for completion the following sections provide the basic equations related to each of the four possible memadmittance systems.

Note: I am making the distinction between “memristive” and “memresistive” below to emphasize the distinction between memristive systems (which may also cover memcapacitors and meminductors) and memristive systems directly applied to resistors which I am calling “memresistors”.

 

1) Memresistor-Memcapacitor

 This system may be classified as a memadmittance system equivalent to a parallel connected memresistor and memcapacitor.
 
 

2) Memresistor-Meminductor

This system may be classified as a memadmittance system equivalent to a parallel connected memresistor (written in terms of memconductance) and meminductor.
 

 

3) Memcapacitor-Meminductor

This system may be classified as a memadmittance system equivalent to a parallel connected memcapacitor and meminductor.
 
 
 
 
 

4) Memresistor-Memcapacitor-Meminductor

This type of system may be classified as a memadmittance system equivalent to a parallel connected memresistor, memcapacitor, and meminductor.
 
 
 
 

Memimpedance Systems

The following sections describe the basic equations related to each of the four possible memimpedance systems.

Note: I am making the distinction between “memristive” and “memresistive” below to emphasize the distinction between memristive systems (which may also cover memcapacitors and meminductors) and memristive systems directly applied to resistors which I am calling “memresistors”.
j = flux, v = voltage (time derivative of flux), q = charge, i = current (time derivative of charge).

 
 

1) Memresistor-Memcapacitor

 
This type of system may be classified as a memimpedance system equivalent to a series connected memresistor and memcapacitor.
 
 

2) Memresistor-Meminductor

This type of system may be classified as a memimpedance system equivalent to a series connected memresistor and meminductor.
 

3) Meminductor-Memcapacitor

This type of system may be classified as a memimpedance system equivalent to a series connected meminductor and memcapacitor.
 
 

4) Memresistor-Meminductor-Memcapacitor

This type of system may be classified as a memimpedance system equivalent to a series connected meminductor, memresistor, and memcapacitor.
 
  

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