On Design of Memristive Amplifier Circuits

Amplifiers are essential building blocks of a majority of the mixed signal circuits that are used in the development of cognitive computing architectures. Their implementation and use is challenged by the second order effects that dominate the MOSFET operations with reduction in the technology size and scale. The ability to program the amplifiers once fabricated becomes an even more challenging problem as it warrants the use of multiple circuit components that lowers circuit performance and in turn outweighs the advantages of generalisation abilities. In this paper, a reconfigurable set of amplifier circuits are proposed based on quantised conductance devices in combination with MOSFET devices. The presented circuits form the basic configurations for the memristor based amplifiers, and show promising performance results in terms of power dissipation, on-chip area and THD values.


Introduction
The programmability of analog amplifiers is a challenging, and emerging topic of interest to developing circuits and systems required for futuristic neuromorphic computing architectures [1] [2].Traditionally, the primary objective of the amplifiers has been to amplify the voltage or current signals and also to match the impedance between circuits [3]- [5].The ability to amplify signals is critical in sensory signal processing and neuromorphic networks to improve the quality and functionality of the modules.Neuromorphic networks are arranged in layers of neurons that are interconnected in an hierarchical manner, with the ability to transmit and transform the signal from input to output for a meaningful objective [6]- [8].The ability of the networks to process the signals through learning weights over the connectivity between the network layers requires signal deformations to be minimal [9]- [11].Increase in the number of the neurons and layers necessitates amplifiers to be integrated into the signal conditioning and processing stages of the neuronal circuit designs [12]- [15].Since often the signal amplifications required are different for different layers of the network the amplifier designs would need to be tuned to fit into the signaling requirements for each layer [16] [17].The problem becomes even more challenging when the analog neuron networks are required to grow in size and functionality as the need for number of neurons increases with number of layers and inputs [18] [19], which in turn puts additional burden on the need to ensure further signal conditioning to ensure accuracy and speed of learning.
Reconfigurability of the analog circuits in terms to adjusting the open loop gain and impedances offer the designers additional flexibility to improve the reliable operation of multilayered networks [20]- [22].Similar to neuromorphic networks, another example of application where they are vital is the memory processing and sensing [23] [24].The analog to digital converters [25] [26] and data read/write circuits [27] make use of amplifiers that require different gains and bandwidths.While the goal is to ensure good performance with minimal change in design, the performance benchmarks drive designers to come up with significantly different circuit topologies for achieving parametric targets of gains, bandwidth and higher noise tolerance under low power conditions [28].The objective of this paper is to advance the idea of programmability of amplifier circuits with very little modification to circuit topology to ensure the structural and functional reproducibility of the amplifiers across the different layers of networks.To achieve this goal, we use simplistic amplifier configurations that make use of memristor like devices to configure the amplifier parameters.The low area requirements and low leakage currents of memristor like devices followed up with the quantization of conductance with a memristive state enable programmability of the circuit.

Quantised Conductance Amplifier Configuration
Owing to increased second order effects and power considerations, current chip fabrication techniques are approaching their fundamental limits.Resistive switching devices such as memristors offer a promising alternative as large scale, highly dense, programmable electronic memory network components [29].Resistive switching enables quantised conductance by reversible formation and dissolution of nanometre-scale conductive filaments, which constrain the motion of electrons.Quantum effects begin to dominate and electron transport is quantised in multiples of the fundamental quantum of conductance ( ) where e is the charge of electron and ℎ is Planck's constant) [29], thus leading to the resistive switching phenomenon where an external electric field coupled with the previous resistance state of the device determine the new resistance state.
The physical model of the memristor from [30] [31] shown in Figure 1 consists of a two-layer thin film size ( 10 TiO , sandwiched between platinum contacts.One of the layers doped with oxygen vacancies behaves as a semiconductor.The second, undopedregion, has an insulating property.As a consequence of quantised conductance process described above, the width w of the doped region is modulated depending on the amount of electric charge passing through the memristor.With electric current passing in a given direction, the boundary between the two regions is moving in the same direction.The total resistance of the memristor, M, is a sum of the resistances of the doped and undoped regions, where is the width of the doped region, referenced to the total length D of the 2 TiO layer, and OFF R and ON R are the values of the memristor resistance for 0 w = and w D = .The ratio of the two resistances is usually given as 2 3 10 10 − [30].The quantised conductance type [29] memristive devices offer different levels of conductance values within its state, that enables the discretized control of resistor values.Since resistors play a major role in the biasing and control of MOSFET amplifier parameters, the replacement of resistors with memristors can offer several advantages as compared to a resistor based design.The resistors in integrated circuits are implemented as semiconductor resistors or as MOSFET pseudo-resistors that have a disadvantage of large reverse currents undesired in a resistor.In addition, a large resistance value is often associated with large area of semiconductor resistor or limited by the threshold voltage control in case of MOSFET pseudo-resistors.In contrast, quantised conductance memristors do not have the issue of reverse leakage currents and do not require to be scaled to achieve higher resistor values.The presented reconfigurable topology of common source, common drain and differential amplifier is built by using a combination of quantised conductance memristor like devices and MOSFET devices.

Common Source Amplifier
Figure 2 shows the common source amplifier that uses memristors   ) where o r is the internal output resistance and m g is the transconductance of the MOSFET.Clearly, the ability to program the values of the D M enables the control of the amplifier open loop gain.The input resistance of the common source amplifier in r is defined in terms of the memristors 1 M and 2 M and is expressed in the Equation ( 4).The output resistance of the circuit is given as:

Common Drain Amplifier
Circuit of the common drain amplifier could be seen on Figure 3, where two memristors 1 M and 2 M are used in the same way as it was for the common source amplifier and S M is the variable memristance that is used for the control of the gain of the amplifier.The gain of the amplifier for the infinite output resistance o r is defined as follows: where r s is the reciprocal of the transconductance of the MOSFET m g .Equation (6) shows that the gain of the common drain amplifier could be controlled by changing the values of the S M .Similar to the common source configuration the output resistance of the common drain is given as: The output resistance of the circuit is defined as:

Differential Amplifer
Figure 4 represents the differential amplifier configuration with the variable memristive element .

SS M
Transistors 1 T and 2 T as well as the other two transistors 3 T and 4 T are in parallel.

Experimental Results
The quantised conductance device offers different resistor values in its working states, and we make use of this to adjust the gain of the amplifiers.The ability of the memristor like devices to be programed to different logic state enable the practical realisation of these amplifiers as tunable open loop amplifiers within integrated circuits paradigm.Our quantised conductance memristive device offer resistances of 1.72 kΩ, 1.99 kΩ, 2.15 kΩ, 3.23 kΩ, 6.45 kΩ and 8.60 kΩ.The simulations were done using BSIM models and memristors SPICE model for simulating the quantized conductance devices.The parameters from the IBM process technology for different technology size were used for the BSIM models to emulate realistic implementations.Figure 8 shows the dependency of output resistance on memristance values for three different configurations.It could be observed that the results are in agreement with the corresponding equations describing behaviour of the amplifiers.
The circuit performance in terms of the area required for implementation of the circuit, power consumption and Total Harmonic Distortion (THD) for each of the three configurations are presented in Table 1.Area for common source, common drain and differential amplifiers were calculated taking into account each transistor and memristor of the circuit.These values are lower than that uses semiconductor resistors or pseudo-resistors of MOSFETs.In addition, the simplicity in the design without the need to increase the circuit complexity makes it a useful alternative in mainstream circuit design.

Conclusion
In this paper, we presented basic integrated amplifier circuits that is programmable for output signal gains using     memristors like quantised conductance devices and MOSFET.The results of the simulations showed the improved performance of the common source, common drain and differential amplifiers in terms of the area, power and THD when implemented using the memristor elements instead of the conventional resistors or pseudo-resistors.The idea of controlling the resistance of the amplifier in an integrated circuit and the use of memristor element in the amplifier design is the new ideas suggested in this work.The proposed approach can have a widespread use in the implementation of analog amplifier circuits in sensory signal processing circuits, analog neuromorphic networks, and memory driven circuits and systems.

M and 2 M
to form the memristive po- tential divider circuit to bias the transistor T, and D M controls the gain of the amplifier, and also plays a role in the output impedance of the amplifier.The open loop voltage gain of the amplifier is given as:

Figure 5 (
a) shows the frequency response of the common source amplifier for discrete switching levels of memristive device for 0.18 μ technology of the transistor.The ability to change the resistance of the memristive device allows controlling the gain of the amplifier.

Figure 5 (Figure 5 .
Figure 5. (a) Graph for common source amplifier showing volatge gain for various values of M D ; (b) Graph for common source amplifer showing voltage gain for various technologies at memristor value of M D = 1.72 kΩ.

Figure 6 (
a) shows the change in gain magnitude of the common drain amplifier for the range of frequencies using different values of memristance for the 0.18 μ technology.Similar to the common source configuration, it is possible to control the voltage gain magnitude of common drain amplifier by implementation of memristor devices.Figure 6(b) shows the voltage gain magnitude at M S = 1.72 kΩ for the technologies having different scales.It can be seen that it is possible to have similar effect of memristor device despite the variations in technology scales.

Figure 7 (
a) shows frequency response of the differential gain magnitude for various memristor device values M SS applied with the 0.18 μ technology of the transistors.The variations in the value of memristor device make it possible to alternate the value of the differential gain for the differential amplifier.

Figure 7 (
b) shows the effect of technology scaling on the differential gain magnitude at the M SS = 1.72 kΩ.Because of the similarity in gain magnitude and bandwidth, it is possible to have practical implementation of memristor devices in differential amplifiers for different purposes.

Figure 6 .
Figure 6.(a) Graph for common drain amplifier showng volatge gain for various values of M S ; (b) Graph for common drain amplifer showing voltage gain for various technologies at memristor value of M S = 1.72 kΩ.

Figure 7 .
Figure 7. (a) Graph for differential amplifier showng volatge gain for various values of M SS ; (b) Graph for differential amplifer showing voltage gain for various technologies at memristor value of M SS = 1.72 kΩ.

Figure 8 .
Figure 8. Graph showing output resistance versus memristance values for each of three amplifer.

Table 1 .
The table indicating the performance measures for different amplifier configurations able type.