Optimization Method for Conductance Modulation in Ferroelectric Transistor for Neuromorphic Computing

The learning accuracy of neuromorphic computing that mimics the biological brain, is affected by the conductance‐modulation characteristics of an artificial synapse. In ferroelectric‐based devices, these characteristics are implemented using a distribution of polarization values. Therefore, the distribution in a ferroelectric thin film with various external voltage signals is investigate. As polarization switching proceeds with voltage pulse, the domains of the switched polarization become larger. In ferroelectric‐gate field effect transistors, the channel layer assumed to lie beneath the ferroelectrics experiences a local conductance change, according to the polarization distribution of the ferroelectric layer. It is found that small clusters with high conductivity become large clusters in the channel layer as the polarization switching proceeds. When the additional pulses are applied, the high conductive regions eventually connect (i.e., percolate) in the channel layer and the conductance of the layer is greatly increased. Adjusting the height of the applied voltage can slow down or speed up this phenomenon. Also, the nanosecond voltage pulses are employed and the width of the conductive pathway is adjusted. It enables to fine‐tune the conductance of the channel layer. It demonstrates that conductance modulation is optimized with an appropriate voltage pulse train pattern.


Introduction
[3][4][5][6][7][8][9] The hardware has a crossbar array structure to implement neuromorphic devices.Synapses through that neurons are interconnected in the brain are mimicked by the cross points in the crossbar array structure. [1]3][4][5][6][7][8][9] In ferroelectric transistors, the accumulation or depletion of charge carriers in the channel layer is induced by the polarization value of the ferroelectric layer. [2,3,5]In Figure 1, for example, the positive polarization regions of the ferroelectric layer induce the accumulation of charge carriers and a higher conductance in the channel layer.The potentiation/depression characteristics indicate the manner in which the conductance increases/decreases when external voltage signals are applied across the device.Linearity, which refers how linearly the conductance increases/decreases in potentiation/depression characteristics, is important for high learning accuracy. [2,3,5,10,11]][14][15][16] Although a total polarization of the film is zero in Figure 1, the accumulation in the channel layer is induced depending on a distribution of the polarization.In addition, it is related to the percolation theory because highly conductive regions connect the source and drain electrodes.Therefore, it is important to consider the distribution in the ferroelectric layer for the neuromorphic computing.However, a direct real-time investigation of the distribution is difficult because of the covering of the top electrode over the ferroelectrics.Kim et al. simulated the polarization switching dynamics with a practical distribution of ferroelectric properties by modeling the capacitor as a 2D plane. [17]They assigned the free energy values to every point on the grid of the plane.When the free energy function is changed due to the application of external voltage signal, the polarization distribution starts to change toward new stable state in the ferroelectric film.The equations for the simulation were described in "Results and Discussion" part.As a result, the simulation allowed us to investigate the polarization distribution in the ferroelectric thin film.Although the approach has the limitation that the phenomena occurring effectively in the normal direction of the film can be observed, the domain dynamics operating the ferroelectric transistor can be studied.Various voltage signals were applied across the ferroelectrics in the simulation, and we investigated the polarization maps over the film area for each signal.Then, the conductance behavior was calculated in the channel layer, which was assumed to lie beneath the ferroelectric layer.
In this work, a polarization map in the ferroelectric Hf 0.6 Zr 0.4 O 2 (HZO) thin film was simulated for the application of various voltage signals.[19] Coefficients of the free energy function and effective fields were calculated from the local coercive fields E c and the local internal bias fields E bias obtained using the first-order reversal curves (FORCs) method, respectively. [17,20]s several voltage pulses were applied across the HZO thin film one by one, the area of switched polarization gradually widened.To study the conductance modulation characteristics, we assumed that there was a source electrode, a drain electrode, and a channel layer around the HZO thin film.The conductance of the channel layer was calculated for each voltage pulse train, and eventually, conductance behavior with a very high linearity was demonstrated by optimizing the train pattern.Also, it contributed to the improvement in the learning accuracy.

Polarization Map
To obtain the simulation parameters, a hysteresis loop of the HZO thin film was measured (Figure 2a) and the distribution of switching densities was obtained using the FORCs method (Figure 2b).The FORCs method employs various voltage sweeps to activate local domains.The range of the voltage sweeps determines that domains participate in the switching.The polarization curves were obtained from various applied voltage V sweeps from V r to the maximal voltage V max , where V r was varied from V max to − V max .The switching density  SW (V,V r ) was obtained from Equation 1 by fitting the polarization P(V, V r ). [17,20] SW (V,V r ) was translated into  SW (E bias ,E c ) using Equation 2. t F is the HZO thin film thickness This distribution represents the distributions of E bias and E c .In LGD theory, the free energy is described as a polynomial function of polarization P (Equation 3). [17,18]The coefficients for LGD theory are obtained from the coercive field and the effective field is obtained considering the internal bias field.Then, we have different free energy and assign the values to every point on the grid of x-y plane.In the local site, the stable state of the free energy determines the polarization value.From the polarization data in the x-y plane, the domain configuration is obtained.
In Equation 3, index i runs from one to 21 316 to describe a uniform 146 × 146 square grid in the x-y plane. i and  i are the ferroelectric anisotropy constants (Equation 4), E eff,i is the effective field (Equation 5), and k is the domain coupling constant that compensates for the spatial nonuniformity of P. [17,18] Equation 4 was obtained using the properties of the hysteresis loop with coercive field E c and remnant polarization P r . [17]The value of P r was obtained from Figure 2a and the values of E c were obtained from Figure 2b.
where V F is the voltage across the HZO thin film.The values of E bias were obtained from Figure 2b.The L-K equation describes the dynamics of P. [17][18][19] − Damping parameter  i represents the dynamic sensitivity with that the local free energy reaches its minimum value. [17,19] F during polarization switching was measured using the circuit shown in Figure 3a.V F was obtained as V F = V S − V R , where V S is the voltage of the pulse generator and V R is the voltage across the input impedance of the oscilloscope.On the other hand, the simulation parameters  i ,  i , and E eff were calculated using Equations ( 4) and (5) and we obtained u i from Equation 3. Subsequently, the voltage transient V sim. was simulated using Equation 6 in the circuit shown Figure 3a.Here, k and  i were determined to make V sim.consistent with V F as shown in Figure 3b.Consequently, k was 10 −5 m 3 /F and  i was uniformly distributed with a minimum of 10 Ωm and a maximum of 690 Ωm.
A polarization map was obtained when the external voltage pulses V S were simulated to be applied across the circuit.A square pulse (height, 3 V; width, 0.3 μs) was simulated to be applied every 3 μs for 60 μs as shown in Figure 4a.The polarization maps were obtained when polarization state became stable in the HZO thin film.Figure 4b shows the polarization map of the HZO thin film in the initial state.Figure 4c,d,e,f show the polarization maps after applying the first, second, ninth, and nineteenth pulses, respectively.Initially, the map was mostly covered by the negatively polarized blue areas in Figure 4b.Positively polarized red dots have grown into large area and merged with neighbors just before applying the second pulse, as shown in Figure 4c.As subsequent voltage pulses were applied, the red regions have become wider, as shown in Figure 4d.Therefore, the polarization map can be modulated using the voltage pulses.Meanwhile, Figure 4e,f show practically no further change, which eventually leads to no change in the conductance despite the applied pulses.For an efficient neuromorphic device in which the conductance states are linearly modulated by the number of applied pulses, the pulse train pattern needs to be improved.

Conductance Behavior
In the case of ferroelectric transistors, these polarization distributions induce local conductance changes in the channel layer and eventually determine the overall conductance.To investigate the conductance behavior in the ferroelectric transistor, we assumed that there was a channel layer beneath the HZO thin film and that the source and drain were formed in contact with the channel layer.The thickness of the channel layer was assumed to be 20 nm.The resistivity of the channel layer, r was assumed to change from 5 × 10 −1 Ωm to 3 × 10 −3 Ωm when the polarization was >10 μCcm −2 .Even if these practical parameters are set as somewhat different values, the optimization of the voltage pulse train described below is meaningful since the tendency in the conductance behavior strongly depends on the switching behavior of the ferroelectrics, not these assumptions about the channel layer.When a voltage is applied between the source and drain electrodes, a current is obtained and the conductance is calculated by dividing the current by the voltage.The applied voltage was calculated using Equations ( 7) and ( 8).⃗ j is the current density and  is the conductivity. [21]j = − ⃗ ∇V (7) Figure 5a,b,c show the resistivity map of the channel layer under the HZO thin film in initial state, after applying the first and the nineteenth pulses, respectively.The resistivity map reflects the shape of the polarization map.With the initial pulses, small clusters of high conductance become large clusters.However, the clusters do not connect between the top and the bottom.When more additional pulses were applied, percolation of the clusters eventually occurred as shown in Figure 5c.To calculate the conductance, 3 V was applied between the top and bottom sides of the channel layer shown in Figure 5a,b,c.The voltage distributions calculated from the configurations in Figure 5a,b,c are shown in Figure 5d,e,f, respectively.As the distribution of the resistivity was almost uniform in Figure 5a, the voltage was applied uniformly, as shown in Figure 5d.Because the low-resistivity region is spread over in Figure 5b, the distribution of the applied voltage has a rather complex configuration, as shown in Figure 5e.As sharp voltage drops occurred across the remaining small clusters of high resistivity in Figure 5c, abrupt color changes were found in the corresponding region as shown in Figure 5f.This indicated that the distribution of the applied voltage reflected the shape of the resistivity map.
The distribution of the electric field shows the resistivity configuration more clearly.Figure 6a,b,c show the magnitudes of the electric field in the channel layer in the initial state, and after applying the first and the nineteenth pulses.We confirmed that the magnitude is large in the high resistivity regions but small in the low resistivity regions in each states.Figure 6d,e,f show some of the electric field vectors from Figure 6a,b,c.The current density vectors were obtained by dividing the electric field vectors by the resistivity as shown in Figure 6g,h,i.The fluctuation of arrows represents that the current flows in a complicated manner through the conductive regions in Figure 6h,i.There are only small arrows in Figure 6g, indicating the low current.The conductance values obtained from three states are 0.04, 0.15, and 0.70 μS, respectively.The meaningful increment was observed in the conductance even before the percolation.It indicates that the growth of the cluster should be useful for the conductance modulation.
Figure 7a shows different voltage pulse trains for conductance modulation.Thus far, the voltage pulse train corresponds to the pulse train A. The conductance behaviors were obtained for pulse trains A, B, C, D, E, F, G, and H, as shown in Figure 7b.For pulse train A, the conductance saturates too early because an excessively strong external voltage was applied across the HZO thin film.When pulse train B with a low voltage was applied, the conductance did not increase significantly.Therefore, a combination of high and low voltages is introduced.A gradually increasing pulse train C from 2 to 3 V can be constructed to gradually increase the conductance behavior. [2,3,5]However, the initial increase in conductance is small compared to the later increase for pulse train C.This implies that the early stage of the train should have slightly larger pulses.We employed the log function, and the height of the n-th voltage pulse, V(n) is obtained from Equation ( 9)- (11).
n, t 0 , t V , V ini , V fin , and PN are the pulse number, the time between adjacent pulses, the duration of the pulse, the height of the initial pulse, the height of the final pulse, and the total number of pulses, respectively.The linearity for pulse train D with  = 0.2 V is improved compared with that for pulse train A, B, and C. A smaller value of  resulted in a steeper increase in the early stage and a slower increase in the late stage.Therefore, the conductance behavior can be changed subtly depending on  value. of pulse train E, F, G, and H are 0.6, 0.5, 0.4, and 0.3 V, respectively.The linearity of the potentiation was obtained using Equation 12and 14. [2,3,5,10,11] G )) ( 14) are the maximum number of pulses, the conductance of the potentiation, the conductance of the depression, the maximum conductance, the minimum conductance, the linearity of the potentiation, and the linearity of the depression, respectively.The linearity for pulse train G is better than others, as shown in Table 1.Therefore, the linear conductance behavior was demonstrated by optimizing the train pattern.
Although the potentiation process was focused on so far, a similar approach can be carried out on the depression process as well.A similar pattern of pulse train will result since the structural and electrical properties of ferroelectric HZO are quite symmetric.However, a detailed tuning of the parameters will be required depending on the internal field of the thin film and on the distribution of positive and negative polarizations at the beginning of the depression process.Therefore, the depression process was also simulated for the  fine-tuning operation of the ferroelectric transistor in the next section.

Fine-Tuning Operation Under Nanosecond Pulses
The ferroelectric transistor is known for its fast operation under nanosecond pulses. [22]As polarization switching time depends on the size of the ferroelectric capacitor and the resistor in series with the ferroelectrics, it is even possible to achieve complete switching in less than a nanosecond. [23]It indicates that the polarization switching is essentially sensitive to the very short external signal.Using the advantage, quite a number of conductive states can be obtained by the pulse train composed of nanosecond pulses for fine performance in the learning process.
Since the nanosecond pulses are applied across the ferroelectrics in the non-equilibrium state for the polarization switch-ing, the ferroelectrics has to be brought into the initial state to change the polarization state.Here, the initial state is the equilibrium state in which the polarization is fully negatively poled.When the appropriate pulses are applied across the ferroelectrics in the initial state, the ferroelectrics induces the conductance of the channel layer.To change the conductance, the ferroelectrics is switched fully negatively again, and the other pulses are applied across the ferroelectrics.Therefore, it is enough for the operation with nanosecond pulses to have only the potentiation.However, our study is meaningful in providing the methodological guideline.The results below present that the pulse optimization method in section 2.2 can be utilized universally.Thus, we simulated the depression as well as the potentiation.Also, it was calculated how much the conductance characteristics from several pulse schemes improved the learning accuracy. [24]e simulated that a square pulse (height, 3 V; width, 25 ns) is applied every 30 ns 240 times and subsequently a square pulse (height, −3.66 V; width, 25 ns) is applied every 30 ns 240 times, as pulse train I in Figure 8a.Considering the internal bias field in the film, the height of the negative pulse for the depression is set as −3 V + 2E bias t F .
In Figure 8b, the conductance values were obtained after every twenty pulses were applied to the sample and after 3 μs to reach stable state.For pulse train I, the conductance saturates and diminishes too early in the potentiation and the depression, respectively, which is similar for pulse train A. The pulse train J is composed of gradually increasing voltages from 2 to 3 V and gradually decreasing voltages from −2 V − 2E bias t F to −3 V + 2E bias t F in the potentiation and the depression, respectively.When the gradually decreasing voltage from −2 to −3 V was applied in the depression, the conductance decreased rapidly in the initial part.It would originate from the different components contributing to the polarization switching.The conductance behavior for the pulse train J is non-linear as shown in Figure 8b.It indicates that the optimization method above is very effective in the finetuning operation under nanosecond pulses.The pulse train K is obtained from Equations ( 9)- (11) for the potentiation and the first half of the depression, and from Equations ( 15)-( 17) for the second half of the depression.Employing the exponential func-tion contributed to alleviate the initial steep change of the conductance in contrast to the log function.Figure 8c shows learning accuracies for identifying handwritten digits for the pulse train I, J, and K. [24] The more linear the conductance behavior was, the better the learning accuracy was.The pulse train K resulted in a high learning accuracy of 93.86%, which was close to 94.1% with the ideal device. [2](n) = − exp 16)

Percolation
In the percolation theory, the ratio of the active sites is called as the percolation threshold when the percolation occurs.The threshold is known as ≈0.593 for 2D square site. [25]In the channel layer, the connection of high conductive regions becomes the conductive pathway connecting the source and the drain when the ratio is near the threshold.We investigated the conductance for the ratio of high conductive regions shown in Figure 9.In Section 2.2, all conductance corresponds to the channel layer below or just above the percolation threshold.These conductance values could be found below the threshold ratio in Figure 9.Most results for the fine-tuning operation in Section 2.3 lay above the threshold ratio.Namely, the practical conductance modulation can be performed with either the growth of the cluster below the perco-0.00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -0.5  lation threshold or the widening of the pathway above the percolation threshold.
Last et al. reported the conductivity of a sheet of colloidal graphite paper with holes randomly punched in it. [26]The conductivity behavior is similar to our result in Figure 9.It means the nucleation is more dominant than the domain growth in the polarization switching.Otherwise, if the domain growth were dominant, the behavior would be rather different from the result of Last et al.This agrees well with the fact that the switching dynamics of many ferroelectric thin films are well described by nucleation-limited-switching model. [27,28]he polarization domain switching of the ferroelectric transistor under fast voltage pulses has been explored experimentally. [29]ue to the polarization switching through domain structure, the conductance of the ferroelectric transistor is not simply proportional to the total area of highly conductive channel regions.Thus, we studied the conductance modulation in the view of the percolation.Our simulation includes the interaction between the local sites over the ferroelectric film.We found that the domain nucleation is more dominant than the domain growth in the operation of the ferroelectric transistor.The insight is useful for modulation of the conductance in the ferroelectric transistor.Also, we established the optimization method of the voltage pulse train.The pulse scheme that varies logarithmically or exponentially allows for the fine control of the linearity.

Conclusion
Polarization maps were investigated using various external voltage signals in the ferroelectric thin films.There was mostly a negatively polarized area without any voltage signal.After applying the voltage pulse, positively polarized dots have grown into large domains and coalescence of domains has occurred.With additional voltage pulses, the domains gradually widened.However, the maps did not change further when the too many pulses were applied.This indicates that the polarization map can be modulated by an appropriate voltage pulse train, and the linearity of the conductance behavior can be improved in the channel layer of the ferroelectric transistor.Therefore, we assumed the channel layer beneath the ferroelectric film and simulated the conductance of the channel layer for different pulse scheme.
For the identical pulses or the linearly increasing pulses, the conductance either saturates too early or climbs up late.When the voltage pulses increased logarithmically, linearity was improved.Also, the conductance behavior can be changed in detail by adjusting the coefficient of the log function.Therefore, we demonstrate that the proper pace for polarization switching development is implemented in the ferroelectric thin film and the linearity of the conductance behavior is improved by optimizing the voltage pulse train pattern.
Utilizing the sensitivity of the ferroelectrics to the very short external signal, 480 conductance states were obtained with the nanosecond pulses.When the combination of the log and the exponential functions was employed, the linearity was improved.We found that the optimization method for the voltage pulse train could also be applied in the fine-tuning operation.The optimized voltage pulse train contributed the high learning accuracy of 93.86%.
Finally, we investigated the conductance for the ratio of the high conductive regions.The conductance behavior is consistent with the result of Last et al.It indicates that the nucleation is more dominant than the domain growth in the ferroelectric polarization switching.Our work gives the insight about the polarization switching in ferroelectric film and can contribute to improving the learning accuracy of neuromorphic computers.

Experimental Section
The ferroelectric HZO thin film was deposited with a thickness of 9.7 nm on a TiN-coated Si substrate using atomic layer deposition.The TiN top electrode was deposited by sputtering with a shadow mask.The top electrode was square-shaped with a side length of 146 μm.The capacitor was annealed at 600 C for 1 min at 4,5 Torr in a nitrogen atmosphere using a rapid thermal process for crystallization.
The hysteresis loop of the capacitor was measured by applying a onecycle triangular wave with an amplitude of 3 V and frequency of 1 kHz using a Radiant Technologies Precision LC II ferroelectric tester.The polarization curves were obtained from various applied electric field sweeps from a reversal electric field to the maximal electric field E max , where the reversal electric field was varied from −E max to E max , using the ferroelectric tester for the FORCs method.The voltage transient was measured from the circuit, which is a series of the capacitor and a Tektronix MDO4104C digital oscilloscope, by applying a voltage pulse using a Tabor Electrics WW1071 pulse generator.After negatively poling, a square wave with an amplitude of 3 V and period of 20 μs was applied.All the measurements were performed after the previous application of the triangular wave for 10 s using the ferroelectric tester.
A 2-layer multilayer perceptron simulator (+NeuroSim) was performed to obtain the learning accuracy for the pulse trains. [24]The Modified National Institute of Standard and Technology database was used, and the neural network was composed of 400, 100 and 10 neurons in input, hidden, and output layers, respectively.The stochastic gradient descent was used for the training algorithms and the sigmoid function was used for the activation function.

Figure 1 .
Figure 1.Schematic structure of the ferroelectric transistor.Positive polarization regions of the ferroelectric layer induce high conductive regions through that charge carriers flow easily in the channel layer.

Figure 2 .
Figure 2. a) Hysteresis loop and b) distribution of switching densities in the HZO thin film.

Figure 3 .
Figure 3. a) Schematic of the measurement circuit for k and  i .b) The voltage transient across the HZO thin film capacitor during polarization switching.

Figure 4 .
Figure 4. a) The voltage transient V sim.across the HZO thin-film capacitor under the external voltage V S .The polarization map of the HZO thin film b) in the initial state and after applying c) the first, d) the second, e) the ninth, and f) the nineteenth pulses.(This was done in simulation.).

Figure 5 .
Figure 5.The resistivity map of channel layer a) in the initial state, and after applying b) the first, and c) the nineteenth pulses.The distribution of applied voltage across channel layer d) in the initial state and after applying e) the first, and f) the nineteenth pulses.(This was done in simulation.).

Figure 6 .
Figure 6.Magnitude of electric field of the channel layer a) in the initial state, and after applying b) the first, and c) the nineteenth pulses.Some of d-f) the electric field vectors, and g-i) the current density vectors from a), b), and c) were presented.(This was done in simulation.).

Figure 7 .
Figure 7. a) Various voltage pulse trains.b) Conductance behaviors for various pulse trains.(This was done in simulation.).

Figure 8 .
Figure 8. a) Various trains composed of nanosecond pulses.b) Conductance behaviors for the trains.c) Learning accuracies for the trains.(This was done in Ratio of high conductive regions

Figure 9 .
Figure 9. Conductance for the ratio of high conductive regions in the channel layer.

Table 1 .
Linearity values for each train.