Modeling of hysteretic Schottky diode-like conduction in Pt/BiFeO3/SrRuO3 switches

The hysteresis current-voltage (I-V) loops in Pt/BiFeO3/SrRuO3 structures are simulated using a Schottky diode-like conduction model with sigmoidally varying parameters, including series resistance correction and barrier lowering. The evolution of the system is represented by a vector in a 3D parameter space describing a closed trajectory with stationary states. It is shown that the hysteretic behavior is not only the result of a Schottky barrier height (SBH) variation arising from the BiFeO3 polarization reversal but also a consequence of the potential drop distribution across the device. The SBH modulation is found to be remarkably lower ( 0.5 eV). It is also shown that the p-type semiconducting nature of BiFeO3 can explain the large ideality factors (>6) required to simulate the I-V curves as well as the highly asymmetric set and reset voltages (4.7 V and −1.9 V) exhibited by our devices.

The hysteresis current-voltage (I-V) loops in Pt/BiFeO 3 /SrRuO 3 structures are simulated using a Schottky diode-like conduction model with sigmoidally varying parameters, including series resistance correction and barrier lowering.The evolution of the system is represented by a vector in a 3D parameter space describing a closed trajectory with stationary states.It is shown that the hysteretic behavior is not only the result of a Schottky barrier height (SBH) variation arising from the BiFeO 3 polarization reversal but also a consequence of the potential drop distribution across the device.The SBH modulation is found to be remarkably lower (<0.07 eV) than previously reported (>0.5 eV).It is also shown that the p-type semiconducting nature of BiFeO 3 can explain the large ideality factors (>6) required to simulate the I-V curves as well as the highly asymmetric set and reset voltages (4.7 V and À1.9 V) exhibited by our devices.V C 2014 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4894116]][3][4][5][6][7][8][9][10] These structures exhibit rectifying and tunable conduction characteristics, do not require electroforming, and show high endurance (>10 5 cycles) and data retention (>10 5 s) at room temperature. 5,6In defining the ON (high current) and OFF (low current) conduction states, these switches take advantage of the polarization reversal property exhibited by the multiferroic BFO film when subjected to opposite electric fields. 6,7Although the connection between the orientation of the ferroelectric domains and conduction is well known, there are still many specific aspects of the electron transport mechanism that require a deeper investigation. 2,8Several models have been proposed to the date to account for the I-V characteristics of these devices, mainly space charge limited conduction, Poole-Frenkel, Fowler-Nordheim, and Schottky-like conduction, but no consensus on which is the dominant mechanism has been reached yet. 10In this letter, a compact representation for the minor and major I-V loops in Pt/BFO/SRO structures based on a Schottky diode-like conduction model in combination with sigmoidally varying parameters is explored.It is shown that the Schottky barrier height (SBH) modulation cannot explain by itself the large hysteretic loops and the remarkable asymmetry of the average coercive voltages.We found that the large band gap of BFO plays a crucial role in this connection.In addition, thanks to the well-behaved rectifying characteristics of our devices, a precise value of the SBH modulation is provided.
The devices under investigation are Au/Pt/BFO/SRO structures fabricated onto SrTiO 3 (STO) substrates.A 50 nm-thick SRO bottom electrode was grown on the substrate prior to a pulsed laser-deposited BFO layer (100 nm-thick).Both the BFO and SRO layers are epitaxially grown.An Au(100 nm)/ Pt(10 nm) top electrode is deposited on the BFO layer through a shadow mask (100 lm Â 100 lm) by E-beam evaporation.In order to control the Bi content, the BFO films are deposited from source targets with controlled Bi/Fe ratio.The Bi/Fe ratio in the film was estimated to be 0.76 6 0.05 by inductively coupled plasma atomic emission spectroscopy.Details about the fabrication and characterization of the Bi-deficient BFO films, together with a confirmation of their ferroelectricity and p-type semiconducting nature can be found in Refs.6 and 7.The I-V characteristics were measured with the top electrode (Au) grounded.
According to previous reports, 6,[11][12][13] the I-V characteristic of a metal-ferroelectric contact is given by a Schottky diode-like expression where A is the conducting area, A * ¼ 4pem * k 2 /h 3 the effective Richardson constant, T the temperature, m * the electron/ hole effective mass, e the electron charge, k the Boltzmann constant, and h the Planck constant./ B is the effective barrier height (see Fig. 1(a)), V B the voltage drop across the Pt/BFO junction barrier and n the so-called ideality factor.Assuming a first order barrier lowering effect for reverse biases (V < 0), 14 / B can be expressed as where / is the SBH, b > 0 a constant, and H the Heaviside function: Author to whom correspondence should be addressed.Electronic mail: enrique.miranda@uab.cat0003-6951/2014/105(8)/082904/4/$30.00 For typical parameter values of a rectifying structure, this effect is only significative for large positive bias (V ) 0).The fraction of downward polarized domains 0 k 1 as a function of the applied bias is modelled according to the sigmoidal expression where r is the switching rate, V S the set voltage, V R the reset voltage, and _ V the time derivative of V (see Fig. 1(b)).It can be demonstrated that assuming a box-type hysteretic model for a single ferroelectric or magnetic domain, the derivative of k yields approximately the Gaussian distribution of coercive voltages around V S and V R associated with multiple domains. 15Using (4), the state of the system is described by the vector X ¼ (I S ,a,R S ) which can be represented in a 3D space by the parametric equation where I S ¼ AA * T 2 exp(À//kT) and a ¼ e/nkT.X m ¼ (I Sm ,a m ,R Sm ) and X M ¼ (I SM ,a M ,R SM ) are the end points of the line segment shown in Fig. 1(c).To circumvent the uncertainties both in A * (because of m * ) and A (because of SBH inhomogeneities), 13 we consider within this approach I S instead of / as one of the relevant descriptors of the hysteresis loop.The solution of the implicit equation ( 1) is given by the compact expression 16 IðVÞ % ð1=aR S ÞWfaR S I S exp ½aðV þ R S I S exp ½ÀjVHðÀVÞÞ ÀjVHðÀVÞg À I S exp ½ÀjVHðÀVÞ; ð6Þ where j ¼ b/kT and W is the Lambert function.The equality in Eq. ( 6) strictly holds for the stationary ON (X M ) and OFF (X m ) states.Figures 2(a) and 2(b) show typical major hysteresis I-V loops obtained by the application of double voltage sweeps in the range À3 V !þ5 V !À3 V.The curve illustrated in Figure 2(a) corresponds to a fresh device, whereas the curve shown in Figure 2(b) corresponds to a highly cycled device.The solid lines were calculated using Eq. ( 6).
The agreement is very good but it is evident that Eq. ( 3) is a very simplistic approximation.R s , which represents the resistance of the BFO layer, might be a function of the voltage or the current as well.Notice also that the possible role played by the dead layers at both interfaces has also been neglected. 6,17Although the rectifying behavior of the devices is clearly observed in both figures, the reverse current in the second case is notably different, being dominated by a conducting phase of the form I P ¼ G P V, where G P is a parallel conductance.The origin of this Ohmic component is still unknown but could be related to a local reduction of the SBH at the domain walls close to the Pt/BFO interface caused by the accumulated damage. 5,18Let us focus the attention on the curve shown in Fig. 2(a).First, we can calculate the SBH modulation from the reverse current as without making any inference about the SBH value itself.Estimated values of / % 0.7-0.8eV were reported for different Bi-deficient BFO films. 6For the curve illustrated in Fig. 2(a), D/ % 0.07 eV is obtained at room temperature (kT ¼ 26 meV) which is far lower than previously published values: 0.6, 1 1.38, 3 0.5 eV. 4 Notice that n does not enter into (7) as proposed in Ref. 12.In the case of Fig. 2(b), the value of D/ extracted from the direct currents is negligible.One frequent problem in determining D/ is that the diode current for V < 0 is often higher than that reported here, 3,9 so that D/ is calculated indirectly assuming an arbitrary fixed location for the polarization charge (usually 1 nm from the Pt/BFO interface 12 ).Moreover, notice that V S ¼ 4.7 V and V R ¼ À1.9 V, which seem to indicate a large asymmetry in the coercive fields for both polarities.Remarkably, jV S jÀjV R j ¼ 2.8 V, which coincides with the BFO band gap (E g ¼ 2.8 eV (Ref.19)).As a consequence of this asymmetry, the ideality factors (n ¼ e/akT) for the OFF (n % 9.9) and ON (n % 6.5) currents, are extremely large and dissimilar (see Fig. 3(a)).We attribute these values to the potential drop V 0 in the BFO/SRO junction, which was not explicitly considered in the previous treatment.We suggest that part of the applied voltage V also drops across this second junction so that a transformation V !V À V 0 will be assessed next.Since the presence of interfacial layers and the non-equilibrium thermodynamic conditions (because of the current flow) rule out the application of a classical band bending approach based on the solution of the Poisson equation for a ferroelectric semiconductor, 20 we consider instead the semi-empirical voltage correction, 21 where E g ¼ 2.8 eV is the BFO band gap and g ¼ 0.55 V À1 and d ¼ 2 V are fitting constants that take into account the formation of the depletion and inversion layers (see the insets in Figs.3(a) and 3(b)).d ¼ 2 V corresponds to a moderately p-type doped semiconductor. 4 In this way, as shown in Fig. 3(a), we are able to shift the ON and OFF experimental currents to a lower voltage range so that the only difference in between them solely resides in the SBH modulation effect (D/ % 0.07 eV) induced by the polarization reversal.With this voltage correction the asymmetry of the average coercive voltages for both polarities is importantly reduced in consistency with the symmetry of the experimental polarizationelectric field (P-E) loops.The resulting lower ideality factor (n % 2.5) is compatible with the presence of an interfacial layer or with recombination in the depletion region of the BFO layer. 14Figure 3(c) illustrates the distributions of coercive voltages (dk/dV) around V S and V R with and without the voltage correction given by Eq. ( 8).The derivatives of the sigmoidal curves (solid lines) are compared to Gaussian distributions (symbols) located at V ¼ 4.7 V and À1.9 V for the original curves and at V ¼ 2.2 V and À1.6 V for the corrected curves, all with standard deviation r ¼ 0.32 V.The agreement between these two representations supports the idea of a boxtype hysteresis loop for the individual domain with normally distributed coercive voltages. 15Although the asymmetry between V S and V R is largely reduced in this way, it is clear that a complete treatment would require a self-consistent solution of the Poisson and current continuity equations across the device.Notice that the proposed model not only can account for the gradual transition from the ON to the OFF state at reverse bias but also the current crossover in the  forward bias case (see inset of Fig. 2(a)).This latest feature is not affected by the introduction of (8).Finally, Fig. 4 shows fitting results for the minor I-V loops.In this case, double voltage sweeps starting at À3 V with decreasing maximum positive voltage from 4 V to 1.5 V (see inset of Fig. 4) were applied to the device.The maximum voltage is kept below 5 V in order to reduce the damage caused to the structure.Notice that while the OFF state remains constant throughout the experiment indicating a total recovery of the structure after the erasing procedure, the ON state depends on the maximum applied voltage.The agreement between the model results and the experimental data is also very good in this case.Interestingly, it has been recently suggested that these intermediate resistance states can be used to fabricate a multilevel memory device. 7n summary, a compact representation of the hysteretic I-V characteristics in Pt/BFO/SRO resistive switches using a Schottky diode-like model was presented.The model is based on the solution of the diode equation using the Lambert W function.The ON and OFF current states are achieved by means of sigmoidally varying parameters.We showed that modeling the potential distribution across the structure is crucial for the understanding of the device behavior.

FIG. 1 .
FIG. 1.(a) Schematic of the Pt/BFO barrier including an interfacial layer.(b) Plot of the fraction of polarized domains as a function of the applied voltage.(c) System trajectory in the parameter space.X M and X m are the stationary states corresponding to the ON and OFF current levels, respectively.

FIG. 3 .
FIG. 3. (a)Modification of the experimental I-V curve using the correction term Eq. (8).The horizontal arrows indicate the shift of the coercive voltages.The inset shows the voltage correction Eq. (8) as a function of the applied bias V. (b) Band bending effect at the BFO/SRO interface for positive and negative biases.(c) Probability distribution (dk/dV) of the coercive voltages of the individual domains as a function of the applied voltage V with and without the voltage correction (8) (solid lines).Symbols correspond to fitting results using a Gaussian distribution with standard deviation r ¼ 0.32 V.

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Miranda et al.Appl.Phys.Lett.105, 082904 (2014) This work was supported by the Ministerio de Econom ıa y Competividad of Spain under Project TEC2012-32305 (with funding of the EU under the FEDER program) and by JSPS KAKENHI (No. 26286055), Japan.