Bias stress instability in organic transistors investigated by ac admittance measurements F. V. Di Girolamo, M. Barra, V. Capello, M. Oronzio, C. Romano, and A. Cassinese Citation: Journal of Applied Physics 107, 114508 (2010); View online: https://doi.org/10.1063/1.3425795 View Table of Contents: http://aip.scitation.org/toc/jap/107/11 Published by the American Institute of Physics JOURNAL OF APPLIED PHYSICS 107, 114508 共2010兲 Bias stress instability in organic transistors investigated by ac admittance measurements F. V. Di Girolamo,1,2,a兲 M. Barra,1,2 V. Capello,2 M. Oronzio,2 C. Romano,2 and A. Cassinese1,2 1 CNR-SPIN, University of Naples Federico II, Piazzale Tecchio, 80125 Naples, Italy Department of Physics Science, University of Naples Federico II, Piazzale Tecchio, 80125 Naples, Italy 2 共Received 16 February 2010; accepted 8 April 2010; published online 3 June 2010兲 In this paper, the bias stress effect 共BSE兲 in organic field-effect transistors has been analyzed by an alternative experimental approach based on ac admittance 共Y = G + jC兲 measurements. conductance 共C兲 and capacitance 共G兲 curves have been recorded as a function of frequency at different times of the bias stress experiments and simultaneously fitted through a transmission line circuit, able to separately model the conducting properties of the channel and contact regions. The determination of the time behavior of the model fitting parameters is assumed as the starting point for a quantitative analysis of the BSE occurrence. This experimental procedure clarifies that both channel resistance 共Rch兲 and contact resistance 共Rc兲 are largely affected by the BSE, while the channel capacitance 共Cch兲, related to the charge accumulation sheet, and the contact capacitance 共Cc兲 result almost unchanged. © 2010 American Institute of Physics. 关doi:10.1063/1.3425795兴 I. INTRODUCTION One of the main issues affecting the operation stability of organic field effect transistors 共OFETs兲 concerns the socalled bias stress effect 共BSE兲. This phenomenon is caused by the prolonged application of the gate-source 共VGS兲 voltage and consists in the time decay of drain-source 共IDS兲 current at any fixed drain-source 共VDS兲 voltage.1,2 The physical origin of BSE has been mainly ascribed to the presence of active trapping sites. These traps are related to intrinsic structural defects 共e.g., grain boundaries,3 organic–dielectric interface4兲 and/or to defect-attracted impurities 共e.g., water or oxygen兲.5 So far, two main experimental approaches, both in dc regime, have been used to investigate the BSE occurrence. The former relies on the analysis of time shift of the transistor threshold Vth voltage, which can be determined from the plot of the transfer-curves through geometrical criteria.1 The latter is based on the direct observation of transistor IDS time decay upon static voltage polarization 共fixed VDS and VGS兲.6 In both cases, the time behavior of the measured parameters can be modeled by formulas based on the stretched exponential function.7,8 This occurrence reflects the dispersive nature of BSE, basically accounting for a wide distribution of characteristic times in the elementary trapping processes. The physical localization of these traps 共e.g., at the interface with dielectric,4 in the film bulk or at the contact regions near the source-drain electrodes9,10兲 have been studied by different techniques. In particular, the presence of traps at the interface between the dielectric substrate and the organic semiconductor has been assessed by considering metal-insulator-semiconductor 共MIS兲 structures. Indeed, frequency-dependent admittance measurements, performed at different bias voltages on MIS capacitors, have been used to determine the interface state density and the corresponda兲 Author to whom correspondence should be addressed. Electronic mail: firstname.lastname@example.org. 0021-8979/2010/107共11兲/114508/6/$30.00 ing relaxation times.11 A similar approach has been followed also to examine the role of the interfacial states in threshold voltage instability effects.12 On the other hand, inspired by the general observation that the structural order of organic films is worse near the transistor electrodes,5 the contribution of the contact resistances 共Rc兲 to BSE has been recently investigated too. This goal has been mainly pursued by using Kelvin probe microscopy9 or the so-called transmission line 共TL兲 method.10 Both these techniques have demonstrated that contact resistances Rc is increased because of the BSE, being its variation extremely sensitive on the contact configuration 共staggered or coplanar兲 and on the electrode material and geometry. In this paper, ac admittance measurements have been directly performed on organic sexithiophene 共T6兲 OFETs, with the main goal to gain insights about the relevance of Rc in the BSE. Admittance measurements have been carried out with shortened source and drain as a function of frequency and dc bias voltages. The same configuration measurement has been recently utilized to investigate the dynamic behavior of OFETs and to discriminate the contribution of the active channel and of the drain-source contact regions in the overall device response.13 Here, we have further developed this approach by recording admittance measurements at the different times of bias stress experiments. The subsequent time evolution of the electrical parameters related to the active channel and to the contact regions suggested an alternative procedure to assess BSE influence. II. DEVICE FABRICATION AND DC CHARACTERIZATION OFETs in bottom-contact 共coplanar兲 bottom-gate configuration have been fabricated in a high vacuum 共Pr ⬇ 10−7 / 10−8 mbar兲 chamber by evaporating T6 films on 107, 114508-1 © 2010 American Institute of Physics 114508-2 J. Appl. Phys. 107, 114508 共2010兲 Di Girolamo et al. FIG. 1. 共Color online兲 共a兲 Typical transfer-curve in linear region 共VDS = −5 V兲 for a T6 transistor. In the inset, the time decay of the IDS dc current due to the BSE 共VGS = −40 V , VDS = −5 V兲. Si++共500 m兲 / SiO2共200 nm兲 substrates provided of gold source-drain interdigitated electrodes. More details about the transistor layout can be found elsewhere.14–16 During the deposition, the entire chamber was warmed at 100 ° C. The evaporation rate was 1 nm/min and the film thickness 60 nm. By this procedure, polycrystalline T6 films, with domains composed by steplike circular aggregates and showing lateral size up to 1 m, are obtained. In these films, T6 molecules are prevalently c-axis oriented 共the molecular long axis is perpendicular to the SiO2 surface兲 in agreement with other results reported in the literature.16 Electrical characterizations were performed in vacuum 共10−5 mbar兲 and in darkness. A representative transfer-curve in linear region 共VDS = −5 V兲 is reported in Fig. 1共a兲. The charge carries mobility has been estimated to be about 1 ⫻ 10−2 cm2 / Vⴱ s by using the standard metaloxide-semiconductor field-effect transistor 共MOSFET兲 equations.17 For most devices, the slope of transfer-curve reduces at high VGS, similarly to what reported in Fig. 1. This feature is basically related to the effect of Rc, whose contribution 共completely neglected in the MOSFET model兲 gets more remarkable when IDS increases and the device total resistance lowers.18 BSEs on OFET operation have been first analyzed in dc regime15 by recording the IDS time decay curves upon static polarization 共VGS = −40 V , VDS = −5 V兲 up to 1000 s. The experimental curves 共IDS versus time兲 have been fitted by the stretched exponential function 共see the inset in Fig. 1兲 冋 冉 冊册 I = I0 exp − t ␤ . 共1兲 Typical stress parameters extracted for the curve in Fig. 1 are = 5100⫾ 200 s and ␤ = 0.40⫾ 0.05. Since Eq. 共1兲 is rigorously valid only if any Rc effect can be excluded, the aforementioned approach has to be considered suitable to describe the BSE dynamics only in first approximation. III. AC ADMITTANCE MEASUREMENTS The admittance measurements 共Y = G + jC兲 have been performed by electrically shortening the source and drain FIG. 2. 共Color online兲 Capacitance 共a兲 and conductance 共b兲 measurements vs dc gate bias 共VB兲 at different frequencies. electrodes and connecting them to the low terminal of a LCR meter 共Agilent 4284A兲, while the gate contact is connected to the high one. Before the ac measurements, the device active area was isolated by scribing the organic material around it.13 The input signal at LCR meter terminals was given by the superimposition of a small amplitude ac signal 共Vac = 0.1 V兲 and dc bias 共VB兲 voltages. Capacitance-frequency 共C-f兲 and conductance-frequency 共G-f兲 curves 关see Figs. 2共a兲 and 2共b兲, respectively兴 have been simultaneously recorded by sweeping the frequency between 100 Hz and 100 KHz and setting VB in the range from ⫺40 and 40 V; C-VB and G-VB curves are collected by exchanging the roles of VB and frequency 关Figs. 3共a兲 and 3共b兲兴. C-VB curves 关Fig. 2共a兲兴 show some meaningful physical features. First, the capacitance rapidly increases going from the depletion region 共VB = 40 V兲 to the accumulation one 共VB = −40 V兲, resembling the dc current behavior in the transfer-curves. Indeed, when the device is depleted of mobile charges, the measured capacitance is given only by the parallel sum of the capacitances between drain-source electrodes and gate contact. Conversely, when the charges are accumulated at the interface with the oxide layer, a further and strongly frequencydependent capacitive contribution arises. In the quasistatic limit 共low frequency region兲, the value ⌬C = 关CACC共VB = −40 V兲 − CDEP共VB = 40 V兲兴 is theoretically equal to the channel capacitance Cch, given by the product between the active channel area and the sum of the oxide and interface capacitances per unit area.13 In the G-VB curves 关Fig. 2共b兲兴, the formation of the accumulation layer gives rise to a characteristic peak, getting smoother and smoother at increasing frequencies. The highly dispersive nature of the measured admittance measurements is represented with more evidence 114508-3 J. Appl. Phys. 107, 114508 共2010兲 Di Girolamo et al. where ␥ = 1 / L共jRchCch兲 and Z0 = 共Rch / jCch兲1/2. In this expression, ZTL represents the input impedance related to the active channel TL, while ZRC is referred to the RC loops for the contact regions. An explicit expression for C and G can be obtained by some simple mathematical manipulations of Eq. 共2兲. Indeed, by defining 1 ␣ = 共CchRch兲1/2 , 2 共3兲 c = R cC c , 共4兲 FIG. 3. 共Color online兲 Capacitance 共a兲 and conductance 共b兲 curves as a function of frequency at different VB voltages. by the C-f and G-f curves 关Figs. 3共a兲 and 3共b兲, respectively兴, yielding clearer the difference between the accumulation and the depletion regimes. The experimental admittance data have been compared with the theoretical predictions deduced by the equivalent circuit in Fig. 4共a兲. The transistor active channel is described by a TL, where the channel capacitance 共Cch兲 and channel resistance 共Rch兲 are distributed elements. This approach was initially proposed to accurately describe the response of very thin gate MOSFET devices and to determine the channel charge density and mobility.19 More recently, the TL model was considered to gain insights into the characterization of bottom-contact polymer thin-film transistors, being extended with the adoption of two lumped loops 共contact resistance Rc and capacitance Cc兲 to shape the electrical properties of the source-drain contact regions.13 According to the configuration of the ac measurements, the electrical short between drain and source renders the circuit symmetric and an equivalent open circuit is located exactly in the middle of the transistor channel. Consequently, by using the impedance transport formulas, the overall circuit admittance can be written as Y = G + jC = = 冋 冤 1 ZTL + ZRC 册 Z0 冉 冊 L tanh ␥ ⫻ 2 + Rc共1 − jRcCc兲 1 + 共 R cC c兲 2 冥 −1 , 共2兲 FIG. 4. 共Color online兲 共a兲 TL model utilized to shape the experimental admittance curves; 共b兲 active layer capacitance, and 共c兲 conductance measurements 共symbols兲 and fitting curves 共solid lines兲 at different VB. 114508-4 J. Appl. Phys. 107, 114508 共2010兲 Di Girolamo et al. Rc , 1 + 共 C cR c兲 2 c = 共5兲 R = 1 冑2 ␣ tanh I = ⫻ 1 冑2 ␣ tan ⫻ ␣ ␣ ␣ 冉 1 + tanh ␣ 冑2 − tanh ␣ 冑2 − tanh 冉 ␣ 冑2 1 + tanh tan ␣ 冑2 冑 冉 ␣ tan 2 ␣ tanh 冑 冊 ␣ 冑2 2 + tan 冑 冊 ␣ 冑2 + tan 冑2 + tanh 冑2 tan 冑2 ␣ 冉 ␣ 冑2 tan 冑2 tan 冊 , C= , 2 2 Cch Ic共R + cI兲 + CchR关1 − cCch共I − cR兲兴 , 关1 − cCch共I − cR兲兴2 + 关Cchc共R + cI兲兴2 共8兲 2 Rc共R + cI兲 − CchI关1 − cCch共I − cR兲兴 2Cch . 关1 − cCch共I − cR兲兴2 + 关Cchc共R + cI兲兴2 These formulas are used to fit the capacitance and conductance experimental curves related to the sole channel contribution, which are obtained by subtracting the C and G values in depletion mode 共VB = 40 V兲 from the C-f and G-f curves recorded at the different VB voltages. The fitting procedure is based on the determination of the four parameters: Rch, Cch, Rc, and Cc. In this study, best fittings of the curves have been achieved by minimizing the 2 through a specialized software routine 关MINUIT 共Ref. 20兲兴, which is able to fit C-f and G-f curves simultaneously. It should be highlighted that in the previous reports the data fitting was limited only to the capacitance data set.13 The procedure discussed above has been applied to the experimental 共symbols兲 curves reported in Figs. 4共b兲 and 4共c兲, which represent the channel capacitance and conductance extracted from the data in Fig. 3. The corresponding fittings are the solid lines in Figs. 4共a兲 and 4共b兲, while the resulting fitting parameters are summarized in the Table I. As it can be observed, when the OFET is in the full accumulation region 共VB between ⫺20 and ⫺40 V兲 the theoretical curves fit reasonably well the experimental data. The differences between theory and experiments get more and more evident at higher voltages 共VB = −10 and 0 V兲 when the accumulation charge sheet is only partially formed 共see also Fig. 2兲. From Table I, only Rch seems to be largely dependent on the VB value. Indeed, it varies of about 30%, 共9兲 when VB goes from ⫺40 to ⫺20 V, and shows a linear dependence on 1 / VB, in agreement with the basic MOSFET model.17 Cch is almost bias voltage independent, while both contact parameters 共Rc and Cc兲 increase of about 10%. The device total resistance RTOT = 共Rch + 2Rc兲 extracted from ac measurements results to be larger than RTOT evaluated by dc transfer-curves. This discrepancy, less then a factor of 2 at VB = −40 V, gets more and more relevant for increasing VB. This occurrence can be partially explained by considering that ac measurements are performed with shortened drainsource electrodes 共VDS = 0兲, thus neglecting any possible dependence of Rc and Rch on VDS. For completeness, a further analysis of the experimental data has been carried out by using an alternative model, recently proposed by Lenski et al.21 In their work, the authors demonstrated the effectiveness of the TL model in correctly describing the admittance response of a pentacene fieldeffect device, only if a complex and frequency-dependent sheet resistance 共Rsh兲 is used to shape the electrical properties of the active channel. In this case, no further Rc-Cc lumped loop was considered for the contact regions. The frequency-dependence of the sheet conductance 共Gsh = 1 / Rsh兲 was modeled in the framework of the so-called universal dielectric response 共UDR兲 by the expression TABLE I. Fitting parameters obtained for C and G curves versus frequency and at different VB voltages. Cch 共pF兲 Cc 共pF兲 Rc 共k⍀兲 Rch 共k⍀兲 2 冊 C and G can be written as 共6兲 G= ␣ 共7兲 ␣ 2 2 ␣ 冑2 − tanh 冑2 VB − 40 V VB = −35 V VB = −30 V VB = −25 V VB = −20 V VB = −10 V VB = 0 V 206⫾ 2 752⫾ 36 195⫾ 20 764⫾ 16 1.3 204⫾ 2 763⫾ 36 205⫾ 24 814⫾ 20 1.4 202⫾ 2 773⫾ 36 206⫾ 24 874⫾ 20 1.4 199⫾ 2 792⫾ 36 217⫾ 28 964⫾ 24 1.5 196⫾ 2 820⫾ 40 223⫾ 28 1089⫾ 24 1.6 185⫾ 2 940⫾ 80 254⫾ 40 1800⫾ 20 2.4 141⫾ 2 37⫾ 16 497⫾ 12 1310⫾ 200 4 114508-5 J. Appl. Phys. 107, 114508 共2010兲 Di Girolamo et al. FIG. 5. 共Color online兲 Comparison between C-f 共a兲 and G-f 共b兲 measurements 共symbols兲 in the full accumulation region 共VB = −40 V兲 and the corresponding fitting curves obtained by Model I 共solid line兲 and Model II 共dashed line兲. Gsh = Gdc + A ⫻ s , 共10兲 where Gdc is the dc sheet conductance and s a characteristic exponent, usually lower than 1. The UDR behavior is a more general law which was introduced to accurately describe the electrical conducting properties of various classes of disordered materials, going from purely dielectric compounds to amorphous inorganic22 and organic23 semiconductors. The UDR approach has been here followed by fitting the channel capacitance and conductance curves by the Eqs. 共8兲 and 共9兲, where the contributions related to the contact loop has been neglected and the channel resistance Rch has been defined as Rch = 冉 冊冉 冊 冉 冊冉 L w 1 L = Gsh w 冊 1 . Gdc + A ⫻ s 共11兲 It should be noted that no frequency-dependence has been considered for Cch.21 Fitting curves of C-f and G-f plots in the full accumulation region 共VB = −40 V兲, obtained by this procedure 共MODEL II兲 and compared with the results of the initial model 共MODEL I兲 accounting for the parallel Rc-Cc, are shown in Figs. 5共a兲 and 5共b兲, respectively. The best fitting curve by the Model II provides the folA = 共68⫾ 10兲 lowing parameters: Cch = 156⫾ 2 pF, ⫻ 10−9 共S*s / rad兲, and s = 0.29⫾ 0.01. The comparison between the fitting curves reveals that, differently from Model I, the UDR approach is unable to provide a good agreement between the theoretical and experimental data, even in the full accumulation region. Hence, the idea that contact regions carries out a striking role in FIG. 6. 共Color online兲 Capacitance 共a兲 and conductance 共b兲 measurements 共symbols兲 at VB = −40 V recorded at different times of the bias stress experiment and the corresponding fitting curves 共solid lines兲 by Model I. Conductance curves are shifted, for clarity, by multiplying the data at t = 150 s, t = 1000 s, and t = 3000 s, respectively, by a factor of 1.5, 2.25, and 3.375. 共c兲 Time variation in fitting ac parameters and of dc total resistance normalized at the initial values. affecting the device electrical behavior is supported by these observations and will be even strengthen from here downwards. IV. ADMITTANCE SPECTROSCOPY INVESTIGATION OF BSE Once tested the reliability of the fitting procedure based on the Model I, it was applied to fit the admittance measurements performed at different times of a bias stress experiment, with the aim to gain more insights into the occurrence of this type of electrical instability. This characterization has been performed stressing the devices by the application of VGS = −40 V for 3000 s and recording several C-f and G-f curves in the full accumulation region 共VB = −40 V兲 at different times, as shown in Figs. 6共a兲 and 6共b兲, respectively. 114508-6 J. Appl. Phys. 107, 114508 共2010兲 Di Girolamo et al. TABLE II. Fitting parameters obtained for C and G curves at different times of the bias stress experiment. Cch 共pF兲 Cc 共pF兲 Rc 共k⍀兲 Rch 共k⍀兲 2 t=0 s t = 150 s t = 500 s t = 1000 s t = 2000 s t = 3000 s 199⫾ 2 738⫾ 36 185⫾ 20 757⫾ 20 1.2 195⫾ 2 733⫾ 36 205⫾ 24 904⫾ 20 1.4 193⫾ 2 732⫾ 36 218⫾ 24 993⫾ 24 1.6 191⫾ 2 732⫾ 36 228⫾ 28 1062⫾ 24 1.7 190⫾ 2 733⫾ 36 238⫾ 28 1139⫾ 28 1.9 189⫾ 2 734⫾ 36 246⫾ 32 1195⫾ 28 2.0 Figures 6共a兲 and 6共b兲 confirm that the experimental data can be reproduced with good agreement by the theoretical fitting curves. The corresponding fitting parameters are reported in Table II and their values, normalized to the initial value 共t = 0 s兲, are plotted in Fig. 6共c兲. From the analysis of these data, it comes out that BSE is much more pronounced for the resistive parameters, while the capacitances seem to be only slightly affected. In particular, Cc is almost unchanged, supporting the idea that it is basically related to the formation of a strongly disordered and semi-insulating region near the contacts. On the other hand, the variations in Rch and Rc after 3000 s of bias stress are about 35% and 25%, respectively. Rch, which is inversely proportional to Vth in the MOSFET model, experiences the most pronounced change. However, the change 共25%兲 of Rc is not negligible, in good agreement with the recent experiments performed by different approaches.9,10 In Fig. 6共c兲, Rc and Rch curves are also compared with the variation in RTOT共t兲 = 关VDS / IDS共t兲兴, extracted from the IDS time decay, obtained in dc regime under VGS = −40 V stress. The behavior of the parameters deduced by ac analysis is slightly different form those evaluated in dc regime even in this case. This discrepancy can be justified by considering that the microscopic processes relevant in ac and dc transport are partially different. Indeed, charge elementary motions involved in ac signal transmission cannot give contribution to the formation of the percolative conducting paths, which instead provides the overall dc electrical transport.21,24 Proceeding along this subject, admittance measurements thus represent an investigation method capable to give information about the trapping of charges which do not contribute to percolative paths and are not detectable through dc measurements, eventually clarifying if they have an influence on device response. In this regard, comparing dc and ac stress properties of devices with different morphologies should be useful. V. CONCLUSIONS In conclusion, admittance measurements have been used as an alternative tool to probe the OFET response at different times of bias stress experiments. The experimental capacitance and conductance data have been simultaneously fitted through a circuit model, with frequency-independent electrical parameters, able to give insights into the conducting properties of both the active channel and drain-source contact regions. The time variations in contact and channel parameters have been examined, bringing to the conclusion that bias stress instability degrades the channel Rch and the con- tact resistance Rc in a comparable way, meanwhile the accumulated charge density is almost unchanged. The striking role played by contact resistances is also confirmed by the ineffectiveness of a model considering a frequencydependent channel conductivity in place of the circuital RC loop accounting for the electrical properties of the film regions near the source-drain electrodes. It follows that, for practical application, the Rc variations due to the BSE are not negligible and should be accurately taken into account to model the transistor response modifications. ACKNOWLEDGMENTS The authors wish to thank P. D’Angelo and F. Biscarini for stimulating discussions. A. Maggio and S. Marrazzo are also acknowledged for their technical support. 1 S. G. J. Mathijssen, M. Cölle, H. L. Gomes, E. C. P. Smits, B. de Boer, I. McCulloch, P. Bobbert, and D. de Leeuw, Adv. Mater. 19, 2785 共2007兲. 2 H. L. Gomes, P. Stallinga, F. Dinelli, M. Murgia, F. Biscarini, D. M. de Leeuw, M. Muccini, and K. Mullen, Polym. Adv. Technol. 16, 227 共2005兲. 3 D. Knipp, R. A. Street, A. Volkel, and J. Ho, J. Appl. Phys. 93, 347 共2003兲. 4 K. Suemori, S. Uemura, M. Yoshida, S. Hoshino, N. Takada, T. Kodzasa, and T. Kamata, Appl. Phys. Lett. 93, 033308 共2008兲. 5 A. Benor, A. Hoppe, V. Wagner, and D. Knipp, Org. Electron. 8, 749 共2007兲. 6 T. Miyadera, S. D. Wang, T. Minari, K. Tsukagoshi, and Y. Aoyagi, Appl. Phys. Lett. 93, 033304 共2008兲. 7 R. B. Wehrspohn, S. C. Deane, I. D. French, I. Gale, J. Hewett, M. J. Powell, J. Robertson, J. Appl. Phys. 87, 144 共2000兲. 8 R. B. Wehrspohn, S. C. Deane, I. D. French, and M. J. Powell, J. NonCryst. Solids 266–269, 459 共2000兲. 9 T. Richards and H. Sirringhaus, Appl. Phys. Lett. 92, 023512 共2008兲. 10 S. D. Wang, T. Minari, T. Miyadera, Y. Aoyagi, and K. Tsukagoshi, Appl. Phys. Lett. 92, 063305 共2008兲. 11 M. Çakar and A. Turut, Synth. Met. 138, 549 共2003兲. 12 I. Torres, D. M. Taylor, and E. Itoh, Appl. Phys. Lett. 85, 314 共2004兲. 13 B. H. Hamadani, C. A. Richter, J. S. Suehle, and D. J. Gundlach, Appl. Phys. Lett. 92, 203303 共2008兲. 14 M. Riccio, A. Irace, G. Breglio, L. Rossi, M. Barra, F. V. Di Girolamo, and A. Cassinese, Appl. Phys. Lett. 93, 243504 共2008兲. 15 F. V. Di Girolamo, C. Aruta, M. Barra, P. D’Angelo, and A. Cassinese, Appl. Phys. A 96, 481 共2009兲. 16 C. Aruta, P. D’Angelo, M. Barra, G. Ausanio, and A. Cassinese, Appl. Phys. A: Mater. Sci. Process. 97, 387 共2009兲. 17 S. Sze, Physics of Semiconductor Devices 共Wiley, New York, 1981兲. 18 D. Natali, L. Fumagalli, and M. Sampietro, J. Appl. Phys. 101, 014501 共2007兲. 19 P. M. D. Chow and K. L. Wang, IEEE Trans. Electron Devices 33, 1299 共1986兲. 20 See, for example, wwwinfo.cern.ch/asdoc/minuit/minmain.html 21 D. Lenski, A. Southard, and M. Fuhrer, Appl. Phys. Lett. 94, 232103 共2009兲. 22 S. R. Elliott, Adv. Phys. 36, 135 共1987兲. 23 M. Barra, M. Biasiucci, A. Cassinese, P. D’Angelo, A. C. Barone, A. Carella, and A. Roviello, J. Appl. Phys. 102, 093712 共2007兲. 24 K. Ryu, I. Kymissis, V. Bulović, and C. G. Sodini, IEEE Electron Device Lett. 26, 716 共2005兲.