A Simplified Analytical Technique for High Frequency Characterization of Resonant Tunneling Diode

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1 [Downloaded from on Saturday, November 9, 4 at 8::9 (UTC) by Redistribution subject to AECE license or copyright.] Volume 4, Number 4, 4 A Simplified Analytical Technique for High Frequency Characterization of Resonant Tunneling Diode Ahmed Ahmed Shaaban DESSOUKI, Rania Mohamad ABDALLAH, Moustafa Hussein ALY Port Said University, Faculty of Engineering, Port Said, Egypt Arab Academy for Science,Technology and Maritime Transport, Alexandria, Egypt Abstract This paper proposes a simplified analytical technique for high frequency characterization of the resonant tunneling diode (RTD). An equivalent circuit of the RTD that consists of a parallel combination of conductance, G (V, f), and capacitance, C (V, f) is formulated. The proposed approach uses the measured DC current versus voltage characteristic of the RTD to extract the equivalent circuit elements parameters in the entire bias range. Using the proposed analytical technique, the frequency response - including the high frequency range - of many characteristic aspects of the RTD is investigated. Also, the maximum oscillation frequency of the RTD is calculated. The results obtained have been compared with those concluded and reported in the literature. The reported results in literature were obtained through simulation of the RTD at high frequency using either a computationally complicated quantum simulator or through difficult RF measurements. A similar pattern of results and highly concordant conclusion are obtained. The proposed analytical technique is simple, correct, and appropriate to investigate the behavior of the RTD at high frequency. In addition, the proposed technique can be easily incorporated into SPICE program to simulate circuits containing RTD. represent the small signal behavior of RTD. The simplest one consists of a parallel combination of RTD conductance and capacitance ((G-C) equivalent circuit) shown in Fig. -a. But it has been shown experimentally that the real and imaginary parts of RTD admittances (i.e. the measured RTD conductance and capacitance) change with frequency, instead of having a constant value. So, this simple model is not applicable for RTD at high frequencies [7]. The other two categories are a series-inductance model [8-9] and a parallel-inductance model (Fig.-b) []. The parallel-inductance equivalent circuit is a modification of the simple (G-C) equivalent circuit obtained by introducing an inductance (L) in series with the RTD conductance. Measurements of both DC I-V characteristic and microwave frequency S-parameters of AlAs/InGaAs/InAs RTDs done in [] support the validity of parallel-inductance equivalent circuit. Index Terms MATLAB, negative differential conductance (NDC), resonant tunneling diode (RTD), small signal model, SPICE. I. INTRODUCTION The double barrier quantum well (DBQW) resonant tunneling diode (RTD) is an excellent candidate for nanoelectronic circuit applications at high switching speed due to its small capacitance and high current density [-]. The negative differential conductance (NDC) of the RTD at room temperature was first demonstrated experimentally by Chang, Esaki, and Tsu in 974 []. RTDs are one of the few quantum-transport devices that operate effectively at room temperature. Most of the work on resonant tunneling (RT) devices has been entirely focused on electronic high speed applications [4]. Due to the high speed of its operation along with the presence of NDC in its I-V curve at room temperature, RTD can generate oscillations at very high frequencies which can lead to attractive microwave applications like oscillators []. Therefore, an understanding of the behavior of the RTD at high frequencies is very important. Also, the use of the resonant tunneling (RT) effect in the optical and infrared regions has been proposed by several groups [6], due to its functional characteristics, high speed response and low radio frequency power consumption. Since the discovery of the double barrier RTD, some slightly different equivalent circuits have been introduced to Figure. Small signal equivalent circuits of RTD: (a) G-C equivalent circuit and (b) parallel-inductance equivalent circuit. The main motivation for this paper comes from the theoretical and the simulation results obtained in [7], [-9]. Our aim in this work is to propose a simplified, correct, and appropriate analytical technique to investigate the behavior of the RTDs at high frequency. Also, to verify the previously published theoretical and simulation results in [], [], [8] using the proposed technique. The previously published results were obtained either by simulating the RTD at high frequency using a computationally complicated quantum simulator [], [-6] or through difficult RF measurements [7], [], []. The most of RTD small signal model equivalent circuit elements extraction approaches were based on fitting the Digital Object Identifier.46/AECE AECE

2 [Downloaded from on Saturday, November 9, 4 at 8::9 (UTC) by Redistribution subject to AECE license or copyright.] Volume 4, Number 4, 4 equivalent circuit model parameters with either measured Sparameter data [7], [], [] or using a computationally complicated numerical analysis approach [], [-6]. For such aims we need to modify the elements values of (G-C) simple equivalent circuit (Fig. -a) to be changed with frequency instead of having a constant value. This is to verify the simulation and experimental results that; the real and imaginary parts of RTD admittances change with frequency. The rest of the paper is organized as follows. In section II, the analytical expressions to calculate the values of (G-C) simple equivalent circuit elements are formulated to be changed with the frequency instead of having a constant value. In section III, the parameters extraction approach and its results introduced in [9] are reviewed for completeness purpose of our work and for reader's convenience. Section IV, contains the simulation results and its discussion. In the beginning of this section, the proposed approach to extract the modified (G-C) simple equivalent circuit elements parameters is verified through a set of comparisons between the measured and the calculated values of the modified simple equivalent circuit elements parameters (the conductance (G), the capacitance (C), and the inductance (L)). Using the proposed analytical technique, the frequency response including the highfrequency range of many characteristic aspects of the RTD is investigated in the rest of this section. In section V, the maximum frequency of oscillation of the RTD using the small signal equivalent circuits is calculated. The paper is concluded in section VI. II. MODIFICATION OF (G-C) SIMPLE EQUIVALENT CIRCUIT In this section, starting with the parallel-inductance model equivalent circuit; analytical expressions for the parameters of the modified equivalent circuit, shown in Fig., are derived. These analytically expressions are frequency dependent and can be expressed as: RD Z real Rs LC CRD () () Yimag L C L CRD RD Rs LCRs L CRD Rs For Rs : RD L (4) Yin Yreal jyimag Geq jbeq (6) (7) where, ω is the angular frequency, RD is the inverse of conductance (G), the capacitance (C) is the summation of the geometrical depletion capacitance (Co) and the quantum capacitance (Cq), the quantum inductance (L) arises from the charge storage in the well, and Rs is the metal semiconductor contact resistance [-]. Figure. Modified simple small signal equivalent circuits of RTD. Then, the equivalent conductance Geq(V, f), the susceptance Beq(V, f) and the equivalent capacitance Ceq(V, f) of the modified simple equivalent circuit, shown in Fig., can be extracted as: Geq Ceq 4 RD Rs LCRs C RD Rs L C Rs RD Rs LCRs L CRD Rs RD RD L (8) LC L CRD RD L L C L CR Yreal () and () RD RD L L C L CRD Yimag Beq L L C CRD Z imag LC CRD Yreal (9) D RD L () III. THE PROPOSED APPROACH TO CALCULATE THE EQUIVALENT CIRCUIT ELEMENTS PARAMETERS In this paper, we used the parameters extraction approach and the RTD device and its parameters introduced in [9] which is our previous work. Therefore, the parameters extraction approach and its results introduced in [9] are reviewed for completeness purpose of our work and for reader's convenience. A. Elements' Parameters Description From Eq. (6) and Eq. (8), the equivalent conductance 88

3 [Downloaded from on Saturday, November 9, 4 at 8::9 (UTC) by Redistribution subject to AECE license or copyright.] Volume 4, Number 4, 4 Geq (V, f) and the equivalent capacitance Ceq (V, f) of the modified simple equivalent circuit are calculated. We have to first calculate its parameters; the capacitance (C), quantum inductance (L) and the differential conductance (G) as a function of the applied voltage. The differential conductance (G) is defined as: di G () dv where I and V are the current and potential difference between adjacent bias points, respectively. The capacitance (C), is defined as: () C Co Cq where Co is defined as area () Co LW LB LD w p D where LW is the width of the quantum well, LB is the width of the barrier, LD is the width of the depletion region, and εw, εp and εd are the dielectric constants of the quantum well, barrier, and depletion region, respectively [-]. The quantum capacitance (Cq) depends on the differential conductance and is given by [9], [-]. Cq c G (4) where τc is the escape rate through the collector barrier. The quantum inductance (L) depends not only on the geometry of the device, but also on the scattering mechanism in the system. The quantum inductance (L) can be expressed by [- ] as: () L G where the time constant (τ) is the electron lifetime which changes exponentially with the thickness of the barrier. From the previous formula, it is noted that the inductance (L) becomes negative in the NDC region due to the negative conductance. The metal semiconductor contact resistance value (Rs) depends on the metal and material of the contact. A good result for the metal semiconductor contact to InGaAs material is Rs = Ω []. B. Elements Parameters Extraction The proposed approach starts with fitting the RTD DC analytical model [7] with the measurement of the DC current voltage characteristics of a given RTD [] in the entire bias range using MATLAB. Then, the equivalent circuit elements parameters (the quantum capacitance, quantum inductance and the conductance) are expressed as a set of related analytical mathematical expressions as a function of the applied voltage as follows. From the described model in [7], the RTD I-V model is given by: I V area J exp V J Gaussian V J mod V where J exp V A exp B V exp B V (6) (7) J Gaussian V C exp D V E J mod V C tan exp D V E D V E tan D V E (8) (9) where A, B, C, D, and E are used to achieve different RTD I-V characteristics and are extracted using the parameters extraction MATLAB routine. C controls the height of the added inverse tangent term. D controls the slope of this function, and consequently, can change the slope of the NDC. E controls the voltage of the middle point of the tangent function. Thus, it always has a value between the peak and valley voltages [7]. The derivative of the I-V curve is given by:, () J exp V AB exp B V exp B V ' J Gaussian V DC V E exp D V E V E exp D V E () ' J mod V D V E D V E D C () D and finally the differential conductance is calculated from, ' ' G area J exp V J Gaussian V J mod V () After calculating the conductance, the capacitance (C) is calculated using Eq. () to Eq. (4) and the inductance (L) is calculated using Eq. (). The geometrical capacitance (Co), τc and τ used in the above calculation are estimated by [] to be: Co = 9. ff, τc =.79 ps and τ =.8 ps. IV. SIMULATION RESULTS AND DISCUSSIONS A. Elements Parameters Extraction Simulation Results In the proposed approach, we have endeavored to calculate the RTD small signal equivalent circuit parameters, G, L and C, as a function of the bias voltage and to compare the results to those obtained on an identical RTD measured by []. The device we have studied in this paper has an area of (.6.6 µm) and the RTD (I-V) characteristics exhibits a peak- to-valley current ratio (PVCR) of. with a peak current of IP =.69 ma, peak voltage of VP =. V and valley voltage of VV =.7 V, as shown in Fig. (dashed line) []. To extract device model parameters from the measured I V characteristics, one first needs to obtain the fitting I V characteristics. Using the parameters extraction MATLAB routine, the measured RTD I-V characteristic can be fitted to the model given in [7] as shown in Fig.. The figure shows a good agreement between the measured (dashed line) RTD I-V curve [] and the fitting (solid line) RTD I-V curve. 89

4 [Downloaded from on Saturday, November 9, 4 at 8::9 (UTC) by Redistribution subject to AECE license or copyright.] Volume 4, Number 4, 4. Fitting curve Measured curve Voltage (V).6.7 Conductance (ms) Using Eq. () to Eq. (), we can calculate the differential conductance (G) as a function of the bias voltage as shown in Fig. 4. The figure shows a comparison between the calculated conductance (G) using the proposed approach and the measured one []. A fair agreement between the calculated conductance and the measured data is obtained especially near the NDC region. Measured Conductance Calculated Conductance Figure. Reciprocal of the inductance (L) with a good agreement between the (circles) measured data and the calculated (line). 4 Calculated Capacitance Measured Capacitance Figure 6. The capacitance (C) versus bias with close agreement between (circles) measured data and the calculated (line). - Fig. 4 through Fig. 6 show fair agreements between the calculated values and measured data of the equivalent circuit elements (the quantum capacitance, quantum inductance and the conductance) for different voltages. On the other hand, at some voltage values, there are mismatches between the calculated values and the measured data due to the inaccuracy in the fitting process at these voltages and the values of τc and τ that assumed to be constants as in [] Figure 4. Comparison of the (line) calculated conductance (G) with the measured (circles).. The quantum inductance (L) calculated from Eq. () is compared to that measured by [] and a good agreement of the reciprocal of inductance (L) is observed, as shown in Fig.. 9 Calculated (/L) Measured (/L) 6-6 Figure. Comparison between the measured I-V RTD curve [] and the RTD fitting I-V curve. 8 Capacitance (ff) Current (ma).. Fig. 6 shows a fair agreement between the calculated total capacitance using Eq. () and the measured capacitance in []. Reciprocal of Inductance /L (/nh) The extracted model parameters are: A =.669 e+8 A/m, B =.4 V-, C = 8.8 e+8 A/m, D =.466 V-, E =.47 V, C =.9879e+8 A/m, D =.47 V-, E =.48 V. B. Characterization of RTD at High Frequency Simulation Results Using the proposed analytical technique, the frequency responses including the high-frequency range of many characteristic aspects of the RTD are reproduced, investigated and compared with those concluded and reported in the literature.

5 [Downloaded from on Saturday, November 9, 4 at 8::9 (UTC) by Redistribution subject to AECE license or copyright.] Volume 4, Number 4, 4.. Freq (Hz). x (a) Freq (Hz) -. x (b) - Fig. 7 compares the calculated real and imaginary parts of the input impedance using our simulation values (solid lines) of the quantum capacitance, quantum inductance and conductance with those calculated ones using the measured data (dashed lines). Various bias values are used in positive differential regions (PDRs) and negative differential region (NDR), i.e. V= (inaccurate fitting point), V =.4 (accurate fitting point) in the first PDR, V =.4 (accurate fitting point) in the NDR and V =.7 (accurate fitting point) in the second PDR. It is observed that, good agreements occur at the accurate fitting voltage values and bad agreement at the inaccurate fitting voltage values. So it is recommended to enhance the fitting accuracy, as much as possible, to get better results. Also, a similar pattern of results and highly concordant conclusion were obtained as compared with []. Fig. 8 shows the real and imaginary parts of the input impedance over the bias range at different frequencies over the frequency range (4 MHZ to GHZ) to distinguish the variation in the input impedance with frequency. Input impedance peaks are observed near the (I-V) peak and valley voltages as in [] f=ghz f=ghz f=ghz Freq (Hz) (d). - x Figure 7. Real and imaginary of input impedance for the RTD at different voltages: (a) V, (b).4v, (c).4v and (d).7v x (c) -. Freq (Hz) f=ghz f=ghz f=ghz -6.. Figure 8. Real and imaginary of input impedance versus bias voltage at different frequencies (Rs = Ω). 9

6 [Downloaded from on Saturday, November 9, 4 at 8::9 (UTC) by Redistribution subject to AECE license or copyright.] Volume 4, Number 4, 4 To clarify the frequency dependence of the conductance (the real part of admittance) and the susceptance (the imaginary part of admittance), especially in the NDR, the frequency responses of the conductance and susceptance at different bias points, at.4v (NDR),.4V (NDR) and. V (PDR), are shown in Fig. 9 and Fig., respectively. Conductance (ms) V=.4V (NDR) V=.4 V (NDR) V=.V (PDR) Freq (GHz) Figure 9. Conductance versus frequency at different bias voltages (Rs = ). Susceptance (ms) V=.4V (NDR) V=.4V (NDR) V=.V (PDR) An examination of Fig. 9 indicates that the RTD conductance in the NDR is strongly frequency dependent compared to the one in the PDR. Therefore, it can be approximated by a frequency independent resistor in the positive differential region as in []. Similarly for small Frequencies, Fig. indicates that the RTD susceptance varies linearly with the frequency and therefore it can be approximated by a frequency independent capacitor as in []. Fig. shows that the total current has a non-trivial dependency with frequency as reported in []. The figure exhibits the total current when the device is biased at three different regions;.v (first PDR),.4V (NDR) and.6 V (second PDR). We noted that the current is extremely dependent on frequency at the NDR compared with other regions. Finally, for completeness purpose and to further clarify the frequency dependence of the conductance G(V,f) and the capacitance C(V,f), especially in the NDR, Fig. and Fig. show the equivalent conductance and capacitance versus the applied voltage for different frequencies, respectively. It is clear that both the conductance and the capacitance are frequency dependent in the region of maximum NDR, and almost frequency independent outside the NDR. These observations were reported experimentally in [8] and through the simulation in []. 4 8 Real (yin) (ms) f=ghz f=ghz f=ghz - Freq(GHz) Figure. Susceptance versus frequency at different bias voltages (Rs = ) Bias Voltage(V) Equivalent Capacitance (ff) Current (ma) V=.4V (NDR) V=.V (PDR) V=.6V (PDR) 4 Freq(GHz) Figure. Total current versus frequency at different bias voltages (Rs= ) Figure. Real part of the input admittance (conductance) versus the bias voltage at different frequencies. 7. f=ghz f=ghz f=ghz Figure. Equivalent capacitance versus the bias voltage at different frequencies.

7 [Downloaded from on Saturday, November 9, 4 at 8::9 (UTC) by Redistribution subject to AECE license or copyright.] Volume 4, Number 4, 4 V. RTD MAXIMUM FREQUENCY OF OSCILLATION Z real Rs Geq (4) Geq Ceq Letting the real part of the impedance be equal to zero yields the maximum oscillation frequency. The calculated RTD maximum oscillation frequency versus the bias voltage (V) is shown in Fig. 4. Max freq (GHz) Rs= ohm Rs= ohm Rs= ohm Max freq (GHz) To ensure that the proposed technique is able to characterize the high-frequency performance of the RTD in high-speed applications, the maximum frequency of the oscillation of RTD is investigated. The results obtained are compared with those reported in the literature [-]. A similar pattern of results and highly concordant conclusion are obtained as shown in the rest of this section. The upper limit for the frequency at which the RTD can be operated is given by the maximum frequency of oscillation (fmax) which is defined as the frequency above which the magnitude of the diode S is below db. This definition indicates that at the frequency above the maximum frequency of oscillation, the power gain of the RTD is below one, i.e. the RTD can no longer be treated as an active device []. An equivalent definition can also be found in [4], [-] where the maximum frequency of oscillation is defined as the frequency at which the real part of the impedance of the RTD becomes zero. For the small signal equivalent circuit shown in Fig., the real part of the input impedance is given by RD (K ohm) - - Figure. The calculated RTD maximum oscillation frequency versus RD at different Rs. VI. CONCLUSION A simplified analytical technique was developed to characterize the behavior of the RTD at high frequency. The results of the proposed technique have been compared with those of the computationally complicated quantum simulators. This comparison was done through the characterization of the high frequency response of many characteristic aspects of the RTD. A similar pattern of results and highly concordant conclusion were obtained. The proposed technique is simple, correct, effective, and appropriate to predict the behavior of the RTDs at high frequencies. Also, it is a SPICE-friendly and will be useful for analyzing and designing of compact and high frequency oscillators with RTDs. As a future work, a microwave oscillator based on a resonant tunneling device could be designed and built in simulation to verify the proposed model by calculating the oscillation frequency and comparing it with the estimated one. REFERENCES [] [] Figure 4. The calculated maximum frequency versus bias voltage (Rs = ohm).. To make it clear how the differential resistance ( RD) and the series resistance (Rs) affect the maximum frequency of oscillation, fmax is plotted versus RD at different values of series resistance Rs as shown in Fig.. The figure shows that the maximum frequency of oscillation can be increased by either by reducing the value of RD or by reducing Rs []. To reduce RD, it is necessary to achieve a high peak current, which can be achieved by reducing the barrier thickness. Furthermore, RD is dependent on the applied voltage. The minimum value of RD can be obtained by biasing the RTD near the center of the NDR region. [] [4] [] [6] [7] J. P. Sun, G. I. Haddad, P. Mazumder, and J. N. Sch /ulman, "Resonant tunneling diodes: models and properties," Proceedings of the IEEE, Vol.86, Issue 4 pp , April P. Mazumder, S. Kulkarni, M. Bhattacharya, J. P. Sun, and G. I. Haddad, "Digital circuit applications of resonant tunneling devices," Proceedings of the IEEE, Vol. 86, Issue 4, pp , April L. L. Chang, L. Esaki, and R. Tsu, Resonant tunneling in semiconductor double barriers, Appl. Phys. Lett., vol. 4, no., pp. 9-9, June H. Mizuta and T. Tanoue, The physics and applications of resonant tunneling diodes, Chapter, Cambridge University Press, Cambridge 99. ISBN: E. R. Brown, T. C. L. G. Sollner, C. D. Parker, W. D. Goodhue, and C. L. Chen, "Oscillations up to 4 ghz in gaas/alas resonant tunneling diodes," Applied Physics Letters, vol., p. 777, J. Figueiredo, B. Romeira, T. Slight and C. Ironside, Resonant Tunnelling Optoelectronic Circuits: Advances in Optical and Photonic Devices, chapter, January, INTECH, Croatia. D. R. Chowdhury (8), Experimental study and modelling of AC characteristics of Resonant Tunneling Diodes, PhD-Thesis, Chapter, Section., Technical University of Darmstadt, Germany, 8. 9

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