Investigation of Loss Mechanisms in Thin Film Barium Strontium Titanate Capacitors

Transcription

1 UNIVERSITY OF CALIFORNIA Santa Barbara Investigation of Loss Mechanisms in Thin Film Barium Strontium Titanate Capacitors A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering by Nadia K. Pervez Committee in Charge: Professor Robert A. York, Chair Professor Susanne Stemmer Professor Umesh Mishra Professor Elliott Brown Lee-Yin Chen, Ph. D. December 2006

2 The Dissertation of Nadia K. Pervez is approved: Professor Susanne Stemmer Professor Umesh Mishra Professor Elliott Brown Lee-Yin Chen, Ph. D. Professor Robert A. York, Committee Chairperson December 2006

3 Investigation of Loss Mechanisms in Thin Film Barium Strontium Titanate Capacitors Copyright c 2006 by Nadia K. Pervez iii

4 Abstract Investigation of Loss Mechanisms in Thin Film Barium Strontium Titanate Capacitors Nadia K. Pervez In recent years there has been significant progress in the development of thin film ferroelectrics and related materials for tunable RF/microwave applications. A significant impediment to the widespread deployment of this technology remains. Circuit designers need accurate models for simulations. Understanding the different contributions to device performance is necessary for the development of these models. This requires the separation of intrinsic film properties from total device properties, as well as an understanding the effect of geometric scaling upon the devices. In this work, the electrical properties of thin film barium strontium titanate (BST) capacitors are studied from the perspectives of film optimization and analysis of loss contributions. Unoptimized films are found to be disordered, as indicated by the presence of unusually long time scale behavior, with leakage current transients persisting past 10 5 seconds. Reevaluation of growth conditions results in materials with improved properties: tunabilities greater than 90%, loss tangents lower than 0.67%, and transients shorter than 5 seconds. The combination iv

5 of low and high frequency measurements, along with loss calculations from measured capacitance data, are used to isolate the properties of the electrodes and electrode-film interfaces; microwave frequency losses are shown to be dominated by this grossly frequency dependent extrinsic contribution. Geometric scaling of device properties is shown to be a function of frequency, with larger area devices having lower loss at low frequencies (MHz) and smaller area devices having lower loss at high frequencies (GHz). Parasitic contributions to both low and high frequency equivalent circuit models are discussed, along with measurement considerations. v

6 Contents Abstract List of Figures List of Tables iv ix xiii 1 Introduction Preface Tunable dielectrics Barium strontium titanate Overview Device characterization Characterization goals Device structure Equivalent circuit AC measurements DC measurements Conclusion BST Growth Background Tunability Film growth Pressure series Argon/oxygen series Interfacial capacitance Conclusion vi

7 4 Loss calculation using relaxation Loss Mechanisms in BST Universal relaxation Low frequency High frequency Why does this work? Conclusion Geometry dependent quality factors Introduction Low frequency High frequency De-embedding failure Conclusion DC Leakage Introduction Time-dependent leakage currents Relaxation currents Ferroelectric transformation Isothermal current transient spectroscopy Dipolar defects Conclusion Conclusions and Future Work Summary Optimized films for circuits Low frequency geometry dependence Electrode edges Transient leakage analysis Anisotropic thin film characterization Bibliography 139 Appendices 147 A BST deposition procedure 148 B Device Process 152 vii

8 C Growth Log 155 viii

9 List of Figures 1.1 C V tuning curve showing a capacitance change of 9.88:1 under the application of a 40 V DC bias Monolithically integrated BST capacitor and AlGaN/GaN HEMT. The BST capacitor is used to block DC; a SiN capacitor for this purpose would have been as large as the entire circuit shown Unit cell of barium titanate in its (a) paraelectric state and (b) ferroelectric state (a) Parallel plate capacitor, and (b) interdigital capacitor Schematics of BST parallel plate capacitor showing (a) side view and (b) top view Equivalent circuit for (a) a capacitor with finite loss and (b) and a metallic strip Equivalent circuit for a parallel plate capacitor Frequency dependence of Q showing very low frequency, low frequency, and high frequency behaviors Example of a bridge circuit used to measure an unknown impedance Z L Illustration of 2-port network Smith charts showing S 11 for (a) a BST capacitor from 10 MHz to 20 GHz and (b) a low loss BZN capacitor from 1.5 GHz to 20 GHz [32] Comparison of (a) network analyzer and (b) impedance analyzer C V and Q V measurements at 100 MHz Comparison of (a) force voltage-measure current (FVMI) (b) force current-measure voltage (FIMV) configurations Comparison of coaxial to triaxial connections for (a) low current and (b) high current measurements ix

10 3.1 Perovskite unit cell Useful and maximum tunability MHz useful and maximum tunability of films in pressure series MHz tuning curves for films grown with different Ar/O 2 flow rates (sccm) MHz Q-factors for four films: (a) 90/10, (b) 80/20, (c) 70/30, and (d) 60/40 sccm Ar/O MHz maximum tuning curves for two most stoichiometric films, 90/10 and 80/20 sccm Ar/O Transient DC leakage current density at a constant field of 200 kv/cm for films grown with different Ar/O 2 flow rates (sccm) Inverse of capacitance density vs. film thickness Capacitance and Q vs. frequency for a BST parallel plate capacitor showing a fractional power law capacitance Capacitance and Q vs. frequency for a STO parallel plate capacitor showing a fractional power law capacitance. Device courtesy of Mr. Sean Keane Capacitance and Q vs. frequency for a BZN parallel plate capacitor showing a fractional power law capacitance. Device courtesy of Dr. Jiwei Lu Loss calculation procedure: (a) measured capacitance data (red dashed) and data fit (solid black) using extrapolated capacitance at 1 Hz, (b) measured capacitance (red solid) together with measured Q (dashed blue) and Q calculated from capacitance fit in (dashed green) Comparison of measured and calculated Q values at 1 MHz for 3 different measurement series (Ar/O 2, thickness, and temperature) Parallel plate (a) device and (b) corresponding shorted structure Measured Q vs. frequency with and without measured R s corrections, together with Q calculated from capacitance relaxation Low frequency measured Q plotted together with low and high frequency calculated Q values Illustration of the available calculation range given an integration range and a three decade window on either side of the calculation frequencies Measured low frequency Q vs. frequency for devices with different top electrode areas Measured high frequency Q vs. frequency for devices with different top electrode areas. Reactance has been corrected for L s x

11 5.3 (a) BST capacitor structure: cross section layout on left, top view photograph on right showing top electrode area A and mesa ledge with L m ; and (b) photograph of array of fabricated devices with different top electrode areas and different mesa ledge widths Illustration of (a) top electrode area A and mesa ledge width L m, and (b) surface conduction pathway MHz Q as a function of top electrode geometry P/A and mesa ledge width L m MHz Q as a function of the top electrode perimeter to area ratio (P/A) for L m = 5, 7.5, and 10 µm. Measured data points are indicated by solid dots while fits to Equation 5.5 are shown as dashed curves MHz Q as a function of the top electrode perimeter to area ratio (P/A) for L m = 12.5 and 15 µm. Measured data points are indicated by solid dots while fits to Equation 5.5 are shown as dashed curves Leakage current density, measured at 2 seconds under a 2 V bias, as a function of top electrode geometry P/A and mesa ledge width L m Incremental length of transmission line with incremental series resistance R, shunt capacitance C, and shunt conductance G MHz capacitance density as a function of top electrode geometry P/A and mesa ledge width L m Simulated Q vs. frequency for devices with different top electrode areas with a geometry-independent series resistance R s =1Ω Measured Q vs. frequency with and without measured R s corrections, together with Q calculated from capacitance relaxation. R s 0.4 Ω Electrode resistances in (a) device and (b) shorted device structures. Distributed resistances indicated by arrows Sweep direction dependence in (a) a 100 nm Ba 0.3 Sr 0.7 TiO 3 film and (b) a 100 nm SrTiO 3 film Voltage step application time dependence in a 100 nm Ba 0.5 Sr 0.5 TiO 3 film Voltage step size dependence in a 100 nm Ba 0.5 Sr 0.5 TiO 3 film Time dependent leakage current density for a Ba 0.3 Sr 0.7 TiO 3 film at an applied bias of 200 kv/cm Time dependent leakage current density for a variety of films at an applied bias of 200 kv/cm Time dependent leakage current density at an applied bias of 200 kv/cm for devices on the same sample, measurements taken 2.5 years apart xi

12 6.7 Transient leakage responses of devices with 7 different top electrode areas (in µm 2 ) under a 200 kv/cm applied bias Series of J V sweeps constructed from J t measurements on a 250 nm thick Ba 0.3 Sr 0.7 TiO 3 film J t measurements on a 250 nm thick Ba 0.3 Sr 0.7 TiO 3 film Transient leakage response of a 100 nm Ba 0.3 Sr 0.7 TiO 3 film to the rising and falling edges of an applied 200 kv/cm step /C versus temperature for the (a) unoptimized film and (b) the optimized film Capacitance vs. temperature for unoptimized BST film with long transients Capacitance vs. temperature for optimized BST film without long transients J vs. t for thin film BZN under a constant bias of 20 V. Device courtesy of Dr. Jiwei Lu Jt versus t responses of a Ba 0.3 Sr 0.7 TiO 3 film under a 2 V bias at (a) 300 K and (b) 463 K J versus t responses of a Ba 0.3 Sr 0.7 TiO 3 film under constant bias at 300 K (2 V), 463 K (1 V), and 500 K (1 V) Two devices from the edgecaps mask, (top) device with top electrode edge above an ordinary dielectric and (bottom) device with top electrode edge on BST or another tunable dielectric (a) CPW, (b) microstrip, and (c) modified CBCPW transmission lines Equipotential lines for three configurations of modified CBCPW: (a) in-plane fields: similar to CPW, (b) out-of-plane fields: similar to microstrip, and (c) both in-plane and out-of-plane fields xii

13 List of Tables 3.1 Compositional and electrical film properties for five films grown at the same total pressure of 45 mtorr, but with different argon/oxygen flow rates Properties of films from the thickness series Growth conditions for high tunability films Intrinsic film loss mechanisms Extrinsic film loss mechanisms MHz capacitance fit parameters and Q calculation results for the temperature series MHz capacitance fit parameters and Q calculation results for the Ar/O 2 series MHz capacitance fit parameters and Q calculation results for the thickness series Comparison of integral in Equation 4.10 over all frequencies with truncated contributions for a six decade window centered at the frequency of interest Data fit parameters from Figures 5.6 and C.1 Growth log from 2/21/03 to 3/21/ xiii

14 Chapter 1 Introduction 1.1 Preface A varactor is a tunable capacitor whose value can be changed through the application of a DC bias. These devices add functionality to communications circuits by making them voltage tunable. Examples of such circuits include tunable filters, phase shifters, and matching networks. Thus a wide variety of communications applications, both civilian and military, stand to benefit from advances in varactor technologies. The subject of this dissertation, thin film barium strontium titanate (BST), is a material with a field-dependent permittivity than can be used to make varactors for RF/microwave applications. BST has been studied extensively over the past decade, first for high-κ and DRAM applications [1, 2] and later for tunable applications. A significant effort has been put forth towards materials growth and optimization [3 10], as well as 1

15 Chapter 1. Introduction the design and fabrication of circuits [11 14]. Reviews on the properties of BST films can be found in [15 17]. The goal of this particular thesis is to investigate loss mechanisms in thin-film BST with the goal of bridging the gap between low frequency (MHz) film characterization and high frequency (GHz) device design. 1.2 Tunable dielectrics The term tunable dielectric refers to a material whose permittivity can be changed through the application of an external DC bias. This situation should be contrasted with an ordinary dielectric, whose permittivity remains constant under the application of an external bias. When used in place of an ordinary dielectric capacitor, a tunable dielectric varactor adds flexibility and functionality to communications electronics. The properties of a tunable dielectric are specified by the tunability n = ɛ max ɛ min, (1.1) defined by the change in permittivity (or capacitance, for a parallel plate capacitor) due to an applied bias, as illustrated in Figure 1.1, where a 40 V DC bias causes the device capacitance to drop by a factor of Alternately, another quantity called relative tunability, n r =1 1 n = ɛ max ɛ min ɛ max, (1.2) 2

16 Chapter 1. Introduction C (F) C Q n = 9.88: V Q Figure 1.1: C V tuning curve showing a capacitance change of 9.88:1 under the application of a 40 V DC bias. may be used instead. The other important design parameter is the quality factor, Q = [tan(δ)] 1 = X R, (1.3) the ratio of the capacitor s reactance to its resistance, which is a measure of loss in the dielectric. Some authors choose to define the product of the relative tunability and quality factor as a figure of merit for the material [18]. We specifically avoid this convention because it is not motivated by physical design parameters. Supposing that tunability and quality factor were equally important for all applications, a design-motivated figure of merit would contain the square root of the tunability, rather than the tunability. Figures of merit aside, the ideal varactor should 3

17 Chapter 1. Introduction have a very high tunability as well as a very high quality factor. Unfortunately these two properties are nearly always at odds for all technologies and the choice between very high tunability and very low loss must be made with specific consideration to the circuit application. Aside from tunable dielectrics, other varactor technologies include microelectromechanical systems (MEMS) and varactor diodes. Tunable dielectrics offer a number of advantages over these technologies. The best tunable dielectrics offer similar performance to the best varactor diodes, but without forward bias conduction. Tunable dielectrics have been monolithically integrated into integrated circuits [19, 20]. While varactor diode processes are clearly compatible with their respective semiconductor technologies, the difference in doping profiles for optimum performance of both transistors and varactors requires two different device-layer film growths. Thus high performance varactor diodes have no inherent advantage over tunable dielectrics for monolithic integration. In addition to tunable circuit applications, tunable dielectrics can be used to make nonlinear devices, and since they have high permittivities they can be used for applications where large capacitances are desirable, such as the blocking capacitor in Figure 1.2. At present we can divide the known tunable dielectrics into two categories, materials with and without ferroelectric instabilities. The first category includes 4

18 Chapter 1. Introduction BST capacitor Figure 1.2: Monolithically integrated BST capacitor and AlGaN/GaN HEMT. The BST capacitor is used to block DC; a SiN capacitor for this purpose would have been as large as the entire circuit shown. 5

19 Chapter 1. Introduction the subject of this dissertation, barium strontium titanate (BST). It includes both materials which undergo a ferroelectric transition (such as barium titanate) and those whose ferroelectric transition is suppressed by quantum fluctuations (such as strontium titanate). The second category of tunable dielectrics is comprised of materials whose tunability is thought to result from the rearrangement of offcenter ions [21]. Cubic pyrochlore bismuth zinc niobate (BZN) is one such nonferroelectric tunable dielectric [22 24]. 1.3 Barium strontium titanate Bulk barium titanate is ferroelectric with a T c of 393 K. [25]. Above T c, in the paraelectric phase, the structure is cubic, as illustrated in Figure 1.3(a). Below T c the cations in the unit cell shift, resulting in a net dipole moment, as shown in Figure 1.3(b). BST is a solid solution of BaTiO 3 and SrTiO 3. At room temperature, bulk strontium titanate is paraelectric. The nominal thin film BST compositions studied in this work, Ba 0.5 Sr 0.5 TiO 3 and Ba 0.3 Sr 0.7 TiO 3, are paraelectric at room temperature with ferroelectric transitions just below room temperature. This close proximity to the ferroelectric transition results in a large dielectric nonlinearity and therefore high tunability. 6

20 Chapter 1. Introduction (a) Ti 4+ Ba 2+ O 2- (b) Figure 1.3: Unit cell of barium titanate in its (a) paraelectric state and (b) ferroelectric state. 7

21 Chapter 1. Introduction The ferroelectric instability is a structural phase transition mediated by softening of a transverse optical phonon mode. At a second order transition, ω TO 0 and there is no restoring force to preserve the cubic symmetry. Thus the material spontaneously polarizes as in Figure 1.3(b). The Lyddane-Sachs-Teller relationship [26] relates the softening of ω TO to a diverging low frequency dielectric constant: ɛ(0) ɛ(ω) = ω2 LO. (1.4) ω 2 TO High tunabilities are typically associated with high permittivities, and therefore a softer transverse optical phonon mode. The highest tunabilities are therefore achieved at temperatures just above the ferroelectric transition. Intrinsic multiphonon losses are therefore higher for these high tunability materials, and represent a fundamental limitation on loss performance which would not be present for non-ferroelectric materials like BZN. However these intrinsic losses do not rule out ferroelectric materials for low loss applications; in high frequency circuits metalization and electrode losses are dominant. For some applications this means that a higher tunability film may offer better overall loss performance than a very low loss film. 8

22 Chapter 1. Introduction 1.4 Overview The overview of this dissertation is as follows. Chapter 2 discusses the techniques used to perform both AC and DC electrical characterization of our BST parallel plate capacitors. Chapter 3 discusses the optimization of sputtering deposition conditions for high tunability BST film. Chapters 4 and 5 discuss the AC loss behavior of our BST films, with Chapter 4 emphasizing the utility of using measured capacitance relaxation to predict losses, and Chapter 5 discussing the geometric scaling of device properties. Chapter 6 discusses the effect of transient relaxation behavior on device DC leakage characteristics. Chapter 7 summarizes conclusions and future work. Detailed information regarding the preparation of high tunability capacitors is found in the Appendices. Appendix A describes the BST deposition procedure. Appendix B describes the capacitor fabrication procedure. Appendix C contains the logs for the sputtering chamber used to produce the high tunability films from February 2003 to March

23 Chapter 2 Device characterization 2.1 Characterization goals Broadband characterization of electronic materials can be performed for a variety of different reasons; the study of fundamental materials properties and the evaluation of a material s suitability for specific applications are examples of different characterization goals. In general, measurements probe total device properties as opposed to intrinsic materials properties. It is therefore important to understand the different contributions to device properties, for both device optimization and materials optimization. Before embarking upon a discussion of the measurements we use to characterize our BST capacitors, it is useful to discuss our choice of a parallel plate capacitor topology, from the perspectives of characterization and circuit design. 10

24 Chapter 2. Device characterization (a) metal film metal (ground) substrate (b) metal film substrate Figure 2.1: (a) Parallel plate capacitor, and (b) interdigital capacitor. 2.2 Device structure In thin films dielectric properties are measured using either stacked or planar structures. At frequencies below 100 MHz, vertically stacked structures such as the parallel plate capacitor in Figure 2.1(a) are advantageous because in the absence of significant parasitic factors they allow for the direct measurement of film properties. At microwave frequencies parasitic resistances and inductances from the metal electrodes, along with distributed circuit effects, significantly contribute to the measured device properties. In particular, the parasitic inductance creates a self-resonance which limits the frequency range of measurements. Thus the measured device properties may be quite different from the film properties; 11

25 Chapter 2. Device characterization (a) Pt Pt BST Pt Pt sapphire substrate (b) L m A Figure 2.2: Schematics of BST parallel plate capacitor showing (a) side view and (b) top view. careful measurements and modeling are required to separate these effects from film properties. Planar structures such as the interdigital capacitor in Figure 2.1(b) are easier to fabricate as they do not require patterning or etching of the thin film. They are also better suited to microwave frequency measurements, particularly in the case of high permittivity thin films, because the electrode separation can be made larger, resulting in a smaller capacitance and therefore a higher self-resonance frequency. One important difference between vertically stacked and planar structures is that the former characterize of out-of-plane electrical properties while the latter characterize in-plane properties. This distinction is especially important in thin films because strain due to substrate-lattice mismatch can cause anisotropy. Thus the in-plane dielectric properties of a thin film may be different from the out-of-plane properties. 1 1 In Chapter 7 a series of devices inspired by conductor-backed coplanar waveguides are proposed as a means for the electrical characterization of anisotropic thin films. 12

26 Chapter 2. Device characterization (a) (b) Figure 2.3: Equivalent circuit for (a) a capacitor with finite loss and (b) and a metallic strip. A simple stacked parallel-plate device structure was used for these measurements. This structure was chosen because it has a well-defined capacitor area and is easy to fabricate. This simple structure also has lower electrode resistances and inductances when compared to standard microwave parallel plate capacitors [27]. These devices are probed directly from above with coplanar probes. This constrains the top electrode to be at least as large as the signal probe footprint. The probes used in our experiments were the Cascade Microtech ACP40-GSG-50, with 50 µm 2 pitch (probe tip spacing) and a signal probe footprint of 30 x 30 µm 2, and I40-GSG-100, with 100 µm 2 pitch and a signal probe footprint of 12 x 12 µm 2. 13

27 Chapter 2. Device characterization C Rs Ls Rp electrode, electrode-film interfaces film, film surface Figure 2.4: Equivalent circuit for a parallel plate capacitor. 2.3 Equivalent circuit The equivalent circuit for a parallel plate capacitor is shown in Figure 2.4. In this circuit, the impedances contains both intrinsic and extrinsic contributions. Specifically, the series resistance and inductance (R s, L s ) include both the intrinsic properties of the metal electrodes, as well as the contact resistances from the metal-film interfaces and inductance due to the electrodes. The capacitance and parallel resistance (C, R p ) include both intrinsic film properties along with any parallel parasitics such as fringing capacitance, inductance due to the electrical length of the device, and any contributions relating to the device mesa. 14

28 Chapter 2. Device characterization The total impedance of the device is Z = jωl s + R s + R p (1 jωr p C) 1+ω 2 R p 2 C 2. (2.1) It should be noted that the terms in (2.1), particularly C and R p, are all potentially frequency-dependent. In particular, in our devices the AC resistance contribution to R p behaves as an imaginary capacitance, following a fractional power law similar to the capacitance [28 30]. At low frequencies, when the impedance of the capacitor is high, the electrode contributions are negligible, and the equivalent circuit can be approximated by a parallel resistor-capacitor (R p -C) combination. Thus the low frequency Q, dominated by the parallel resistance, is given by Q = ωr p C. (2.2) R p contains contributions from both the film s DC and AC resistances. At very low frequencies, where the DC resistance dominates, R p is roughly constant. At higher frequencies where the film s AC resistance dominates, R p varies as ω n, where n 1. This results in a weakly frequency-dependent Q. At high frequencies the impedance of the capacitor is small so the equivalent circuit can be approximated as a series resistor-inductor-capacitor (R s -L s -C) combination, making the high frequency Q dominated by the series resistance. After factoring out the series inductance, the high frequency Q becomes Q = 1 ωr s C, (2.3) 15

29 Chapter 2. Device characterization f 1-n n~1 Log Q f 1/f very low frequency low frequency Log f high frequency Figure 2.5: Frequency dependence of Q showing very low frequency, low frequency, and high frequency behaviors. where R s has a weak frequency dependence. Thus, the variation in Q as a function of frequency can be illustrated as in Figure AC measurements A number of different measurement techniques are available for on-wafer broadband characterization. We will limit our discussion of this topic to the methods employed in our lab: bridge measurements and reflection measurements. In a bridge circuit like the one in Figure 2.6, a variable impedance is changed until no potential difference appears across the center of the circuit (indicated by nodes A and B in Figure 2.6). The bridge is then said to be balanced. The Agilent 4294A 16

30 Chapter 2. Device characterization Z known1 Zvariable V probe A B Zknown2 Z DUT Figure 2.6: Example of a bridge circuit used to measure an unknown impedance Z L. precision impedance analyzer uses a set of four auto-balancing bridges to perform measurements from 40 Hz to 110 MHz. At higher frequencies, where distributed circuit phenomena become important, reflection measurements can be used to measure impedances. Vector network analyzers, such as the Agilent 8722D and PNA series, measure the magnitude S 21 port 1 S 11 S 22 port 2 S 12 Figure 2.7: Illustration of 2-port network. 17

31 Chapter 2. Device characterization and phase of voltages transmitted and reflected from a 2-port network, such as the one illustrated in Figure 2.7. The two-port scattering parameters (also known as s-parameters) are referred to as S ij, where i refers to the port where the voltage is measured and j refers to the port where the voltage is sourced. In our measurements, the impedance of the device under test Z L is not matched to the characteristic impedance of the network analyzer, Z 0. The reflection Γ associated with this step discontinuity is Solving for the device impedance, we find Γ= Z L Z 0 Z L + Z 0. (2.4) ( ) 1+Γ Z L = Z 0, (2.5) 1 Γ where Γ = S 11. For the majority of high frequency instruments Z 0 = 50 Ω. Thus to determine the device impedance we use the measured one-port reflection parameter together with the instrument s characteristic impedance. Network analyzer calibration is performed to move the instrument s reference plane to the tips of the coplanar probes used for on-wafer measurements. There are a number of different two-port calibration schemes that can be performed; for coplanar probes, Cascade Microtech provides impedance standard substrates appropriate for the following methods: short-open-load-through (SOLT), linereflect-match (LRM), and through-reflect-line (TRL) [31]. TRL is limited in that 18

32 Chapter 2. Device characterization it requires multiple line standards for wide frequency bands. It is also inappropriate for low frequencies where the necessary lines would be prohibitively long (below 5 GHz). LRM and LRRM (with an additional reflect standard) are considered by Cascade Microtech to be the best calibration methods for broadband measurements below 20 GHz. In these methods, the reflect standards do not need characterization, and the reactance of the load standard does not need to be known. The load match does not even need to be exactly 50 Ω. In the context of our measurements, a significant drawback of LRM/LRRM is that neither can be used for one-port calibrations. SOLT, the most popular calibration method, is not the best calibration method, but it is appropriate in situations when other calibration methods do not apply, such as a two-port calibration with two different types of probes, probes that are not in the ground-signal-ground configuration, and one-port calibrations. In SOLT and its one-port variation SOL, the parasitic reactances of the open, short, and through standards must be known, and the load must be a trimmed 50 Ω resistor. For our measurements we perform one-port SOL calibrations. The quality of the calibration is assessed through measurement of short and open standards while viewing the trace data on a Smith chart display. The short should be a small resistance with a small inductive component. The open should be a large resistance with a small capacitive component. In many instances the calibration 19

33 Chapter 2. Device characterization procedure must be repeated multiple times until these results are obtained. Occasionally these results cannot be obtained because the coplanar probe parasitics are different than those specified by the manufacturer. In that situation we usually switch to a different probe, although it is possible to manual change the values of the probe parasitics used during calibration. Microwave frequency dielectric characterization using capacitor structures has inherent difficulties even before considering the impact of parasitic electrode contributions on device performance. One-port reflection measurements of these devices are very sensitive to calibration errors because the devices are reactive loads with reflection coefficients close to unity. Figure 2.8 shows reflection measurements on thin-film BST and low-loss cubic pyrochlore bismuth zinc niobate (BZN) parallel plate capacitors. The responses of both devices follow the bottom half of the unity reflection circle; self-resonance in Figure 2.8(a) is indicated by the response crossing into the upper (inductive) half of the Smith chart. In addition to the problems created by the devices high reflection coefficients, phase quantization limits the precision with which Q can be determined. Q is defined as the inverse of the loss tangent, which is the tangent of the phase difference between the voltage and current. For an ideal resistor the voltage and current are completely in phase, while for a lossless capacitor they are in quadrature. A network analyzer s phase precision limits the maximum Q that the instrument 20

34 Chapter 2. Device characterization (a) (b) Figure 2.8: Smith charts showing S 11 for (a) a BST capacitor from 10 MHz to 20 GHz and (b) a low loss BZN capacitor from 1.5 GHz to 20 GHz [32]. 21

35 Chapter 2. Device characterization C Q Impedance Analyzer 100MHz C (F) Q (a) V C Q Network Analyzer 100MHz C (F) Q (b) Figure 2.9: Comparison of (a) network analyzer and (b) impedance analyzer C V and Q V measurements at 100 MHz. V 22

36 Chapter 2. Device characterization is capable of measuring. For example, 0.1 phase precision limits the maximum Q to 573, and 0.5 corresponds to a maximum Q of 115. Thus the combination of phase quantization and poor calibration can severely limit the network analyzer s ability to accurately measure Q. Figure 2.9 shows the same capacitance and Q versus voltage measurement performed on both a network analyzer and an impedance analyzer. Note that while the capacitance curves are identical, the Q curves are not. At higher voltages the network analyzer measurement is off by 2 orders of magnitude, and in fact the Q values become negative as the small measured resistance becomes negative. 2.5 DC measurements In addition to the AC measurements described in the preceding section, DC measurements are used to characterize conduction mechanisms and transient behavior of the devices. Two different methods can be used to evaluate leakage current behavior: force voltage-measure current (FVMI) and force current-measure voltage (FIMV), as shown in Figure In FVMI a voltage is applied between the terminals of the device and the resulting leakage current is measured. This technique is most appropriate for high resistance loads like our BST capacitors, where the resistance of the cables in the measurement setup is much lower than 23

37 Chapter 2. Device characterization (a) V applied + DUT I measured (b) I applied DUT V measured Figure 2.10: Comparison of (a) force voltage-measure current (FVMI) (b) force current-measure voltage (FIMV) configurations. the device under test. For low resistance loads FIMV, where a current is applied and the voltage that develops between the device electrodes is measured, is more appropriate as it allows for a four-probe configuration to factor out cable resistances. The Agilent 4155B semiconductor parameter analyzer and Keithley 6517A electrometer used for measurements in our lab both have triaxial cable connections. Since the probes used for measurement have coaxial connectors, it is nec- 24

38 Chapter 2. Device characterization (a) (b) Figure 2.11: Comparison of coaxial to triaxial connections for (a) low current and (b) high current measurements. essary to use an adapter to convert from triax to coax. A triaxial cable has three concentric conductors, where the inner conductor carries the signal, the middle conductor, guard, is held at the same potential as the inner conductor, and the outer conductor is grounded. In a coaxial cable the inner conductor carries the signal while the outer conductor is grounded. Thus when connecting triax to coax, the inner conductors are connected together, but the outer conductor of the coax may be connected to either guard or ground. For low current measurements, the outer coax conductor should be connected to guard as it eliminates the leakage current contribution from the coaxial cable. It should be noted that in this configuration the measurement voltage appears on the outer conductor of the coax, eliminating the aforementioned leakage pathway, but creating a possible hazard 25

39 Chapter 2. Device characterization in the lab. Thus for high current measurements the outer coax conductor should be grounded. 2.6 Conclusion Parallel plate capacitors, interdigital capacitors, and transmission lines can be used to characterize thin film dielectrics such as BST. We elect to use parallel plate devices as our BST test structures for two reasons. The first reason is that their properties approach film properties at low frequencies, making them useful for basic film characterization. The second reason is that the parallel plate capacitor structure is the preferred topography for circuit applications because of its high tunability and compact size. A combination of DC, low frequency, and high frequency measurements are used to characterize the response of BST capacitors. The loss behavior in BST capacitors can be divided into three regimes: very low frequency where the film s DC conductivity dominates, low frequency where the film s AC conductivity dominates, and high frequency where the electrode resistance dominates. 26

40 Chapter 3 BST Growth 3.1 Background Barium strontium titanate is a solid solution of barium titanate and strontium titanate, both with the ABO 3 perovskite structure illustrated in Figure 3.1. The films for this work were deposited by RF-magnetron sputtering, an energetic physical vapor deposition technique. In RF-magnetron sputtering, an RF plasma is used to knock atoms off of a target. These atoms then land on the substrate, which must be maintained at a high enough temperature for the adsorbed atoms to be able to migrate on the surface and crystallize properly. At the same time the temperature must be low enough that these species are not evaporated. Oxygen has a low sticking coefficient, making films sputtered from stoichiometric oxide targets oxygen deficient [33]. Regardless of the deposition method employed, per- 27

41 Chapter 3. BST Growth ovskite structures have a tendency to form oxygen vacancies [34]. To compensate for both of these factors, oxygen is added to the argon sputtering ambient. In our experiments we have found that bottom electrode coverage significantly affects substrate temperature and therefore the resulting film quality. The use of pre-patterned electrodes results in a lower overall substrate temperature, causing higher titanium incorporation through decreased evaporation. 1 The pressure of the sputtering ambient, in our case a mixture of argon and oxygen, also affects the stoichiometry. In particular, the pressure has a large impact on titanium incorporation as it is much smaller than the other cations. It should be mentioned that a key difference between bulk and thin film materials is strain. It is well known that strain couples to dielectric and ferroelectric properties. Thin film strain changes T c [35], and even changes the nature of the ferroelectric phase transition in BST from first order (discontinuous) to second order (continuous) [36]. The perovskite structure easily accommodates impurities present in the deposition system, making a clean system with a low base pressure a necessity. These impurities may act as dopants or cause oxygen vacancies, which can each delocalize up to two electrons [38]. Oxygen vacancies can affect the material prop- 1 Our experiments have shown that a BST film deposited concurrently on to platinized and bare half-wafers of sapphire etch at different rates. The films on bare sapphire etch much faster. When compared to films on blanket platinized substrates, BST films deposited on the pre-patterned electrodes etch faster, have lower zero-bias permittivities, and lower tunabilities. 28

42 Chapter 3. BST Growth Ti O Ba or Sr Figure 3.1: Perovskite unit cell. erties in at least three ways. In isolation, oxygen vacancies can act as electron traps. In higher concentrations they can also cause nano-polar regions to form within the material. A high concentration of nano-polar regions can make an ostensibly paraelectric material ferroelectric, while a lower concentration may leave local polarizations uncorrelated, causing a broad distribution of energetic barriers (relaxation times) within the material [37]. 3.2 Tunability BST has a field dependent permittivity. Tunability, n, is the measure of this field dependence; it is defined as the ratio of the maximum permittivity at zero 29

43 Chapter 3. BST Growth C Q rollover Q 140 maximum tunability 10.15:1 C (F) useful tunability 5.36: Q V catastrophic device failure Figure 3.2: Useful and maximum tunability. bias, to the permittivity at some field E: n = ɛ(0) ɛ(e). (3.1) The relative tunability or percent tuning, n r, is defined as the ratio of the change in permittivity at field E to the zero bias permittivity: n r = ɛ(0) ɛ(e). (3.2) ɛ(e) Tunability values are typically obtained by progressively increasing the swept voltage until catastrophic failure. While this is the convention followed in the lit- 30

44 Chapter 3. BST Growth erature, it is not a practical method for characterization of the useful portion of the tuning curve. Device Q-factor decreases dramatically above a certain critical field, which is associated with an exponential increase in leakage currents. Determination of the maximum sustainable field in the films for a practical application would require careful, long-duration stress testing, but a rough approximation would be the field at which the leakage losses approach the intrinsic dielectric loss. With the needs of circuit designers in mind, we define a quantity called the useful tunability as the tunability measured using the portion of the ɛ r E curve below the field where Q starts to decrease. Figure 3.2 illustrates this concept by showing the difference between the useful and maximum tunabilities for a C V measurement. While much work on growth optimization has focused on reducing material losses in BST for microwave devices [4, 5], tunability is also a critical parameter in determining overall circuit loss in an application. Figures-of-merit involving products of intrinsic material quality and tunability are sometimes proposed for comparing materials, but these are not always useful predictors of circuit performance. In the microwave range, losses in the capacitor s electrode metalization become significant, mitigating the advantages of low loss-tangent films. In this work, we choose to emphasize improved tunability as a means for reduction of circuit losses. Higher tunabilities allow for designs with fewer cascaded tuning el- 31

45 Chapter 3. BST Growth ements, directly reducing the net circuit loss. Phase shifter circuits, for example, require a predetermined amount of phase delay, and a number of individual phase shifting units are cascaded in order to achieve this goal. When the film tunability is increased, the amount of phase shift per unit is increased, thereby decreasing the required number of units [13]. 3.3 Film growth The role of the oxygen ambient during RF-magnetron sputtering of BST films has been specifically studied for its influence on leakage [6], lifetime [7], and film texturing [39]. We chose to optimize the electrical tunability of BST capacitors with platinum electrodes with respect to the oxygen partial pressure during sputter-deposition, leading to significantly higher tunabilities than previously reported. The effects of oxygen concentration during sputtering appear to be complex; the ambient oxygen partial pressure influences not only the oxygen incorporation into the films, but also the titanium non-stoichiometry and the A/B site ratio which are strongly linked to the zero-field permittivity and loss [40]. Additionally, evidence of changes in the platinum electrode morphology when exposed to oxygen at high temperatures has been reported [41]; thus the resulting film texture and interface quality are influenced by the oxygen partial pressure. 32

46 Chapter 3. BST Growth The films in these experiments were co-sputtered from two three inch ceramic targets in an argon/oxygen ambient. The two targets had slightly different compositions, one being Ba 0.5 Sr 0.5 TiO 3 (stoichiometric) and the other Ba 0.5 Sr 0.5 Ti 1.02 O 3 (2% excess titanium). The targets were located 30 off axis and the target to substrate distance was The RF power on each gun was 150 W. The heater set temperature of 800 C corresponded to a surface temperature of 700 Con the platinized c-plane sapphire substrate with a 500 nm backside titanium layer. The heater was ramped up from room temperature at 50 C/minute and down at 20 C/minute. The argon and oxygen flows were started when the sample reached 200 C, and continued post-deposition until the samples cooled below 200 C. The guns were ramped up to 150 W over a five minute period. The sample holder was rotated during deposition at 10 rpm. The deposition times for the different growths in these experiments were held constant at 64 minutes. Parallel plate capacitors were fabricated using samples grown on c-plane sapphire substrates with 200 nm platinum ground planes in a two-layer mask process. The first layer defined BST mesas for a device isolation etch in buffered hydrofluoric acid. The second mask layer opened windows for lift-off of mesa and ground plane electrodes. 150 nm thick platinum electrodes were deposited by electron-beam evaporation. A more detailed description of the device fabrication procedure can be found in Appendix B. 33

47 Chapter 3. BST Growth 12 7 Maximum Tunability (100MHz) Useful Tunability (100MHz) Pressure (mtorr) Figure 3.3: 100 MHz useful and maximum tunability of films in pressure series. 3.4 Pressure series In the first growth optimization experiment, the pressure series, the sputtering ambient s argon/oxygen flow rates were kept constant at 90/10 sccm respectively, while the overall pressure was varied from 35 mtorr to 50 mtorr in 5 mtorr increments. This corresponded to oxygen partial pressures of 3.5, 4.0, 4.5, and 5.0 mtorr, respectively. All other growth conditions were held constant. 34

48 Chapter 3. BST Growth Capacitance and Q tuning curves were measured on µm 2 capacitors using an Agilent 4294A impedance analyzer with an oscillation amplitude of 500 mv. Maximum and useful tunabilities were then calculated. Figure 3.3 shows average results for maximum and useful tunability from the pressure series at 100 MHz. The film grown at 45 mtorr clearly outperforms the other films in this series. For this reason, 45 mtorr was chosen as the overall pressure for the second growth optimization experiment. 3.5 Argon/oxygen series In the second experiment, the argon/oxygen series, the total ambient pressure was held constant at 45 mtorr while the oxygen partial pressure was varied. All other growth conditions were held constant. The argon/oxygen flow rates for the films in this set were 90/10, 80/20, 70/30, 60/40, and 50/50 sccm corresponding to oxygen partial pressures of 4.5, 9.0, 13.5, 18.0, and 22.5 mtorr, respectively. Films were simultaneously deposited on to both platinized and bare c-plane sapphire substrates with backside titanium layers. The films grown on platinized substrates were processed into parallel plate capacitor test structures, while the films grown on bare sapphire were sent out for Rutherford backscattering spectroscopy (RBS) analysis. The resulting film thicknesses were determined by surface profilometry 35

49 Chapter 3. BST Growth Table 3.1: Compositional and electrical film properties for five films grown at the same total pressure of 45 mtorr, but with different argon/oxygen flow rates. Ar/O 2 P O2 Ba/ (Ba+Sr)/ ɛ r Q Tunability Useful (sccm) (mtorr) Sr Ti (1 MHz) (100 MHz) tunability 90/ :1 6.46:1 80/ :1 8.14:1 70/ :1 4.20:1 60/ :1 4.60:1 50/ :1 3.63:1 on the electrical test samples, and ranged from 113 nm to 145 nm with lower oxygen partial pressures corresponding to higher deposition rates. Capacitance and Q tuning curves were measured using an Agilent 4294A impedance analyzer for low voltage measurements at 1 MHz and 100 MHz with a 500 mv oscillation amplitude. High voltage (>40 V) measurements at 100 MHz were obtained using an HP 8722D network analyzer with an external bias tee. The high voltages were sourced using a Keithley 6517A electrometer with a series current limiting resistance of approximately 50 kω. DC leakage current measurements were performed using an Agilent 4155B semiconductor parameter analyzer. Compositional data was determined through Rutherford backscattering spectroscopy (RBS) analysis on the samples grown on bare sapphire. Table 3.1 summarizes the electrical measurement and RBS analysis results. Electrical data in each row was verified on multiple devices from each sample. In materials such as BST with the ABO 3 perovskite structure, the ratio of A to B site ions is an important predictor of the defect density and quality of the film 36

50 Chapter 3. BST Growth structure [42]. RBS sensitivity to oxygen is poor, making the most useful results of the analysis the A to B site ratio, and in the case of a solid solution such as BST, the ratio of the different cations occupying the A site as well. It should be noted that although the samples for RBS and electrical measurements were grown together, the absence of a platinum ground plane on the RBS samples resulted in a lower surface temperature during deposition, and therefore less titanium evaporation (i.e., higher incorporation). As the oxygen partial pressure was increased from 4.5 mtorr to 22.5 mtorr the titanium non-stoichiometry increased, as indicated by the decrease in the A to B site ratio from unity to Figures 3.4 and 3.5 show the effect of the excess titanium on the electrical properties of the film. The ɛ r E curves in Figure 3.6 were obtained by progressively increasing the voltage sweep until catastrophic device failure. The tunability figures reported in Table 3.1 were then determined by taking the ratio of the maximum (zero-bias) permittivity to the minimum permittivity at breakdown. The useful tuning results reported in Table 3.1 were estimated using the field where Q starts to decrease. Maximum tunability measurements depicted in Figure 3.6 show that despite its more desirable zero-bias dielectric properties, the 90/10 film ultimately had a lower tunability (11.6:1 or 91%) than the 80/20 film (13.7:1 or 93%) due to its lower dielectric strength. All of the films in the series asymptotically approach the 37

51 Chapter 3. BST Growth /10 80/20 70/30 60/40 50/50 er E (MV/cm) Figure 3.4: 1 MHz tuning curves for films grown with different Ar/O 2 flow rates (sccm). 38

52 Chapter 3. BST Growth 300 (a) 300 (b) Q Q Q E (MV/cm) (c) E (MV/cm) (d) Q E (MV/cm) E (MV/cm) Figure 3.5: 1 MHz Q-factors for four films: (a) 90/10, (b) 80/20, (c) 70/30, and (d) 60/40 sccm Ar/O 2. 39

53 Chapter 3. BST Growth /10 80/ er /20 breakdown 90/10 breakdown E (MV/cm) Figure 3.6: 100 MHz maximum tuning curves for two most stoichiometric films, 90/10 and 80/20 sccm Ar/O 2. 40

54 Chapter 3. BST Growth same high-field permittivity limit, as shown in Figure 3.4, but the 90/10 film broke down before approaching this limit. Figure 3.5 indicates the onset of significant leakage in the 90/10 film at lower fields, so its useful tuning range is smaller as well. Although additional improvement in the breakdown voltage and leakage resistance was observed with the further incorporation of titanium into the film, the significant decrease in the zero-field permittivity limits the usefulness of such an approach for high tunability applications. Thus the 80/20 film appears to be an optimum combination of tunability and dielectric strength for this series. The large tunabilities in Table 3.1 exceed other reported data [8, 43], due to the simultaneous improvement in both zero-bias permittivity and dielectric strength. In addition to stoichiometry, interface quality is thought to play a strong role in these results. The so-called dead-layer effect a non-tunable interfacial capacitance appears to be influenced by interfacial roughness and surface contamination, and this in turn influences the apparent zero-bias dielectric constant [9, 44]. Interface quality also influences the effective Schottky barrier at the contact, which determines the amount of charge-injection at high-fields. Thus the maximum sustainable voltage, and therefore tunability, is also influenced by interface quality. Figure 3.7 shows the transient leakage behavior [10] of the films at a constant electric field of 200 kv/cm. Three different types of transient behavior were ob- 41

55 Chapter 3. BST Growth /10 80/20 70/30 60/40 50/50 J (A/cm 2 ) t (sec) Figure 3.7: Transient DC leakage current density at a constant field of 200 kv/cm for films grown with different Ar/O 2 flow rates (sccm). 42

56 Chapter 3. BST Growth served, with the 90/10 film having the sort of long transient that will be revisited in Chapter 6. Note that there was no further decrease in leakage at this field for oxygen partial pressures above 13.5 mtorr (70/30 sccm Ar/O 2 ), with the measurements on the films with the three highest oxygen partial pressures being essentially indistinguishable. The initial benefit was quite dramatic, with an order of magnitude difference in leakage between the two films with the lowest oxygen partial pressures. Considering the difference between platinized and bare sapphire, from the RBS results we expect the 90/10 film to be slightly titanium deficient. This would also cause the film to be oxygen deficient, which is consistent with its high permittivity and low breakdown voltage. A possible explanation for the further observed improvement is that small quantities of excess titanium can be accommodated in the grain boundaries, making them more insulating, but large amounts of excess titanium result in accommodation within the grain interiors [45]. 3.6 Interfacial capacitance To evaluate the magnitude of the non-tunable interfacial capacitance ( dead layer ), a thickness series was grown. All of the growth conditions for the films in this series were the same except for the growth time. The shortest growth time attempted (8.5 minutes) did not yield usable test structures, most likely due to 43

57 Chapter 3. BST Growth Table 3.2: Properties of films from the thickness series. Growth time Thickness C 40x50 (pf) Q C/A (minutes) (nm) (1 MHz) (1 MHz) (ff/µm 2 ) poor film coverage. The longest growth time attempted (270 minutes) failed to yield test structures because the film could not be etched. This film had also cracked due to strain from film-substrate lattice mismatch. Table 3.2 lists the properties of the films from which usable devices were obtained. A wide range of values for interfacial capacitance density have been reported in the literature [46 48]. It has been suggested that the origin of this interfacial layer may be a strain gradient due to lattice mismatch at the metal-film interface [49]; the interfacial layer is absent for films grown on lattice matched oxide electrodes such as SrRuO 3 [50]. Figure 3.8, a plot of inverse capacitance density versus film thickness, has a non-zero ordinate intercept corresponding to an interfacial capacitance density of 59 ff/µm 2. 44

58 Chapter 3. BST Growth /density (µm 2 /ff) y = x thickness (nm) Figure 3.8: Inverse of capacitance density vs. film thickness. 45

59 Chapter 3. BST Growth 3.7 Conclusion In summary, we have demonstrated high tunability BST films deposited by RF-magnetron sputtering on platinized sapphire substrates. Through variation of the oxygen partial pressure, we have shown that growth conditions can be optimized to yield very high tunability films with moderate loss, or low loss films with moderate tunability. A slight increase in the oxygen partial pressure beyond our previous growth condition (4.5 mtorr; 90/10 sccm Ar/O 2 ) resulted in a film (9 mtorr; 80/20 sccm Ar/O 2 ) with lower loss and lower leakage. This film exhibited a modest reduction in permittivity which was compensated by an increase in dielectric strength, ultimately allowing for an extremely large tunability of 13.7:1. The films grown at higher oxygen partial pressures had increasing excess titanium incorporation, resulting in lower tunabilities but higher Q-factors. The growth conditions for the optimized high tunability films are listed in Table 3.3. For applications such as phase shifters, the benefits of high tunability dominate over the drawbacks of increased film loss, making our films well suited to such applications. 46

60 Chapter 3. BST Growth Table 3.3: Growth conditions for high tunability films. Target compositions Ba 0.5 Sr 0.5 TiO 3,Ba 0.5 Sr 0.5 Ti 1.02 O 3 Substrate to target distance off axis RF power (per gun) 150 W Heater set temperature 800 C Heater ramp 50 C/min up, 20 C/min down Sputtering gas pressure 45 mtorr Sputtering gas flows 80 sccm Ar, 20 sccm O 2 Substrate single side polished c-plane sapphire Ground plane metalization 200 nm Pt with 3 nm Ti adhesion layer 47

61 Chapter 4 Loss calculation using relaxation 4.1 Loss Mechanisms in BST The presence of soft transverse optical phonons, even above T c, results in several phonon-mediated loss mechanisms that do not occur (at low frequencies) for ordinary dielectrics. These mechanisms are listed in Table 4.1. For a discussion of these loss mechanisms see [51]. Three-quanta loss refers to scattering between states of different phonon branches in a small region of k-space. Four-quanta loss refers to scattering between states of different phonon branches in different regions on k-space. Field-induced quasi-debye loss refers to scattering between states of the same phonon branch due to its finite width; in this process both transverse optical as well as transverse acoustical phonons participate. The dominant loss mechanisms in our films are not the intrinsic multi-phonon processes listed in Table 4.1. Extrinsic loss mechanisms for BST are listed in 48

62 Chapter 4. Loss calculation using relaxation Table 4.1: Intrinsic film loss mechanisms Three-quanta [51] tanδ ωɛ 1.5 T 2 Four-quanta [51] tanδ ωɛ 1.5 T ( 2 ) Field-induced quasi-debye [52] tanδ(e) ω ɛ(0) 1 ɛ(e) Table 4.2: Extrinsic film loss mechanisms Conduction electrons tanδ = σ dc ωɛ Charged defects [17] tanδ ωɛn d Z [1 2 1 Local polar regions [52] tanδ ɛ 4.5 d Universal relaxation [28] tanδ = ( ) ( 1 ɛ HF ɛ cot nπ ) 2 (1+ω 2 /ω 2 c) 2 ] Table 4.2. DC conductivity losses, where σ dc is a constant, dominate at very low frequencies where the conduction electrons can respond to the AC field. Losses due to the motion of charged defects are proportional to the permittivity and are dependent on the properties of the defects (n d is the density of defects, Z is their effective charge, r c is the correlation length of the charge distribution, and ν t is the average transversal velocity of sound). The losses due to local polar regions within the material are quasi-debye in nature, and are proportional to a power of the permittivity dependent on the dimensionality d of the polar defect. Universal relaxation refers to losses that obey a fractional power law in frequency governed by the behavior of the permittivity. It does not refer to a specific loss mechanism, and is present in a wide variety of systems [28]. In our films, at room temperature, universal relaxation dominates the loss behavior in the MHz to GHz regions. 49

63 Chapter 4. Loss calculation using relaxation 4.2 Universal relaxation Universal relaxation refers to behavior where a material s complex dielectric susceptibility has been observed to follow a decreasing power law over multiple decades in frequency [28]. This behavior is observed in a variety of different materials including Ba x Sr 1 x TiO 3,Al 2 O 3,Ta 2 O 5, HfO 2, and SiO 2 [30,53]. Parallel plate capacitors using UCSB-grown films of BST (Figure 4.1), STO (Figure 4.2), and BZN (Figure 4.3) follow this behavior in the MHz range. It appears to be a property of extrinsic disorder rather than an intrinsic material property [54, 55]. The susceptibility shows an f n 1 frequency dependence, where 0 <n<1. 1 For a lossless material, n = 1. A direct consequence of this power law is that if the real component of a material s complex susceptibility obeys a power law, so must the imaginary component. Thus when we find a fractional power law in the real part of the susceptibility and therefore the capacitance we also know the form of the dielectric loss associated with that capacitance. More specifically, if we observe a capacitance of the form of C(f) =C HF + C 0 ( f f 0 ) n 1, (4.1) 1 In the time domain, the corresponding transient leakage current with n 1 is known as Curie-von Schweidler behavior. 50

64 Chapter 4. Loss calculation using relaxation C Q C (F) Q f (Hz) Figure 4.1: Capacitance and Q vs. frequency for a BST parallel plate capacitor showing a fractional power law capacitance. 51

65 Chapter 4. Loss calculation using relaxation C Q C (F) Q f (Hz) Figure 4.2: Capacitance and Q vs. frequency for a STO parallel plate capacitor showing a fractional power law capacitance. Device courtesy of Mr. Sean Keane. 52

66 Chapter 4. Loss calculation using relaxation C Q C (F) 1000 Q f (Hz) Figure 4.3: Capacitance and Q vs. frequency for a BZN parallel plate capacitor showing a fractional power law capacitance. Device courtesy of Dr. Jiwei Lu. 53

67 Chapter 4. Loss calculation using relaxation where C HF, C 0, f 0, and n are constants associated with the data fit, and 0 <n<1, for multiple decades, Q must have the form ( C(f) Q(f) = C(f) C HF ) tan( nπ 2 ). (4.2) While this is not an intrinsic loss mechanism for BST thin films, it has been observed to be the dominant loss behavior up to 20 GHz [30]. Although this behavior has been observed in many materials, thus far little attention has been focused on the corresponding loss predictions using the universal relaxation model. Even when little relaxation is observed when n is very close to 1 the model can still accurately predict loss. The ability to calculate losses from capacitance data may be advantageous in situations where the direct measurement of Q-factors is difficult, such as network analyzer measurements of low-loss films. Reflectiontype measurements of the small resistive contribution in high-q reactive loads are difficult, but reactances are easier to measure. Provided that parasitic electrode inductances at high frequencies can be accounted for, this technique offers an accurate way to indirectly measure film loss through capacitance measurements. In BST we have successfully used power-law capacitance data to predict Q values. Comparison of 1 MHz Q-factors calculated and measured using an impedance analyzer demonstrates the validity of this approach. The calculated values are consistently equal to or slighter higher than measured values, consistent with the expected small contribution of series electrode resistance to measured Q-factors at 54

68 Chapter 4. Loss calculation using relaxation 1 MHz. Based on the successful application of this technique at low frequencies, we performed high frequency measurements on BST thin film capacitors, using the observed dielectric relaxation to calculate the films Q-factors. 4.3 Low frequency To evaluate the accuracy of loss calculations using universal relaxation, measured and calculated values were compared at low frequencies. Capacitance and Q measurements were performed on parallel plate capacitors with a ground-signalground (GSG) coplanar probe using an Agilent 4294A impedance analyzer for measurements from 40 Hz to 110 MHz. The loss calculation procedure was as follows: the measured capacitance data was fit to the form of Equation 4.1 using a low frequency extrapolated data point (typically 3 4 decades below the lower edge of the data) and two fit parameters, C HF and n 1; Q values were then calculated using Equation 4.2. Figure 4.4 illustrates this procedure. Calculation results from three different groups of measurements were compared, where Q variation was expected in each group. The first group of measurements consisted of temperature dependent measurements on devices fabricated from a 72 nm film ( temperature series ). The results of the capacitance fit parameters for the temperature series are listed in Table 4.3. The second group of 55

69 Chapter 4. Loss calculation using relaxation y = m1 + (6.722e-11 - m1)*m0... Value Error m e e-12 m Chisq e-26 NA R NA C n= C HF =1.9e-12 C 1Hz =6.722e (a) f C measured Q measured Q calculated C n= C HF =1.9e-12 C 1Hz =6.722e Q (b) Figure 4.4: Loss calculation procedure: (a) measured capacitance data (red dashed) and data fit (solid black) using extrapolated capacitance at 1 Hz, (b) measured capacitance (red solid) together with measured Q (dashed blue) and Q calculated from capacitance fit in (dashed green). f 56

70 Chapter 4. Loss calculation using relaxation measurements consisted of measurements on films grown under different conditions, with different Ar/O 2 partial pressures ( Ar/O 2 series ). The results of the capacitance fit parameters for the Ar/O 2 series are listed in Table 4.4. The third group of measurements were performed on films with different thicknesses ( thickness series ). The results of the capacitance fit parameters for the thickness series are listed in Table 4.5. Table 4.3: 1 MHz capacitance fit parameters and Q calculation results for the temperature series. Temp. C C 1Hz C HF n Q Q (K) (pf) (pf) (pf) (meas.) (calc.) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 4.4: 1 MHz capacitance fit parameters and Q calculation results for the Ar/O 2 series. Ar/O 2 C C 1Hz C HF n Q Q (sccm) (pf) (pf) (pf) (meas.) (calc.) 90/ ± ± / ± ± / ± ± / ± ± / ± ± Figure 4.5 summarizes the results for all three data sets, comparing the measured Q values at 1 MHz with the calculated Q values. The calculated values are 57

71 Chapter 4. Loss calculation using relaxation Q calculated (1MHz) Ar/O 2 thickness temperature Q measured (1MHz) Figure 4.5: Comparison of measured and calculated Q values at 1 MHz for 3 different measurement series (Ar/O 2, thickness, and temperature). 58

72 Chapter 4. Loss calculation using relaxation Table 4.5: 1 MHz capacitance fit parameters and Q calculation results for the thickness series. Thickness C C 1Hz C HF n Q Q (nm) (pf) (pf) (pf) (meas.) (calc.) ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± in agreement with the measured values. Most of the calculated values at 1 MHz are slightly higher than the measured values. A likely explanation for this discrepancy is the small contribution of series electrode resistance at this frequency; series resistances do not contribute to the calculated Q values. 4.4 High frequency An Agilent 8722D network analyzer was used for capacitance measurements from 50 MHz to 40 GHz, and an Agilent E8362B PNA series network analyzer was used for measurements from 10 MHz to 20 GHz. At GHz frequencies series electrode losses result in a dramatic decrease in measured Q values. Part of this series electrode resistance can be accounted for through the measurement of a shorted structure, such as the one shown in Figure 4.6(b), where the BST has been completely etched away before top electrode deposition. The measured Q 59

73 Chapter 4. Loss calculation using relaxation values are determined using Q = X corr R corr = X ωl R R s, (4.3) where X corr is the measured reactance corrected for the parasitic inductance L calculated from the measured self-resonance, and R corr is the difference between the measured resistance R and the resistance of a shorted structure R s. While this procedure does lead to a correction in Q, it fails to capture the full series resistance contribution. Figure 4.7 shows data with and without the measured series resistance correction, along with the Q calculated directly from the capacitance data. The series resistance correction slightly raises Q, but Q continues to exhibit a gross frequency dependence which is absent in the calculated Q. Since the calculated Q contains only parallel resistances, the discrepancy between the corrected and calculated Q must be due to an unaccounted for series resistance. The greatest benefit of loss calculation from capacitance data is that it is much less arduous to calibrate a network analyzer for a good capacitance measurement than for both good capacitance and Q measurements. Figure 4.8 shows both the low frequency and high frequency Q values for devices of the same geometry from the same sample. While the uncertainty in the high frequency calculation is much higher than in the low frequency calculation, it is still relatively small. Another advantage of the calculation is that only resistances in parallel to the capacitance contribute to the calculated Q. This is highly advantageous for the determination 60

74 Chapter 4. Loss calculation using relaxation (a) (b) Figure 4.6: Parallel plate (a) device and (b) corresponding shorted structure. of film properties at high frequencies (such as in Figure 4.7) where series electrode contributions dominate measured Q values. 4.5 Why does this work? The Kramers-Kronig relations are a Hilbert transform pair that can be used to relate the real and imaginary parts of a complex function G(ω) over an infinite frequency range: Re {G(ω)} = 2 π P.V. 0 ω Im {G(ω )} ω 2 ω 2 dω (4.4) 61

75 Chapter 4. Loss calculation using relaxation 100 Q calc. Q meas. (w/o Rs correction) Q meas. (w/ Rs correction) Q C (F) 10 C f (Hz) Figure 4.7: Measured Q vs. frequency with and without measured R s corrections, together with Q calculated from capacitance relaxation. 62

76 Chapter 4. Loss calculation using relaxation QcalcQ Qcalc Qcalc max Qcalc min Qcalc min max Q f (Hz) Figure 4.8: Low frequency measured Q plotted together with low and high frequency calculated Q values. 63

77 Chapter 4. Loss calculation using relaxation and Im {G(ω)} = 2ω π P.V. 0 Re {G(ω )} ω 2 ω 2 dω, (4.5) where ω is radian frequency, ω is an integration variable, and P.V. denotes the Cauchy principal value of the integral. These expressions can be written in a more convenient form using a dimensionless integration variable x: Re {G(ω)} = 2 π P.V. 0 xim {G(xω)} x 2 1 dx (4.6) and Im {G(ω)} = 2 π P.V. 0 Re {G(xω)} 1 x 2 dx. (4.7) This definition is more convenient for our purposes because the integrals singularity is always at x = 1 regardless of the radian frequency of interest ω. In our case, we would like to calculate loss from a capacitance. Considering a complex capacitance of the form C(f) =C (f)+jc (f), (4.8) we can calculate the imaginary part C (f) from the real part C (f) using Equation 4.7. We can then define Q as Q = C (f) C (f). (4.9) 64

78 Chapter 4. Loss calculation using relaxation For a capacitance of the form of Equation 4.1, Equation 4.7 becomes C (f) = 2 [ ( ) n 1 xf π P.V. 1 C 0 + C HF] 0 f 0 1 x dx 2 = 2 ( ) n 1 f π C x n 1 0 P.V. dx. (4.10) 1 x2 f 0 where C (f) is the imaginary part of the complex capacitance, and C 0, f 0, and n are characteristics of the capacitance data fit. Since it is not possible to measure capacitance data over an infinite frequency range, in practice we must truncate the integration range in Equation Thus the final expression for the imaginary component of the capacitance is C (f) = 2 π C 0 ( f f 0 0 ) n 1 P.V. fmax f f min f x n 1 dx, (4.11) 1 x2 where the quantities f min and f max in the integration limits are the measured data extent, and f is the frequency where we would like to calculate the loss. Table 4.6 compares the value of the integral in Equation 4.10 with the truncated integral in Equation Examining the integral for the limiting cases of n =0.990 and n =0.999, we find that the effects of truncation are small provided that a three decade frequency window on either side of the frequency of interest can be used. Figure 4.9 illustrates the relationship between the integration range and this range of frequencies where C (f) can be calculated. It should be noted that in thin film BST at room temperature, a capacitance of the form of 65

79 Chapter 4. Loss calculation using relaxation integration range calculation range 0 f min 10 3 f min 10-3 f max f max f Figure 4.9: Illustration of the available calculation range given an integration range and a three decade window on either side of the calculation frequencies. Equation 4.1 has been observed from 1 mhz up to 20 GHz [30], corresponding to a calculation range of 1 khz to 20 MHz. Table 4.6: Comparison of integral in Equation 4.10 over all frequencies with truncated contributions for a six decade window centered at the frequency of interest. x n 1 dx x 2 n P.V. x n 1 dx P.V x n 1 dx x n 1 dx 0 1 x x x Calculating Q(f) using Equation 4.2 does not introduce errors due to truncation, but it does introduce errors through the assumption that the power-law capacitance behavior persists outside of the measurement window. This is the situation whenever Kramers-Kronig is applied to data that his been fit to a function. 66

80 Chapter 4. Loss calculation using relaxation Examination of the integration kernel in Equation 4.7 reveals that the integral has a second order pole at the calculation frequency of interest. For x 1, corresponding to evaluation of G(ω) at frequencies much higher than the frequency of interest, the integration kernel becomes 1/x 2, suppressing higher frequency contributions to the integral. For x 1, the kernel approaches 1, which means that lower frequency contributions are present. However, these contributions cannot dominate the value of the integral unless they are associated with a pole that is at least second order, like the pole in the integration kernel. Fortunately the singularities in physical response functions are weakened by thermal broadening. In the case of our experiments, the poles are further broadened by thin film effects. Thus the contributions from low frequency poles are higher order corrections to the value of the integral in the vicinity of the kernel s pole. To lowest order, we need only concern ourselves with the behavior of the capacitance in the neighborhood of where we wish to calculate Q. 4.6 Conclusion We have successfully shown that using the universal relaxation model, we can calculate Q values that agree well with our low frequency measurements. We have also shown that the model can be used to calculate Q values at higher frequen- 67

81 Chapter 4. Loss calculation using relaxation cies where measurement capabilities are limited. The high frequency calculation values show the same trend measured at lower frequencies. A direct benefit of this technique is its insensitivity to series electrode resistance, which is otherwise difficult to de-embed from measured data. 68

82 Chapter 5 Geometry dependent quality factors 5.1 Introduction One puzzling observation that we have encountered with our parallel plate BST capacitors is that quality factor Q varies as a function of device area. At low frequencies, the quality factor of large area devices is nearly constant. The µm 2 device in Figure 5.1 exhibits this behavior. As top electrode area decreases, the magnitude of Q decreases, and the frequency dependence of Q increases. At microwave frequencies, as shown in Figure 5.2, the quality factors for all sizes of devices are grossly frequency dependent, and smaller area devices have higher Q. Area-dependent quality factors have also been observed in similarly sized cubic pyrochlore bismuth zinc niobate parallel plate capacitors [32, 56]. Thus we believe that this behavior may be common to other small-geometry high- 69

83 Chapter 5. Geometry dependent quality factors x15 µm2 20x20 µm2 30x30 µm2 35x35 µm2 45x45 µm2 Q f (Hz) Figure 5.1: Measured low frequency Q vs. frequency for devices with different top electrode areas. permittivity devices. In this paper we discuss the possible role of extrinsic factors in BST parallel plate capacitor characterization, and present simple models for the observed geometry dependence of Q in these devices. To that end, we will begin with a description of the devices used in our experiments and their generic equivalent circuit, followed by a discussion of low frequency properties, and finally high frequency properties. 70

84 Chapter 5. Geometry dependent quality factors Figure 5.2: Measured high frequency Q vs. frequency for devices with different top electrode areas. Reactance has been corrected for L s. 71

85 Chapter 5. Geometry dependent quality factors 5.2 Low frequency Parallel plate devices are often characterized at low frequencies (1 MHz) because the device properties are closer to the film properties. It is important to note that this does not imply that measured low frequency device properties are necessarily intrinsic film properties; contributions of parallel parasitic pathways are still important at low frequencies. In practice, the room temperature behavior of our BST capacitors is consistent with universal relaxation [28 30], which means that the measured device properties are not dominated by intrinsic film properties [17]. The dependence on top electrode area of Q in Figure 5.1 is a clear example of how extrinsic factors can influence device properties at low frequencies. At 1 MHz the devices with larger electrode areas have larger Q, but at 100 MHz, the trend is reversed. To further investigate this low frequency geometry dependence, a set of devices with different top electrode sizes and different mesa ledge sizes were fabricated, as shown in Figure 5.3(b). On the mask used for fabrication, the top electrode areas A ranged from µm 2 to µm 2, while the mesa ledge widths L m, as indicated in Figure 5.3(a), ranged from 5 µm to 15 µm. Electrode and mesa lengths and widths were measured using an optical microscope with the photomask features as a length standard. Electrical measurements on these devices revealed 72

86 Chapter 5. Geometry dependent quality factors (a) Pt Pt BST Pt Pt sapphire substrate L m A increasing A (decreasing P/A) (b) increasing Lm (5 µm to 15 µm) Figure 5.3: (a) BST capacitor structure: cross section layout on left, top view photograph on right showing top electrode area A and mesa ledge with L m ; and (b) photograph of array of fabricated devices with different top electrode areas and different mesa ledge widths. 73

87 Chapter 5. Geometry dependent quality factors (a) (b) A L m Figure 5.4: Illustration of (a) top electrode area A and mesa ledge width L m, and (b) surface conduction pathway. that Q was not only a function of top electrode area, but also possibly of the device mesa ledge width as well, as summarized in Figure 5.5. The measured device admittance Y tot can be expressed as the sum of an areadependent term Y A and a perimeter-dependent term Y P, Y tot = Y A + Y P. (5.1) We would like to characterize the parallel plate capacitance and conductance, which are included in Y A, Y A = g d A + jωca, (5.2) where A is the top electrode area, ω is the radian frequency, g d is the conductance density, and c is the capacitance density. The quality factor associated with Y A is clearly geometry-independent; the presence of geometry dependence in the Q associated with Y tot indicates a non-zero Y P contribution. 74

88 Chapter 5. Geometry dependent quality factors Q L m (µm) P/A (µm -1 ) Figure 5.5: 1 MHz Q as a function of top electrode geometry P/A and mesa ledge width L m. 75

89 Chapter 5. Geometry dependent quality factors A variety of factors contribute to Y P. One well-known contribution is fringing capacitance. Analysis of fringing contributions from BST tunability curves reported in [57] revealed a fringing capacitance C f of the form C f = κ P d (5.3) where P is the top electrode perimeter, d is the film thickness, and κ = 0.6 ff for d = 210 nm. It should be noted that such analysis may pick up other contributions to Im{Y P } that are not tunable by an applied DC bias. The geometry variation in Q resulting from the addition of a fringing capacitance, Q = ωca + ωκp d g d A, (5.4) causes Q to increase as the ratio of perimeter to area (P/A) increases. This is the opposite of the trend observed in Figure 5.5. The observed geometry variation in Q is consistent with the addition of a perimeter-dependent conductance term. A non-zero surface conductivity over the path depicted in Figure 5.4(b) enables a distributed perimeter-dependent conductance GP which appears in parallel with the film conductance g d A, where G is a linear conductance density, P is the top electrode perimeter, g d is the film conductance density, and A is the top electrode area. It should be noted that G is a function of L m ; the functional form of this relationship depends on the surface and bulk material parameters, as will be explained later in this section. With the 76

90 Chapter 5. Geometry dependent quality factors Table 5.1: Data fit parameters from Figures 5.6 and 5.7. ωc G L m g d g d g d G (µm) (m) (Ω 1 m 2 ) (Ω 1 m 1 ) x x x x x x x x x x 10 3 addition of this conductance term Q can be expressed as ( )( ωc Q = 1+ G ) 1 P. (5.5) g d A g d Since both the capacitance ca and film conductance g d A scale with A, the influence of the perimeter dependent conductance term scales with the ratio of perimeter-to-area P/A. With square electrodes, as A decreases, P/A increases. Figure?? shows the results of data fits at 1 MHz. The corresponding conductance values and related quantities are listed in Table 5.1. Leakage current measurements failed to reveal any top electrode geometry dependence, as shown in Figure 5.8. Thus DC leakage over/through the device mesa does not contribute to G. Comparing the results for devices with different mesa sizes, we find that the values of G saturate for the larger device mesas. This behavior is consistent with a distributed parasitic pathway. Figure 5.9 illustrates an incremental length z of a transmission line model for the parasitic pathway. The admittance Y P of the parasitic pathway approaches the character- 77

91 Chapter 5. Geometry dependent quality factors Q Q Q P/A (µm-1) L m =5µm L m =7.5µm L m =10µm P/A (µm-1) P/A (µm-1) G g d = 3.3x10-6 (m) = 210 G g d G g d ωc gd = 2.3x10-6 (m) ωc gd = 194 = 2.2x10-6 (m) ωc gd = 191 Figure 5.6: 1 MHz Q as a function of the top electrode perimeter to area ratio (P/A) for L m = 5, 7.5, and 10 µm. Measured data points are indicated by solid dots while fits to Equation 5.5 are shown as dashed curves. 78

92 Chapter 5. Geometry dependent quality factors Q Q L m =12.5µm L m =15µm G g d P/A (µm-1) G g d P/A (µm-1) = 2.0x10-6 (m) ωc gd = 187 = 2.0x10-6 (m) ωc gd = 189 Figure 5.7: 1 MHz Q as a function of the top electrode perimeter to area ratio (P/A) for L m = 12.5 and 15 µm. Measured data points are indicated by solid dots while fits to Equation 5.5 are shown as dashed curves. 79

93 Chapter 5. Geometry dependent quality factors J (A/cm 2 ) L m (µm) P/A (µm -1 ) Figure 5.8: Leakage current density, measured at 2 seconds under a 2 V bias, as a function of top electrode geometry P/A and mesa ledge width L m. R z C z G z z Figure 5.9: Incremental length of transmission line with incremental series resistance R, shunt capacitance C, and shunt conductance G. 80

94 Chapter 5. Geometry dependent quality factors istic admittance Y 0 of the line for large L m. The characteristic admittance of this structure is G + jωc Y 0 = R (5.6) where R, C, and G are the incremental resistance, capacitance, and conductance of the parasitic conduction pathway, respectively. The propagation constant γ for the line is γ = R(G + jωc). (5.7) For our pathway, C and G are proportional to P, while R is inversely proportional to P. Therefore Y 0 is proportional to P and γ is independent of P. Our pathway can be modeled as a short circuit terminated transmission line of length L m. Thus the input admittance of the parasitic pathway is Y p = Y 0 [tanh(γl m )] 1. (5.8) The conductance correction is GP =Re{Y 0 }Re{[tanh(γL m )] 1 } Im{Y 0 }Im{[tanh(γL m )] 1 }, (5.9) and the capacitance correction is Im{Y P } ω = Re{Y 0} Im{[tanh(γL m )] 1 } ω + Im{Y 0} Re{[tanh(γL m )] 1 }. (5.10) ω 81

95 Chapter 5. Geometry dependent quality factors For most of the devices in our set L m is very large, so Y p Y 0, (5.11) GP Re{Y 0 }, (5.12) and Im{Y P } ω Im{Y 0}. (5.13) ω Thus to determine the characteristic admittance of the parasitic pathway, both capacitance and Q must be examined. Figure 5.10 shows the variation in capacitance density for devices with different top electrode areas and mesa ledge widths. The capacitance density decreases with smaller top electrode areas (larger P/A). While Y P always includes a capacitance correction, the correction should cause the capacitance to increase with larger P/A. This is the opposite of the trend observed in the measured data. Fringing would also result in an increase with larger P/A. This behavior cannot be explained by a parasitic inductance because such a parasitic inductance, either in series or parallel, would need to be unphysically large (i.e., > 100 µh). Another explanation for the observed decrease in capacitance density for larger P/A could be a systematic error in the measurement of the fabricated electrode areas. Since the top electrodes are square, an offset in width/length measurements would produce a term proportional to perimeter as a correction to the top 82

96 Chapter 5. Geometry dependent quality factors electrode area. The sign of this term is negative if the measured length/width is smaller than the actual length/width. For the data in Figure 5.10, an error of approximately 1 µm changes the sign of the perimeter-dependent term, causing capacitance density to increase with larger P/A. The photomask features used as a length standard were fabricated with a ±0.25 µm tolerance. In the photographs used to compare the mask features with the fabricated devices 1 µm (i.e., 1/12 of a 12 µm feature) was easily resolvable. The measured electrode lengths/widths were typically 0.5 µm larger than the mask features. This is consistent with the image-reversal process used to define the electrodes, which tends to result in larger features than the mask, rather than smaller features which a 1 µm error would require. Thus it is unlikely that error in length/width measurements alone are the cause of the capacitance density variation. It is important to note that this does not change the geometry-dependent behavior of Q observed in Figure High frequency The simplified high frequency equivalent circuit for a parallel plate capacitor consists of the series combination of R s, L s, and C. The measured self-resonance frequencies, between 4 and 16 GHz for these devices, were used to approximate 83

97 Chapter 5. Geometry dependent quality factors C/A (ff/µm 2 ) L m (µm) P/A (µm -1 ) Figure 5.10: 1 MHz capacitance density as a function of top electrode geometry P/A and mesa ledge width L m. 84

98 Chapter 5. Geometry dependent quality factors L s. The measured reactance was then corrected for this contribution, leaving the capacitance. The associated quality factors for these devices are shown in Figure 5.2. The behavior at these frequencies is dominated by series resistance, agreeing with (2.3). The geometry dependence at high frequency is evident in Figure 5.2 where smaller area devices have higher Q. If both the capacitance and series resistance were proportional to area, Q would be geometry independent. The presence of a geometry-independent series resistance changes this behavior, giving smaller area devices a higher Q because of their larger reactance. Figure 5.11 shows how Q is expected to vary for devices with different areas but the same series resistance. This is consistent with the behavior in Figure 5.2. Comparison of Figure 5.2 and Figure 5.11 reveals that the geometry-independent series resistance component in the measured data has some frequency dependence. This is evident when the measured data is compared to the 1/f reference line; the traces in Figure 5.2 deviate from 1/f while the traces in Figure 5.11 do not. Thus while the addition of a geometry-independent frequency-independent series resistance qualitatively describes the geometry dependence of the high frequency quality factors, a better model would include some frequency dependence in the series resistance term. 85

99 Chapter 5. Geometry dependent quality factors x15 µm x20 µm 2 30x30 µm 2 35x35 µm 2 45x45 µm 2 1/f reference line Q f (Hz) Figure 5.11: Simulated Q vs. frequency for devices with different top electrode areas with a geometry-independent series resistance R s =1Ω. 86

100 Chapter 5. Geometry dependent quality factors Two geometry-independent sources of series resistance can be identified for the parallel plate structures under consideration: the bottom electrode and the coplanar probe. The bottom electrode resistance is largely determined by the probe pitch, the lateral spacing between the centers of the top electrode and ground plane contacts. This resistance is in-plane and therefore inversely proportional to the bottom electrode thickness. When the device is small compared to the probe pitch (spacing between probe feet) which is usually the case for the small devices used for high frequency measurements, but not necessarily the case for large low frequency devices the series electrode resistance is easily overwhelmed by this non-geometry-dependent contribution. Another source of geometry-independent series resistance is the coplanar probe used for characterization. In principle, the combination of calibration and the measurement of shorted structures (as in Figure 5.13) should remove these contributions from the measured device properties, but there are reasons why this may not always be the case. 5.4 De-embedding failure Series resistances are typically evaluated using a shorted device structure. Our shorted device structure consists of top metal contacts deposited directly upon the ground plane, as shown in Figure While this structure is literally the 87

101 Chapter 5. Geometry dependent quality factors parallel plate structure without a dielectric, it fails to capture all od the series resistances. Figure 5.12 illustrates how a series resistance component remains after de-embedding a shorted structure. The weakly frequency-dependent calculated Q values in Figure 5.12 were obtained by fitting measured broadband capacitance relaxation to a fractional power law. The details and applicability of this technique are discussed elsewhere [28,29]. The behavior of the calculated Q is similar to that of the low frequency Q because the capacitance relaxation in both frequency ranges obeys the same fractional power law [29, 30]. The weak frequency dependence in the calculated Q corresponds to the weak frequency dependence in the capacitance (which has been corrected for L s ). The presence of a series resistance causes Q to decrease strongly with frequency. The de-embedding procedure using a shorted device fails to fully account for the series resistance of the device; the measured shorted resistance R s of approximately 0.4 Ω is smaller than the 1 Ω resistance suggested by the data. This results in a de-embedded Q that continues to decrease as 1/ω. 1 This failure also leads to geometry-dependent de-embedded Q factors. To understand why the shorted device fails to fully capture the series resistances, we must consider the differences between the device under test and the corresponding shorted structure [58]. Two differences between these structures are the contributions of the electrode and the contact resistance at the electrode-film 1 The frequency dependence of the measured quality factors in Figure 5.12 is weaker than 1/ω because of the influence of R p ; this frequency range is the crossover between (2.2) and (2.3). 88

102 Chapter 5. Geometry dependent quality factors 100 Q calc. Q meas. (w/o Rs correction) Q meas. (w/ Rs correction) Q C (F) 10 C f (Hz) Figure 5.12: Measured Q vs. frequency with and without measured R s corrections, together with Q calculated from capacitance relaxation. R s 0.4 Ω. 89

103 Chapter 5. Geometry dependent quality factors interface to the series resistance. The contact resistance is completely neglected in the shorted device structure, while the electrode resistances are only partially accounted for. The spreading resistance in the top contact and directly below the device are different in the shorted structure, as illustrated in Figure The electrode resistances in the device, shown in Figure 5.13(a), can be expressed as ( ) Rs = 1 ( ξ ρ + 1 W L device 2 t b 3 2 ρ + 1 W t b 3 2 ρ t t W ρ t b + t t ), (5.14) where ξ refers to the lateral spacing between the top and side contacts, L and W refer to the lengths and widths of the top and side contacts, t t is the top metalization layer thickness, t b is the bottom metalization layer thickness, and ρ is the resistivity of platinum. The prefactor of 1/2 appears because of the symmetry of the coplanar ground-signal-ground probe. The factors of 1/3 appear because of the distributed nature of the spreading resistance [27]. The resistances in the shorted structure, shown in Figure 5.13(b), can be expressed as ( ) Rs = 1 ( ξ ρ + 1 W L short 2 t b 3 2 ρ t b + t t W ρ t b + t t ). (5.15) Comparing the device with the shorted structure, we find that the spreading resistance of the top contact has combined with the spreading resistance in the ground plane beneath the device (the second and third terms in (5.14), respectively), yielding a spreading resistance through the stack of both metalization 90

104 Chapter 5. Geometry dependent quality factors (a) (b) Figure 5.13: Electrode resistances in (a) device and (b) shorted device structures. Distributed resistances indicated by arrows. 91